Dependencies

Under the same assumptions as in Theorem 5.77, for every countable subset \(T' \subseteq T\) with positive diameter, for \(L(T, c_1, d, p, q, \beta ) {\lt} \infty \) the same constant,

For \(\varepsilon _n = \varepsilon _0 2^{-n}\), with the hypothesis of Lemma 5.30, we have

Let \(X:\mathbb {R}_+ \to \Omega \to E\) be a right-continuous martingale with values in a normed space \(E\). For every \(T\) and \(\lambda {\gt}0\) we have

Let \(X:\mathbb {R}\times \Omega \rightarrow E\) be a right-continuous martingale with values in a normed space \(E\). For every \(\lambda {\gt}0\) and \(p{\gt}1\) and \(\tau \) stopping time a.s. bounded by \(T{\gt}0\), we have

Let \(X:\mathbb {R}\times \Omega \rightarrow E\) be a right-continuous martingale with values in a normed space \(E\). For every \(\lambda {\gt}0\) and \(p{\gt}1\) and \(\tau \) stopping time a.s. bounded by \(T{\gt}0\), we have

Let \(X : \mathbb {R} \rightarrow \Omega \rightarrow E\) be a right-continuous martingale with values in a normed space \(E\). For every \(T, \lambda {\gt}0\) and \(p{\gt}1\) we have

That is, for \(\Vert \cdot \Vert _p\) the \(L^p\) norm, \(\left\Vert \sup _{t\in [0,T]} X_t \right\Vert _p \leq \frac{p}{p-1} \left\Vert X_T \right\Vert _p \: .\)

Let \(E, F, G\) be normed real vector spaces. Let \(B : E \times F \to G\) be a continuous bilinear map. Then for every simple process \(V\) with values in \(\mathbb {R}\), simple process \(W\) with values in \(E\), and stochastic process \(X\) with values in \(F\), we have

Let \(X : T \to \Omega \to \mathbb {R}\) be an adapted stochastic process which is right-continuous in probability and such that the boundedness condition of Theorem 11.3 holds. Then \(X\) has a cadlag modification.

Suppose that \(T\) is a finite set with bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). For all \(\delta {\gt} 0\),

With the same assumptions and notations as in Theorem 5.70, for all \(\delta \in (0, 4\mathrm{diam}(T)]\),

Under the assumptions of Lemma 5.60, for \(\varepsilon _n = \varepsilon _0 2^{-n}\), then for \(m \le k\),

Under the assumptions of Lemma 5.64, for \(\varepsilon _n = \varepsilon _0 2^{-n}\), then for \(m \le k\),

If \(X : ι \to Ω \to ℝ\) is a right-continuous and adapted process with respect to a right-continuous filtration, then for any \(a \in \mathbb {R}\), the random time \(\tau _{X \ge a}\) is a stopping time.

Let \(X : T \rightarrow \Omega \rightarrow E\) a martingale with values in a normed space \(E\). Then \(\Vert X \Vert \) is a sub-martingale.

Let \(X : \Omega \to E\) be an integrable random variable with values in a normed space \(E\). Then, for any sub-\(\sigma \)-algebra \(\mathcal{G}\), we have

Let \(X : T \to E\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\). For \(m \le k\),

Let \(X\) be a submartingale and \(A\) be an elementary predictable set. Then for all \(t \in T\),

Define \(A_0=0\) and for \(t\in \mathcal{D}_n^T\) positive,

A process \(X : T \to \Omega \to E\) is said to be adapted with respect to a filtration \(\mathcal{F}\) if for all \(t \in T\), \(X_t\) is \(\mathcal{F}_t\)-measurable.

A time index set \(T\) is said to be approximable if for any stopping time \(\tau : \Omega \to T \cup \{ \infty \} \), there exists a discrete approximation sequence \((\tau _n)\) of \(\tau \).

Given a stopping time \(\tau : \Omega \to T \cup \{ \infty \} \), a sequence of stopping times \((\tau _n)_{n \in \mathbb {N}}\) is called an discrete approximation of \(\tau \) if \(\tau _n(\Omega )\) is countable for each \(n\) and \(\tau _n \downarrow \tau \) a.s. as \(n \to \infty \).

A function \(f : T \to E\) is of bounded variation on a set \(s\) if its variation \(V_f(s)\) is finite.

By Theorem 5.84, there exists a modification \(B\) of the pre-Brownian process such that all the paths of \(B\) are Hölder continuous of all orders \(\gamma \in (0, 1/2)\). We call \(B\) the Brownian motion on \(\mathbb {R}_+\).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). We define the chaining sequence of \(x\), denoted \((\bar{x}_i)_{i \le k}\), recursively as follows: \(\bar{x}_k = x\) and for \(i {\lt} k\), \(\bar{x}_i = \pi (\bar{x}_{i+1}, C_i)\).

The characteristic function of a measure \(\mu \) on an inner product space \(E\) is the function \(E \to \mathbb {C}\) defined by

This is equal to the normed space version of the characteristic function applied to the linear map \(x \mapsto \langle t, x \rangle \).

The characteristic function of a measure \(\mu \) on a normed space \(E\) is the function \(E^* \to \mathbb {C}\) defined by

A stochastic process \((X_t)\) is of class D (or in the Doob-Meyer class) if it is progressively measurable and the set \(\{ X_\tau \mid \tau \text{ is a finite stopping time}\} \) is uniformly integrable.

A stochastic process \((X_t)\) is of class DL if it is progressively measurable and for all \(t \ge 0\), the set \(\{ X_\tau \mid \tau \text{ is a stopping time with } \tau \le t\} \) is uniformly integrable.

A continuous semi-martingale is a process that can be decomposed into a local martingale and a finite variation process. More formally, a process \(X : \mathbb {R}_+ \to \Omega \to E\) is a continuous semi-martingale if there exists a continuous local martingale \(M\) and a continuous adapted process \(A\) with locally finite variation and \(A_0 = 0\) such that

for all \(t \ge 0\). The decomposition is a.s. unique.

The covariance bilinear form of a measure \(\mu \) on \(F\) with finite second moment is the continuous bilinear form \(C_\mu : F^* \times F^* \to \mathbb {R}\) with

For \(M\) and \(N\) local martingales, the covariation \(\langle M, N \rangle \) is a stochastic process defined by polarization of the quadratic variation:

The covariance bilinear form of a finite measure \(\mu \) with finite second moment on a Hilbert space \(E\) is the continuous bilinear form \(C_\mu : E \times E \to \mathbb {R}\) with

This is \(C_\mu \) applied to the linear maps \(L_x, L_y \in E^*\) defined by \(L_x(z) = \langle x, z \rangle \) and \(L_y(z) = \langle y, z \rangle \).

The covariance matrix of a finite measure \(\mu \) with finite second moment on a finite dimensional inner product space \(E\) is the positive semidefinite matrix \(\Sigma _\mu \) such that for \(u, v \in E\),

This is the covariance bilinear form \(C'_\mu (u, v)\), as a matrix.

For \(T{\gt}0\), let \(\mathcal{D}_n^T = \left\lbrace \frac{k}{2^n}T \mid k=0,\cdots 2^n\right\rbrace \) be the set of dyadics at scale \(n\) and let \(\mathcal{D}^T=\bigcup _{n\in \mathbb {N}}\mathcal{D}_n^T\) be the set of all dyadics of \([0,T]\).

A set \(A \subseteq T \times \Omega \) is an elementary predictable set if it is a finite union of sets of the form \({0} \times B\) for \(B \in \mathcal{F}_0\) or of the form \((s, t] \times B\) with \(0 \le s {\lt} t\) in \(T\) and \(B \in \mathcal{F}_s\).

We call scalar elementary stochastic integral the general elementary stochastic integral of Definition 10.9 with \(V\) being \(\mathbb {R}\)-valued and \(B(r, x) = r \cdot x\) the scalar multiplication. We denote it by \(V \bullet _{\mathbb {R}} X\).

Let \(V \in \mathcal{E}_{T, E}\) be a simple process and let \(X\) be a stochastic process with values in a normed space \(F\). Let \(B\) be a continuous bilinear map from \(E \times F\) to another normed space \(G\). The general elementary stochastic integral process \(V \bullet _B X : T \to \Omega \to G\) is defined by

Let \(E\) and \(F\) be normed real vector spaces. We call linear elementary stochastic integral the general elementary stochastic integral of Definition 10.9 with \(V\) being valued in \(L(E, F)\)-valued and \(B(L, x) = L(x)\) the evaluation map. We denote it by \(V \bullet _L X\).

The (extended-real-valued) variation of a function \(f : T \to E\) on a set \(s\) inside a linear order is the supremum of \(\sum _i \mathrm{edist}(f(u_{i+1}, f(u_i)))\) over all finite increasing sequences \(u : N \to T\) in \(s\). We denote it by \(V_f(s)\) .

The external covering number of a set \(A \subseteq E\) for \(\varepsilon \ge 0\) is the smallest cardinality of an \(\varepsilon \)-cover of \(A\). Denote it by \(N^{ext}_\varepsilon (A)\).

A filtration on a measurable space \((\Omega , \mathcal{A})\) with measure \(P\) indexed by a preordered set \(T\) is a family of sigma-algebras \(\mathcal{F} = (\mathcal{F}_t)_{t \in T}\) such that for all \(i \le j\), \(\mathcal{F}_i \subseteq \mathcal{F}_j\) and for all \(t \in T\), \(\mathcal{F}_t \subseteq \mathcal{A}\).

We denote by \(P_B\) the projective limit of the projective family of the Brownian motion given by Theorem 4.3. This is a probability measure on \(\mathbb {R}^{\mathbb {R}_+}\).

For \(I = \{ t_1, \ldots , t_n\} \) a finite subset of \(\mathbb {R}_+\), let \(P^B_I\) be the multivariate Gaussian measure on \(\mathbb {R}^n\) with mean \(0\) and covariance matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\). We call the family of measures \(P^B_I\) the projective family of the Brownian motion.

The real Gaussian measure with mean \(\mu \in \mathbb {R}\) and variance \(\sigma ^2 {\gt} 0\) is the measure on \(\mathbb {R}\) with density \(\frac{1}{\sqrt{2 \pi \sigma ^2}} \exp \left(-\frac{(x - \mu )^2}{2 \sigma ^2}\right)\) with respect to the Lebesgue measure. The real Gaussian measure with mean \(\mu \in \mathbb {R}\) and variance \(0\) is the Dirac measure \(\delta _\mu \). We denote this measure by \(\mathcal{N}(\mu , \sigma ^2)\).

Let \(v_1, \ldots , v_n\) be vectors in an inner product space \(E\). The Gram matrix of \(v_1, \ldots , v_n\) is the matrix in \(\mathbb {R}^{n \times n}\) with entries \(G_{ij} = \langle v_i, v_j \rangle \) for \(1 \leq i,j \leq n\).

A set \(T\) is said to have a cover with bounded covering numbers if there exists a monotone sequence of totally bounded subsets \((T_n)_{n \in \mathbb {N}}\) of \(T\) such that for all \(n\), \(T_n\) has bounded internal covering number with constant \(c_n\) and exponent \(d_n {\gt} 0\), and such that \(T \subseteq \bigcup _{n \in \mathbb {N}} T_n\).

Let \(\mathrm{diam}(A)\) be the diameter of \(A \subseteq E\), i.e. \(\mathrm{diam}(A) = \sup _{x,y \in A} d_E(x, y)\). A set \(A \subseteq E\) has bounded internal covering number with constant \(c{\gt}0\) and exponent \(t{\gt}0\) if for all \(\varepsilon \in (0, \mathrm{diam}(A)]\), \(N^{int}_\varepsilon (A) \le c \varepsilon ^{-t}\).

We say that a stochastic process \(X : T \to \Omega \to E\) has independent increments if for all \(t_1, \ldots , t_n \in T\) with \(t_1 \le t_2 \le \cdots \le t_n\), the random variables \(X_{t_2} - X_{t_1}, X_{t_3} - X_{t_2}, \ldots , X_{t_n} - X_{t_{n-1}}\) are independent.

We say that a stochastic process is integrable if the map \((t,\omega ) \mapsto X_t(\omega )\) is strongly measurable and for all \(t\), \(X_t\) is integrable. A process has an integrable supremum if \((\sup _{s \le t} \Vert X_s \Vert )_t\) is integrable.

We say that a filtered probability space \((\Omega , \mathcal{F}, P)\) satisfies the usual conditions if the filtration is right-continuous and if \(\mathcal{F}_0\) contains all the \(P\)-null sets.

For \(X : T \to \Omega \to E\) a stochastic process, \(B\) a subset of \(E\) and \(t_0 \in T\), the hitting time of \(X\) in \(B\) after \(t_0\) is the random variable \(\Omega \to T\cup \{ \infty \} \) defined by

in which the infimum is infinite if the set is empty.

Let \(c{\gt}0\). Define the hitting time on \(\mathcal{D}^T_n\)

We say that a stochastic processes \(Y\) is a indistinguishable from \(X\) if \(\mathbb {P}\)-a.e., for all \(t \in T\), \(X_t = Y_t\).

The internal covering number of a set \(A \subseteq E\) for \(\varepsilon \ge 0\) is the smallest cardinality of an \(\varepsilon \)-cover of \(A\) which is a subset of \(A\). Denote it by \(N^{int}_\varepsilon (A)\).

A function \(f : T \to E\) is cadlag if it is right continuous and the limit \(\lim _{s \uparrow t} f(s)\) exists for all \(t \in T\).

A set \(C \subseteq E\) is an \(\varepsilon \)-cover of a set \(A \subseteq E\) if for every \(x \in A\), there exists \(y \in C\) such that \(d_E(x, y) \le \varepsilon \).

A measure \(\mu \) on \(F\) is Gaussian if for every continuous linear form \(L \in F^*\), the pushforward measure \(L_* \mu \) is a Gaussian measure on \(\mathbb {R}\).

A process \(X : T \to \Omega \to E\) is Gaussian if for every finite subset \(t_1, \ldots , t_n \in T\), the random vector \((X_{t_1}, \ldots , X_{t_n})\) has a Gaussian distribution.

Let \(X : T \to \Omega \to E\) be a stochastic process, where \((T, d_T)\) and \((E, d_E)\) are pseudo-metric spaces and \((\Omega , \mathbb {P})\) is a measure space. Let \(p, q {\gt} 0\). We say that \(X\) satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\) if for all \(s, t \in T\), \((X_s, X_t)\) is \(\mathbb {P}\)-a.e. measurable for the Borel \(\sigma \)-algebra on \(E^2\) and

We say a stochastic process \((M_t)_{t \in T}\) is a local martingale if it is locally a cadlag martingale in the sense of Definition 9.5. That is, there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), the process \(M^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is a cadlag martingale.

A stochastic process is a local submartingale if it is locally a cadlag submartingale in the sense of Definition 9.5. That is, there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), the process \(M^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is a cadlag submartingale.

Let \((M, +)\) be a commutative monoid that is also a partial order. It is said to be an ordered monoid if for all \(a, b, c \in M\), we have the following implication:

Let \(\alpha , \beta \) be preorders with \(0\) elements and such that there is a scalar multiplication \((\_ \cdot \_ ) : \alpha \times \beta \to \beta \). Then \(\beta \) is said to be an ordered \(\alpha \)-module (or ordered module if \(\alpha \) is clear from the context) if the following hold:

• \(\forall a \in \alpha , \forall b_1, b_2 \in \beta , 0 \le a \implies b_1 \le b_2 \implies a \cdot b_1 \le a \cdot b_2\);

• \(\forall a_1, a_2 \in \alpha , \forall b \in \beta , 0 \le b \implies a_1 \le a_2 \implies a_1 \cdot b \le a_2 \cdot b\).

A measure \(\mu \) on \(E^T\) is the projective limit of a projective family of measures \(P\) indexed by finite sets of \(T\) if, for every finite set \(I \subseteq T\), the projection from \(E^T\) to \(E^I\) maps \(\mu \) to \(P_I\).

A family of measures \(P\) indexed by finite sets of \(T\) is projective if, for finite sets \(J \subseteq I\), the projection from \(E^I\) to \(E^J\) maps \(P_I\) to \(P_J\).

A set \(c \subseteq E\) is \(\varepsilon \)-separated if for all \(x, y \in c\), \(d_E(x, y) {\gt} \varepsilon \).

Let \(T\) be a linear order with bottom element 0, on which we have a filtration \(\mathcal{F}\) satisfying the usual conditions. We say that a martingale \(M : T \to \Omega \to E\) is square integrable if it is càdlàg and \(\sup _{t \in T} \Vert M_t \Vert _{L^2} {\lt} \infty \) (Lean remark: use eLpNorm (M t) 2).

A stopping time with respect to some filtration \(\mathcal{F}\) indexed by \(T\) is a function \(\tau : \Omega \to T \cup \{ \infty \} \) such that for all \(i\), the preimage of \(\{ j \mid j \le i\} \) along \(\tau \) is measurable with respect to \(\mathcal{F}_i\).

Let \(M \in \mathcal{M}^2\). Then the elementary stochastic integral map \(\mathcal{E} \to \mathcal{M}^2\) defined by \(V \mapsto V \bullet M\) extends to an isometry \(L^2(M) \to \mathcal{M}^2\).

We introduce the constant

Let \(M\) be a continuous local martingale. We define \(L^2_{loc}(M)\) as the space of predictable processes \(X\) such that for all \(t \ge 0\), \(\mathbb {E}\left[ \int _0^t X_s^2 \: d\langle M \rangle _s \right] {\lt} \infty \).

Let \(M\) be a square integrable martingale. We define

in which \(\mathcal{P}\) is the predictable \(\sigma \)-algebra and \(d\langle M \rangle \) is the measure induced by the quadratic variation of \(M\).

We denote by \(\mathcal{M}^2(E)\) or simply \(\mathcal{M}^2\) the space of equivalence classes with respect to indistinguishability of square integrable martingales \(T \to \Omega \to E\) .

We define an inner product on \(\mathcal{M}^2\) by

We define a norm on \(\mathcal{M}^2\) by

Let \(\mu \) be a measure on \(T\) and \(P\) a measure on \(\Omega \). We denote by \(L^2(\mu , P)\) the space \(L^2(T \times \Omega , \mathcal{P}, \mu \times P)\), in which \(\mathcal{P}\) is the predictable \(\sigma \)-algebra, and the measure \(\mu \times P\) is the restriction of the product measure on \(\mathcal{B}(T) \otimes \mathcal{F}\) to \(\mathcal{P}\) .

For a process \(X : ι \to Ω \to ℝ\) and a real number \(a\), define the random time

in which the infimum is infinite if the set is empty.

Assume that \(T\) is a partial order. For \(\mathcal{F}\) a filtration indexed by \(T\) and \(t \in T\), we define the left continuation as

Note that \(\bigsqcup \) denotes the supremum in the lattice of sigma-algebras on \(\Omega \).

Let \(X : T \to \Omega \to E\) be a stochastic process, let \(\mathcal{F}\) be a filtration on \(\Omega \) indexed by \(T\) and let \(P\) be a measure on \(\Omega \). If there exists a function \(Y : \Omega \to E\) which is measurable with respect to \(\mathcal{F}_\infty \) such that for \(P\)-almost surely, \(X_t\) converges to \(Y\) as \(t\) goes to infinity, then we say that \(Y\) is the limit of \(X\). We denote it by \(X_\infty \).

A localizing sequence is a sequence of stopping times \((\tau _n)_{n \in \mathbb {N}}\) such that \(\tau _n\) is non-decreasing and \(\tau _n \to \infty \) as \(n \to \infty \) (a.s.). That is, it is a pre-localizing sequence that is also almost surely non-decreasing.

Let \(P\) be a class of stochastic processes (or equivalently a predicate on stochastic processes). We say that a stochastic process \(X : T \to \Omega \to E\) is locally in \(P\) (or satisfies \(P\) locally) if there exists a localizing sequence \((\tau _n)_{n \in \mathbb {N}}\) such that for all \(n \in \mathbb {N}\), the process \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is in \(P\) (in which \(X^{\tau _n}\) denotes the stopped process). We denote the class of processes that are locally in \(P\) by \(P_{\mathrm{loc}}\).

A function \(f : T \to E\) is of locally bounded variation on a set \(s\) if for every \(a, b\) in \(s\), the variation \(V_f(s \cap [a, b])\) is finite.

Let \(f : T \to E\) be a cadlag function of locally bounded variation on a conditionally complete linear order \(T\) (TODO: and probably more assumptions?). Then by Lemma 13.5, \(f\) can be written as a difference of two cadlag monotone functions, say \(f = f_+ - f_-\) . We denote by \(df\) the signed measure on \(T\) associated to \(f\) defined by \(df = df_+ - df_-\), in which \(df_+\) and \(df_-\) are given by Definition 13.6.

A process has locally integrable supremum if it is locally a process with integrable supremum.

Let \(M\) be a continuous local martingale and let \(X \in L^2_{loc}(M)\). We define the local stochastic integral \(X \cdot M\) as the unique continuous local martingale with \((X \cdot M)_0 = 0\) such that for any continuous local martingale \(N\), almost surely,

Let \((T,d_T)\) be a metric space and let \(J \subseteq T\) be finite, \(a,c \in \mathbb R_+\) with \(a \ge 1\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). An log-size ball sequence for \((J, a, c, n)\) is a sequence of \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) such that

• \(V_0 = J\), \(t_0\) is an arbitrary point in \(J\),

• for all \(i\), \(r_i\) is the log-size radius of \(t_i\) in \(V_i\),

• \(V_{i+1} = V_i \setminus B_{V_i}(t_i, (r_i - 1)c)\), \(t_{i+1}\) is arbitrarily chosen in \(V_{i+1}\).

Let \(V\) be a finite subset of a metric space and let \(t \in V\) and \(a {\gt} 1\), \(c {\gt} 0\). Let the log-size radius of \(t\) in \(V\), denoted by \(r_{V,t}\), be the smallest positive integer \(r\) such that \(\vert B_V(t, r c) \vert \le a^{r}\).

For \(t\in \mathcal{D}_n^T\), define \(M^n_t = S_t-A^n_t\) .

Let \(\mathcal{F}\) be a filtration on a measurable space \(\Omega \) with measure \(P\) indexed by \(T\). A family of functions \(M : T \to \Omega \to E\) is a martingale with respect to a filtration \(\mathcal{F}\) if \(M\) is adapted with respect to \(\mathcal{F}\) and for all \(i \le j\), \(P[M_j \mid \mathcal{F}_i] = M_i\) almost surely.

Let \(X : \mathbb {N} \to \Omega \to E\) be a process indexed by \(\mathbb {N}\), for \(E\) a Banach space. Let \((\mathcal{F}_n)_{n\in \mathbb {N}}\) be a filtration on \(\Omega \) and let \(A\) be the predictable part of \(X\) for that filtration. The martingale part of \(X\) is the process \(M : \mathbb {N} \to \Omega \to E\) defined by \(M_n = X_n - A_n\).

The identity is a measurable equivalence between the continuous functions of \(\mathbb {R}^{\mathbb {R}_+}\) with the subset sigma-algebra obtained from the product sigma-algebra, and \(C(\mathbb {R}_+, \mathbb {R})\) with the Borel sigma-algebra coming from the compact-open topology.

Mathematically this says nothing more than the equality of sigma-algebras of Theorem 6.14 but in Lean we have two different types so we need an equivalence.

We say that a stochastic process \(Y\) is a modification of another stochastic process \(X\) if for all \(t \in T\), \(Y_t =_{\mathbb {P}\text{-a.e.}} X_t\).

The multivariate Gaussian measure on \(\mathbb {R}^d\) with mean \(m \in \mathbb {R}^d\) and covariance matrix \(\Sigma \in \mathbb {R}^{d \times d}\), with \(\Sigma \) positive semidefinite, is the pushforward measure of the standard Gaussian measure on \(\mathbb {R}^d\) by the map \(x \mapsto m + \Sigma ^{1/2} x\). We denote this measure by \(\mathcal{N}(m, \Sigma )\).

Let \(S\) be a finite set of \(E\) and \(x \in E\). We denote by \(\pi (x, S)\) the point in \(S\) which is closest to \(x\), i.e. a point such that \(d_E(x, S) = \min _{y \in S} d_E(x, y)\) (chosen arbitrarily among the minima if there are several).

Let \(X\) be a topological space that is also a preorder. The space \(X\) is set to be order-closed, or to have order-closed topology, if the set \(\{ (x, y) \in X \times X \mid x \le y\} \) is closed.

The packing number of a set \(A \subseteq E\) for \(\varepsilon {\gt} 0\) is the largest cardinality of an \(\varepsilon \)-separated subset of \(A\). Denote it by \(P_\varepsilon (A)\).

Let \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) be a log-size ball sequence for \((J, a, c, n)\). For \(i \in \mathbb {N}\), let \(K_i = \{ t_i\} \times B_{V_i}(t_i, r_i c)\) be the set of pairs \((t_i, s)\) for \(s\) in the ball \(B_{V_i}(t_i, r_i c)\). We define \(K = \bigcup _{i=0}^{\vert J \vert -1} K_i\), set of all pairs from the log-size ball sequence.

Let \(\Omega = \mathbb {R}^{\mathbb {R}_+}\) and consider the probability space \((\Omega , P_B)\) (where \(P_B\) is the measure defined in Definition 4.10). The identity on that space is a function \(\Omega \to \mathbb {R}_+ \to \mathbb {R}\). We reorder the arguments to define a stochastic process \(X : \mathbb {R}_+ \to \Omega \to \mathbb {R}\), which we call the pre-Brownian process.

A process \(X : T \to \Omega \to E\) is said to be predictable with respect to a filtration \(\mathcal{F}\) if it is measurable with respect to the predictable sigma-algebra on \(T \times \Omega \).

Let \(\mathcal{F}\) be a filtration on a measurable space indexed \(\Omega \) by a linearly ordered set \(T\). Let \(S = \{ \{ \bot \} \times A \mid A \in \mathcal{F}_\bot \} \) if \(T\) has a bottom element and \(S = \emptyset \) otherwise. The predictable sigma-algebra on \(T \times \Omega \) is the sigma-algebra generated by the set of sets \(\{ (t, \infty ] \times A \mid t \in T, \: A \in \mathcal{F}_t\} \cup S\).

Let \(X : \mathbb {N} \to \Omega \to E\) be a process indexed by \(\mathbb {N}\), for \(E\) a Banach space. Let \((\mathcal{F}_n)_{n\in \mathbb {N}}\) be a filtration on \(\Omega \). The predictable part of \(X\) is the process \(A : \mathbb {N} \to \Omega \to E\) defined for \(n \ge 0\) by

A pre-localizing sequence is a sequence of stopping times \((\tau _n)_{n \in \mathbb {N}}\) such that \(\tau _n \to \infty \) as \(n \to \infty \) (a.s.).

The law of a stochastic process \(X\) is the measure on the measurable space \(E^T\) obtained by pushing forward the measure \(\mathbb {P}\) by the map \(\omega \mapsto X(\cdot , \omega )\).

A stochastic process \(X\) is said to be progressively measurable with respect to a filtration \(\mathcal{F}\) if at each point in time \(i\), \(X\) restricted to \((-\infty , i] \times \Omega \) is measurable with respect to the product \(\sigma \)-algebra where the \(\sigma \)-algebra used for \(\Omega \) is \(\mathcal{F}_i\).

For \(M\) a local martingale, the predictable quadratic variation of \(M\) is defined as the predictable part of the Doob-Meyer decomposition of the local sub-martingale \(\Vert M \Vert ^2\) . We denote it by \(\langle M \rangle \) .

For \(M\) a local martingale, we denote by \(d\langle M \rangle \) the measure on \(\mathbb {R}_+\) associated to the non-decreasing càdlàg function \(\langle M \rangle \) .

We say that the filtration is right-continuous if for all \(t \in T\), \(\mathcal{F}_{t+} \subseteq \mathcal{F}_t\).

Let \(T\) be equipped with a partial order and \(f : T \to E\) be a function. Then \(f\) is right continuous if for all \(t \in T\), \(\lim _{s \downarrow t} f(s) = f(t)\).

Assume that \(T\) is a partial order. For \(\mathcal{F}\) a filtration indexed by \(T\) and \(t \in T\), we define the right continuation as

Note that \(\mathop{\Large \sqcap }\) denotes the infimum in the lattice of sigma-algebras on \(\Omega \).

Let \((s_k {\lt} t_k)_{k \in \{ 1, ..., n\} }\) be points in a linear order \(T\) with a bottom element 0. Let \((\eta _k)_{0 \le k \le n}\) be bounded random variables with values in a normed real vector space \(E\) such that \(\eta _0\) is \(\mathcal{F}_0\)-measurable and \(\eta _k\) is \(\mathcal{F}_{s_k}\)-measurable for \(k \ge 1\). Then the simple process for that sequence is the process \(V : T \to \Omega \to E\) defined by

Let \(\mathcal{E}_{T, E}\) be the set of simple processes indexed by \(T\) with value in \(E\).

A class of stochastic processes \(P\) is stable if whenever \(X\) is in \(P\), then for any stopping time \(\tau \), the process \(X^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is also in \(P\).

Let \((e_1, \ldots , e_d)\) be an orthonormal basis of \(E\) and let \(\mu \) be the standard Gaussian measure on \(\mathbb {R}\). The standard Gaussian measure on \(E\) is the pushforward measure of the product measure \(\mu \times \ldots \times \mu \) by the map \(x \mapsto \sum _{i=1}^d x_i \cdot e_i\).

Let \(f : T \to ℝ\) be a cadlag monotone function on a conditionally complete linear order \(T\). We denote by \(df\) the measure on \(T\) defined by \(df((a, b]) = f(b) - f(a)\) for all \(a, b \in T\) .

TODO: in Mathlib this is only for \(T = \mathbb {R}\) for now, but will be generalized.

For a continuous semi-martingale \(X = M + A\) and \(V \in L^2_{semi}(X)\) (to be defined) we define the stochastic integral as

in which \(V \cdot M\) is the local stochastic integral defined in 13.57 and \(V \cdot A\) is the Lebesgue-Stieltjes integral with respect to the locally finite variation process \(A\).

Let \(X : T \to \Omega \to E\) be a stochastic process and let \(\tau : \Omega \to T\). The stopped process with respect to \(\tau \) is defined by

Given a stopping time \(\tau \) on a time index \(T\), define \(\mathcal{F}_\tau = \bigcap _{t \in T} \{ A \in \mathcal{F} \mid A \cap \{ \tau \le t\} \in \mathcal{F}_t\} .\)

Let \(\mathcal{F}\) be a filtration on a measurable space \(\Omega \) with measure \(P\) indexed by \(T\). A family of functions \(M : T \to \Omega \to E\) is a submartingale with respect to a filtration \(\mathcal{F}\) if \(M\) is adapted with respect to \(\mathcal{F}\) and for all \(i \le j\), \(P[M_j \mid \mathcal{F}_i] \ge M_i\) almost surely.

The Wiener measure on \(C(\mathbb {R}_+, \mathbb {R})\) with the Borel sigma-algebra is the map of the auxiliary Wiener measure by the measurable equivalence of definition 6.15.

The pushforward of the measure \(P_B\) of Definition 4.10 by the Brownian motion \(B\) is a measure on the continuous functions on \(\mathbb {R}^{\mathbb {R}_+}\), with the sigma-algebra induced by the product sigma-algebra on \(\mathbb {R}^{\mathbb {R}_+}\).

There exists a set \(E\subseteq \Omega \), \(P(E)=0\) and a subsequence \(k_n\) such that \(\lim _n\mathcal{A}^{k_n}_t(\omega )=A_t(\omega )\) for every \(t\in \mathcal{D}^T,\omega \in \Omega \setminus E\).

\(A^n_{\tau _n(c)} \le c\) and if \(\tau _n(c) {\lt} T\) then \(A^n_{\tau _n(c)+T2^{-n}} {\gt} c\).

Let \(a, b {\gt} 0\) with \(a \le b\). If \(\tau _n(b) {\lt} T\) then \(A^n_{\tau _n(b)+T2^{-n}} - A^n_{\tau _n(a)} \ge b - a\).

\((A_t)_{t\in [0,T]}\) is an increasing process.

The sequence \((A^n_T)_{n\in \mathbb {N}}\) is uniformly integrable (bounded in \(L^1\) norm).

Suppose \(T\) is a linearly ordered set equipped with the order topology. If \(T\) is second countable and a stochastic process \((X_t)_{t \in T}\) is right continuous and adapted, then it is progressively measurable.

The predictable part \(A\) is adapted to the filtration \((\mathcal{F}_{n+1})_{n \in \mathbb {N}}\).

The simple processes \(\mathcal{E}_{T, F}\) form an additive commutative group.

If \(E\) is separable and \(X : T \to \Omega \to E\) is a process such that \(X_t\) is \(\mathbb {P}\)-a.e. measurable for all \(t \in T\), then for all \(s, t \in T\), the pair \((X_s, X_t)\) is \(\mathbb {P}\)-a.e. measurable for the Borel \(\sigma \)-algebra on \(E^2\).

Let \(I = \{ t_1, \ldots , t_n\} \) be a finite subset of \(\mathbb {R}_+\). For \(i \le n\), let \(v_i = \mathbb {I}_{[0, t_i]}\) be the indicator function of the interval \([0, t_i]\), as an element of \(L^2(\mathbb {R})\). Then the Gram matrix of \(v_1, \ldots , v_n\) is equal to the matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\).

The covariation \(\langle M, N \rangle \) of local martingales \(M\) and \(N\) is cadlag.

For \(V \in \mathcal{E}_{T, F}\), \(X : T \to \Omega \to E\) a càdlàg process and a continuous bilinear map \(B: E \times F \to G\), the elementary stochastic integral \(V \bullet _B X\) is càdlàg.

The quadratic variation \(\langle M \rangle \) of a local martingale \(M\) is cadlag.

\(a^{r_{V,t}-1} \le \vert B_V(t, (r_{V,t}-1)c) \vert \) .

\(\vert B_V(t, r_{V,t}c) \vert \le a^{r_{V,t}}\) .

The cardinal of the pair set \(K\) of a log-size ball sequence for \((J, a, c, n)\) satisfies \(|K| \le a |J|\).

The central moment of order \(2n\) of a real Gaussian measure \(\mathcal{N}(\mu , \sigma ^2)\) is given by

in which \((2n - 1)!! = (2n - 1)(2n - 3) \cdots 3 \cdot 1\) is the double factorial of \(2n - 1\).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for all \(i \le k\), \(\bar{x}_i\in C_i\).

The characteristic function of a real Gaussian measure with mean \(\mu \) and variance \(\sigma ^2\) is given by \(x \mapsto \exp \left(i \mu x - \frac{\sigma ^2 x^2}{2}\right)\).

Let \(\mu \) be a measure on \(F\) and let \(L \in F^*\). Then

The characteristic function of the standard Gaussian measure on \(E\) is given by

Let \(\mu \) be a measure on a normed space \(E\) and let \(L\) be a continuous linear map from \(E\) to \(F\). Then for all \(L' \in F^*\),

Assume \(T\) has a bottom element and that its closed intervals are compact, and that the filtration is right-continuous. If \(X\) is a càdlàg process of class DL, then it has locally integrable supremum.

If \(X\) is of class DL then it is locally of class D.

A stochastic process of class D is of class DL.

Let \(X\) be a real-valued submartingale with respect to a filtration \(\mathcal{F}\). Then for all \(i \le j\), we have \(0 \le P[M_j - M_i \mid \mathcal{F}_i]\) almost surely.

If \((X_i)_{i \in \iota }\) is a family of (probabilistically) uniformly integrable functions and \((\mathcal{F}_j)_{j \in \kappa }\) is a family of \(\sigma \)-algebras, then the family \((P[X_i \mid \mathcal{F}_j])_{i \in \iota , j \in \kappa }\) is uniformly integrable.

Let \(X : \Omega \to E\) be an integrable random variable with values in a normed space \(E\) and let \(\phi : E \to \mathbb {R}\) be a convex function such that \(\phi \circ X\) is integrable. Then, for any sub-\(\sigma \)-algebra \(\mathcal{G}\), we have

The paths of the Brownian motion are continuous.

Let \(X\) be a normed vector space (over \(\mathbb {R}\)). Let \((x_n)_{n\in \mathbb {N}}\) be a sequence in \(X\) converging to \(x\) w.r.t. the topology of \(X\). Let \((N_n)_{n\in \mathbb {N}}\) be a sequence in \(\mathbb {N}\) such that \(n\leq N_n\) for every \(n\in \mathbb {N}\) (maybe here we could have \(N_n\) increasing WLOG). Let \((a_{n,m})_{n\in \mathbb {N},m\in \left\lbrace n,\cdots ,N_n\right\rbrace }\) be a triangular array in \(\mathbb {R}\) such that \(0\leq a_{n,m}\leq 1\) and \(\sum _{m=n}^{N_n}a_{n,m}=1\). Then \((\sum _{m=n}^{N_n}a_{n,m}x_m)_{n\in \mathbb {N}}\) converges to \(x\) uniformly w.r.t. the triangular array.

Let \(X : T \rightarrow \Omega \rightarrow E\) a sub-martingale. Let \(\phi :E \rightarrow \mathbb {R}\) convex, continuous, and increasing such that \(\phi (X_t)\in L^1(\Omega )\) for every \(t\in T\). Then \(\phi (X)\) is a sub-martingale.

For \(\mu \) a measure on \(F\) with finite second moment and \(L \in F^*\), \(C_\mu (L, L) = \mathbb {V}_\mu [L]\).

Let \(M\) and \(N\) be square integrable martingales. Then

\(\langle M, N \rangle _0 = 0\) .

Let \(E\) and \(F\) be two Hilbert spaces with \(F\) finite dimensional, \(\mu \) a finite measure on \(E\) with finite second moment, and \(L : E \to F\) a continuous linear map. Then the covariance bilinear form of the measure \(L_*\mu \) is given by

in which \(L^\dagger : F \to E\) is the adjoint of \(L\).

Let \(E\) and \(F\) be two finite dimensional inner product spaces, \(\mu \) a measure on \(E\) with finite second moment, and \(L : E \to F\) a continuous linear map. Then the covariance matrix of the measure \(L_*\mu \) has entries

in which \(L^\dagger : F \to E\) is the adjoint of \(L\).

The covariance matrix of the multivariate Gaussian measure \(\mathcal{N}(m, \Sigma )\) is \(\Sigma \).

The covariance matrix of the standard Gaussian measure is the identity matrix.

The set of simple processes is dense in \(L^2(M)\).

Let \(\tau \) be a stopping time bounded by \(t \in T\) and \((\tau _n)\) be a discrete approximation sequence of \(\tau \). Then, the sequence of stopping times \(\tau _n \wedge t\) is also a discrete approximation sequence of \(\tau \).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for all \(i {\lt} k\), \(d_E(\bar{x}_i, \bar{x}_{i+1}) \le \varepsilon _i\).

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\). Let \(m, k, \ell \in \mathbb {N}\) with \(m \le k\) and \(m \le \ell \) and let \(x \in C_k\) and \(y \in C_\ell \). Then

Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(A \subseteq E\) with \(C_n \subseteq A\) and \(x \in C_k\) for some \(k \in \mathbb {N}\). Then for \(m \le k\), \(d_E(\bar{x}_m, x) \le \sum _{i=m}^{k-1} \varepsilon _i\).

Let \((s, t)\) be a pair in the pair set \(K\) of a log-size ball sequence for \((J, a, c, n)\). Then \(d_T(s, t) \le c n\).

Let \(S\) be a finite set of \(E\) and \(x \in E\). Then for all \(y \in S\), \(d_E(x, \pi (x, S)) \le d_E(x, y)\).

Let \(C_\varepsilon \) be a finite \(\varepsilon \)-cover of \(A \subseteq E\) (assuming such a finite cover exists). Then for all \(x \in A\), \(d_E(x, \pi (x, C_\varepsilon )) \le \varepsilon \).

Let \(X : I \rightarrow \Omega \rightarrow \mathbb {R}\) be a non-negative sub-martingale with \(I\) countable. Then for every \(M \in I,\lambda {\gt} 0\) and \(p{\gt}1\) we have

Let \(X:\mathbb {R}\times \Omega \rightarrow \mathbb {R}\) be a right-continuous non-negative sub-martingale. For every \(\lambda {\gt}0\) and \(p{\gt}1\) and \(\tau \) stopping time a.s. bounded by \(T{\gt}0\), we have

Let \(X:\mathbb {R}\times \Omega \rightarrow \mathbb {R}\) be a right-continuous non-negative sub-martingale. For every \(\lambda {\gt}0\) and \(p{\gt}1\) and \(\tau \) stopping time a.s. bounded by \(T{\gt}0\), we have

Let \(X : I \rightarrow \Omega \rightarrow \mathbb {R}\) be a non-negative sub-martingale. Let \(I\) be countable. For every \(M\in I,\lambda {\gt} 0\) and \(p{\gt}1\) we have

That is, for \(\Vert \cdot \Vert _p\) the \(L^p\) norm, \(\left\Vert \sup _{i \le M} X_i \right\Vert _p \leq \frac{p}{p-1} \left\Vert X_M \right\Vert _p \: .\)

\((M^n_t)_{t\in \mathcal{D}_n^T}\) is a martingale.

\((A^n_t)_{t\in \mathcal{D}_n^T}\) is a predictable process.

A set \(A \subseteq T \times \Omega \) is an elementary predictable set if and only if the indicator function \(\mathbb {1}_A\) is a simple process.

\(V \bullet _{\mathbb {R}} (W \bullet _{\mathbb {R}} X) = (V \times W) \bullet _{\mathbb {R}} X\) for real-valued simple processes \(V, W\) and stochastic process \(X\).

The elementary stochastic integrals \(\bullet _B\), \(\bullet _L\) and \(\bullet _{\mathbb {R}}\) are linear in both arguments.

Let \(X\) be a stochastic process, \(V\) a simple process and \(τ\) a stopping time. Then

Let \(E, F, G, H, I, J\) be normed real vector spaces. Let \(B_1 : E \times H \to I\), \(B_2 : F \times G \to H\), \(B_3 : E \times F \to J\) and \(B_4 : J \times G \to I\) be continuous bilinear maps such that for all \(x \in E\), \(y \in F\) and \(z \in G\), \(B_1(x, B_2(y, z)) = B_4(B_3(x, y), z)\). Then for every simple process \(V\) with values in \(E\), simple process \(W\) with values in \(F\), and stochastic process \(X\) with values in \(G\), we have

Let \(X_c\) be a constant process (equal to the same random variable for all times). Then for every simple process \(V\),

\((V \bullet _B X)_0 = 0\) for every simple process \(V\) and stochastic process \(X\).

For simple process \(W\) with values in \(L(E, F)\), simple process \(V\) with values in \(L(F, G)\), and stochastic process \(X\) with values in \(E\), we have

in which \(V \circ W\) is the simple process defined by \((V \circ W)_t = V_t \circ W_t\).

For \(V \in \mathcal{E}_{T, F}\) bounded by a constant \(D\), \(M \in \mathcal{M}^2(E)\) and a continuous bilinear map \(B: E \times F \to G\),

TODO: this can be improved to \(D \: \Vert B \Vert \: \Vert M_t \Vert _{L^2}\)?

If \(\mathcal{F}\) is right-continuous, then for all \(t \in T\), \(\mathcal{F}_t = \mathcal{F}_{t+}\).

\(V_f([0, t]) = \vert df \vert ([0, t])\) .

Let the filtered probability space satisfy the usual conditions. Then every local martingale \(X\) admits a modification that is still a local martingale with cadlag trajectories.

Let the filtered probability space satisfy the usual conditions. Then every nonnegative submartingale \(X\) admits a modification that is still a nonnegative submartingale with cadlag trajectories.

\(N^{ext}_\varepsilon (A) \le N^{int}_\varepsilon (A)\).

For \(B \subseteq A\), \(N^{ext}_\varepsilon (B) \le N^{ext}_{\varepsilon }(A)\).

The predictable \(\sigma \)-algebra on \(T \times \Omega \) is a sub-\(\sigma \)-algebra of the product \(\sigma \)-algebra \(\mathcal{B}(T) \otimes \mathcal{F}\) .

The right continuation of \(\mathcal{F}\) is a filtration.

Suppose that \(J \subseteq T\) is a finite set and that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). Let \(\beta \in (0, (q - d)/p)\). Then

\(\mathbb {R}_+\) has a cover with bounded covering numbers for the sets \(T_n = [0,n)\), constants \(c_n = n\) and exponents \(d_n = 1\).

If \(A\) has bounded internal covering number with constant \(c{\gt}0\) and exponent \(d{\gt}0\), then for all \(B \subseteq A\), \(B\) has bounded internal covering number with constant \(2^d c\) and exponent \(d\).

The unit interval \(I = [0, 1] \subseteq \mathbb {R}\) has bounded internal covering number with constant \(1\) and exponent \(1\): for \(\varepsilon \le 1\), \(N^{int}_\varepsilon (I) \le 1/\varepsilon \).

The Brownian motion has independent increments.

For \(t \in \mathbb {R}_+\), the law of \(B_t\) (the Brownian motion at time \(t\)) is the real Gaussian measure \(\mathcal{N}(0,t)\).

For \(s, t \in \mathbb {R}_+\), the law of \(B_t - B_s\) is the real Gaussian measure \(\mathcal{N}(0,\vert t - s \vert )\).

Let \(X\) be the pre-Brownian process of Definition 6.1. Then, for all \(s, t \in \mathbb {R}_+\), the random variable \(X_t - X_s\) is a Gaussian random variable with mean \(0\) and variance \(|t - s|\).

Assume that the filtration is right-continuous. Let \(X\) be a stochastic process with locally integrable supremum. Then \(X\) is locally of class DL.

Assume \(T\) has a bottom element and that its closed intervals are compact. Assume that the filtration is right-continuous. If a process is càdlàg and locally of class DL, then it has locally integrable supremum.

Under the assumptions of Theorem 5.77, for \(E\) a complete space and \(\beta \in (0, (q - d)/p)\), there exists a modification \(Y\) of \(X\) (i.e., a process \(Y\) with \(\mathbb {P}(Y_t \ne X_t) = 0\) for all \(t\)) such that the paths of \(Y\) are Hölder continuous of order \(\beta \).

For any class of processes \(P\), we have \(P \subseteq P_{\mathrm{loc}}\).

If \(f_n, f : [0, 1] \rightarrow \mathbb {R}\) are increasing functions such that \(f\) is right continuous and \(\lim _n f_n(t) = f (t)\) for \(t \in \mathcal{D^T}\), if \(f\) is continuous in \(t\in [0,T]\) then \(\lim _n f_n(t) = f (t)\).

If \(f_n, f : [0, 1] \rightarrow \mathbb {R}\) are increasing functions such that \(f\) is right continuous and \(\lim _n f_n(t) = f (t)\) for \(t \in \mathcal{D}^T\), then \(\limsup _n f_n(t) \leq f (t)\) for all \(t \in [0, T]\).

If \(Y\) is indistinguishable from \(X\), then \(Y\) is a modification of \(X\).

Let \(T\) and \(E\) be topological spaces and suppose that \(T\) is separable Hausdorff. Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are modifications of each other and are almost surely continuous. Then \(X\) and \(Y\) are indistinguishable.

For \(V \in \mathcal{E}_{T, \mathbb {R}}\) and \(M, N \in \mathcal{M}^2\), we have

\(\langle X \cdot M, Y \cdot M \rangle _{\mathcal{M}^2} = (XY) \cdot \langle M, N \rangle _{\mathcal{M}^2}\).

Let \(X\) be a stochastic process with integrable supremum (Definition 9.31). Then \(X\) is of class DL.

Let \(J \subseteq T\) be a finite set and suppose that \(T\) has finite diameter. For \(k \in \mathbb {N}\), let \(\eta _k = 2^{-k}(\mathrm{diam}(T) + 1)\). For \(X : T \to \Omega \to E\) a stochastic process and \(\beta \in (0, (q - d)/p)\),

Let \(X \in L^2(M)\) such that \(\langle X, V\rangle = 0\) for any simple process \(V\). Then for all \(t\), \(\int _0^t X_s \: d\langle M \rangle _s = 0\).

For \(\mu _1, \ldots , \mu _d\) probability measures on \(\mathbb {R}\) and \(f : \mathbb {R} \to \mathbb {R}\) integrable with respect to \(\mu _i\), we have

The mean of the multivariate Gaussian measure \(\mathcal{N}(m, \Sigma )\) is \(m\).

The mean of the standard Gaussian measure is \(0\).

Let \(X\in L^2(M)\) such that \(\int _0^t X_s \: d\langle M \rangle _s = 0\) for all \(t \ge 0\) a.s.. Then \(X = 0\) \((\mathbb {P} \times d\langle M \rangle )\)-almost everywhere.

Let \(X, Y \in L^2(M)\) such that \(\int _0^t X_s \: d\langle M \rangle _s = \int _0^t Y_s \: d\langle M \rangle _s\) for all \(t \ge 0\) a.s.. Then \(X = Y\) almost everywhere.

Let \(X : T \to \Omega \to E\) be a stochastic process. Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). For \(p \ge 1\) and \(m \le k\),

Let \(X : T \to \Omega \to E\) be a stochastic process. Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). For \(0 {\lt} p \le 1\) and \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(C\) be a finite \(\varepsilon \)-cover of \(J \subseteq T\) with \(C \subseteq J\), with minimal cardinal. Then for \(c \ge 0\),

Note the logarithm has base \(2\).

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). For all \(n \in \mathbb {N}\), let \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\) for \(\varepsilon _n = \varepsilon _0 2^{-n}\), with minimal cardinal. Suppose \(\varepsilon _0 {\lt} \infty \), let \(\delta \in (0, 4 \varepsilon _0]\) and let \(m\) be a natural number such that \(\varepsilon _0 2^{-m} \le \delta \) and \(\delta \le \varepsilon _0 2^{-m+2}\). Then for \(k \ge m\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(\varepsilon {\gt} 0\) and \(C \subseteq T^2\) be a finite set such that for all \((s, t) \in C\), \(d_T(s, t) \le \varepsilon \). Then

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers in \((0, \mathrm{diam}(T))\) and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\), and with minimal cardinality. Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Then for \(p \ge 1\) and \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers in \((0, \mathrm{diam}(T)]\) and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\), and with minimal cardinality. Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Then for \(p \le 1\) and \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). Then for \(p \ge 1\) and \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). For \(0 {\lt} p \le 1\) and \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(J \subseteq T\) be finite, \(a, c \in \mathbb R_+\) with \(a \ge 1\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). Then

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers and \(C_n\) a finite \(\varepsilon _n\)-cover of \(T\) with \(C_n \subseteq T\). Then for \(j {\lt} k\),

\(N_\varepsilon ^{int}(B_1) \ge \frac{1}{\varepsilon ^d}\).

\(N_\varepsilon ^{int}(B_1) \le \left(\frac{2}{\varepsilon } + 1\right)^d\).

If \(\mathrm{diam}(A) \le \varepsilon \) and \(A\) is nonempty, then \(N^{int}_\varepsilon (A) = 1\).

\(N^{int}_\varepsilon (A) \le P_\varepsilon (A)\).

If \(0 {\lt} \varepsilon {\lt} \infty \) then \(N^{int}_\varepsilon (A) \le \frac{V(A + B_{\varepsilon /2})}{V(B_{\varepsilon /2})}\).

For \(B \subseteq A\), \(N^{int}_\varepsilon (B) \le N^{int}_{\varepsilon /2}(A)\).

Assume \(T\) is a linear order endowed with a topology making it first countable and \(E\) is a pseudo-metric space. If \(X\) is a càdlàg process then it maps compact sets of \(T\) to bounded sets.

The standard Gaussian measure on \(E\) is centered, i.e., \(\mu [L] = 0\) for every \(L \in E^*\).

The characteristic function of a Gaussian measure \(\mu \) on \(E\) is given by

Two Gaussian measures \(\mu \) and \(\nu \) on a separable Hilbert space are equal if and only if they have same mean and same covariance.

A Gaussian measure is a probability measure.

For a centered Gaussian measure \(\mu \), \(\mu \times \mu \) is invariant by rotation.

A Gaussian measure \(\mu \) has finite moments of all orders. In particular, there is a well defined mean \(m_\mu := \mu [\mathrm{id}]\), and for all \(L \in F^*\), \(\mu [L] = L(m_\mu )\).

Let \(\mu \) be a Gaussian measure on \(F\) and let \(c \in F\). Then the measure \(\mu \) translated by \(c\) (the map of \(\mu \) by \(x \mapsto x + c\)) is a Gaussian measure on \(F\).

The convolution of two Gaussian measures is a Gaussian measure.

A finite measure \(\mu \) on a Hilbert space \(E\) is Gaussian if and only if for every \(t \in E\), the characteristic function of \(\mu \) at \(t\) is

A finite measure \(\mu \) on \(E\) is Gaussian if and only if there exists \(m \in E\) and \(C\) positive semidefinite such that for all \(t \in E\), the characteristic function of \(\mu \) at \(t\) is

If that’s the case, then \(m = m_\mu \) and \(C = C'_\mu \).

Let \(F, G\) be two Banach spaces, let \(\mu \) be a Gaussian measure on \(F\) and let \(T : F \to G\) be a continuous linear map. Then \(T_*\mu \) is a Gaussian measure on \(G\).

A multivariate Gaussian measure is a Gaussian measure.

The standard Gaussian measure on \(E\) is a Gaussian measure.

The Brownian motion is a Gaussian process.

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are modifications of each other (that is, for all \(t \in T\), \(X_t =_{a.e.} Y_t\)). If \(X\) is a Gaussian process, then \(Y\) is a Gaussian process as well.

The pre-Brownian process \(X\) of Definition 6.1 is a Gaussian process.

The paths of the Brownian motion are locally Hölder continuous of all orders \(\gamma \in (0, 1/2)\).

If \(X : T \to \Omega \to E\) is a function that satisfies the Kolmogorov condition, then for all \(t \in T\), \(X_t\) is \(\mathbb {P}\)-a.e. measurable.

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition, then for all \(s,t \in T\) the function \(\omega \mapsto d_E(X_s(\omega ), X_t(\omega ))\) is \(\mathbb {P}\)-a.e. measurable.

If \(X : T \to \Omega \to E\) is a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\) and \(s, t \in T\) are such that \(d_T(s, t) = 0\), then \(\mathbb {P}\)-a.e. \(d_E(X_s, X_t) = 0\).

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(T'\) be a countable subset of \(T\) such that for all \(s, t \in T'\), \(d_T(s, t) = 0\). Then

The pre-Brownian process \(X\) of Definition 6.1 satisfies the Kolmogorov condition for exponents \((2n,n)\) with constant \((2n - 1)!!\) for all \(n \in \mathbb {N}\). That is, for all \(s, t \in \mathbb {R}_+\), we have

Let \(P\) be a predicate on paths and suppose \(X\) is a stochastic process satisfying \(P\) a.s. Then, defining

for all \(n \in \mathbb {N}\), the sequence \((\tau _n)_{n \in \mathbb {N}}\) is a localizing sequence.

Assume \(T\) has a bottom element and that its closed intervals are compact, and that the filtration is right-continuous. If \(X\) is a real-valued càdlàg and adapted process, then the sequence \(\tau _n := \inf \{ t | X_t \ge n\} \) is a localizing sequence.

If \((\tau _n)_{n \in \mathbb {N}}\) is a pre-localizing sequence, then the sequence defined by \(\tau '_n = \inf _{m \ge n} \tau _m\) is a localizing sequence.

If \(M\) is a cadlag local martingale, then \(\Vert M \Vert ^2\) is a cadlag local sub-martingale.

Every local martingale is locally of class D.

A local martingale is a cadlag martingale if and only if it is of class DL.

Every local submartingale is locally of class D.

A nonnegative local submartingale is a cadlag submartingale if and only if it is of class DL.

Let \((\tau _n)_{n \in \mathbb {N}}\) be a localizing sequence and let \((\sigma _{n,k})_{k \in \mathbb {N}}\) be a localizing sequence for each \(n\). Then, there exists a strictly increasing sequence \((k_n)_{n \in \mathbb {N}}\) such that the sequence defined by \(\tau '_n = \tau _n \wedge \sigma _{n,k_n}\) is a pre-localizing sequence.

The standard Gaussian measure is a probability measure.

The projective family of the Brownian motion is a projective family of measures.

For \(M\) a square integrable martingale, the function \(t \mapsto \Vert M_t \Vert _{L^2}\) is non-decreasing.

If \(M\) and \(N\) are square integrable martingales and \(a \in \mathbb {R}\), then \(M + N\) and \(a M\) are square integrable martingales.

If \(M\) is a square integrable martingale, then \(\Vert M \Vert ^2\) is a submartingale.

For \(M\) a square integrable martingale,

For \(M\) a square integrable martingale, we have \(M_t \to M_\infty \) almost surely and in \(L^2\) as \(t \to \infty \) .

For \(V \in \mathcal{E}_{T, F}\), \(M \in \mathcal{M}^2(E)\) and a continuous bilinear map \(B: E \times F \to G\), the elementary stochastic integral \(V \bullet _B M\) is in \(\mathcal{M}^2(G)\).

The class D is stable.

The class DL is stable.

The class of process with integrable supremum is stable.

The class of process with locally integrable supremum is stable.

The class of cadlag processes is stable.

The class of processes with left limits is stable.

If \(P\) is a stable class of processes, then \(P_{\mathrm{loc}}\) is also stable.

The class of right continuous processes is stable.

\(\tau _n(c)\) is a stopping time.

If \(X : ι \to Ω \to ℝ\) is a progressively measurable process with respect to a right-continuous filtration, then for any \(a \in \mathbb {R}\), the random time \(\tau _{X \ge a}\) is a stopping time.

Real simple processes generate the predictable \(\sigma \)-algebra, i.e., the predictable \(\sigma \)-algebra is the supremum of the comaps by simple processes (seen as maps from \(T \times \Omega \) to \(\mathbb {R}\)) of the Borel \(\sigma \)-algebra.

Let \(( f_n)_{n\in \mathbb {N}}\) be a uniformly integrable sequence of functions on a probability space \((\Omega , \mathcal{F} , P)\). Then there exist functions \(g_n \in convex( f_n, f_{n+1}, \cdots )\) such that \((g_n)_{n\in \mathbb {N}}\) converges in \(L^1 (\Omega )\).

Let \(H\) be a Hilbert space and \((f_n)_{n\in \mathbb {N}}\) a bounded sequence in \(H\). Then there exist functions \(g_n\in convex(f_n,f_{n+1},\cdots )\) such that \((g_n)_{n\in \mathbb {N}}\) converges in \(H\).

Let \((f_n)_{n\in \mathbb {N}}\) be a sequence in a vector space \(E\) and \(\phi : E \to \mathbb {R}_+\) be a function such that \(\phi (f_n)\) is a bounded sequence. For \(\delta {\gt} 0\), let \(S_\delta = \{ (f, g) \mid \phi (f)/2 + \phi (g)/2 - \phi ((f+g)/2) \ge \delta \} \). Then there exist \(g_n\in convex(f_n,f_{n+1},\cdots )\) such that for all \(\delta {\gt} 0\), for \(N\) large enough and \(n, m \ge N\), \((g_n, g_m) \notin S_\delta \).

For \(i,n\in \mathbb {N}\) set \(f_{n}^{(i)}:=f_n \mathbb {1}_{(|f_n|\leq i)}\) such that \(f_{n}^{(i)}\in L^2\). There exists the sequence of convex weights \(\lambda _n^{n}, \ldots , \lambda _{N_n}^{n}\) such that the functions \( (\lambda _n^{n} f_n^{(i)} + \ldots +\lambda _{N_n}^{n} f_{N_n}^{(i)})_{n\in \mathbb {N}}\) converge in \(L^2\) for every \(i\in \mathbb {N}\) uniformly.

Let \((f_n)_{n\in \mathbb {N}}\) be a sequence of random variables with values in \([0, \infty ]\). Then there exist random variables \(g_n \in convex( f_n, f_{n+1}, \cdots )\) such that \((g_n)_{n\in \mathbb {N}}\) converges almost surely to a random variable \(g\).

The space \(\mathcal{M}^2(E)\) is a real vector space.

For \(M \in \mathcal{M}^2(E)\), \(\Vert M \Vert = 0\) if and only if \(M = 0\).

For \(X, Y \in L^2(\mu , P)\), we have

The norm is \(\Vert X \Vert _{L^2(\mu , P)}^2 = P\left[ \int _0^{\infty } \Vert X_t \Vert ^2 \: d\mu \right]\) .

Let \(V \in \mathcal{E}_{T, E}\) and \(\mu \) a measure on \(T\) . Let \(f\) be a right-continuous non-decreasing function such that for all \(s {\lt} t\), \(\mu ((s,t]) = f(t) - f(s)\) (the CDF of \(\mu \), but that’s only defined for \(\mathbb {R}\) in Lean). Then

For \(\mathrm{diam}(T) {\lt} \infty \), \(p{\gt} 0\), \(q {\gt} d {\gt} 0\) and \(\beta \in (0, (q-d)/p)\), the constant \(L(T, c_1, d, p, q, \beta )\) is finite.

Let \(A \subseteq E\) and let \(S \subseteq A\) be an \(\varepsilon \)-separated set. Then \(\vert S \vert V(B_{\varepsilon /2}) \le V(A + B_{\varepsilon /2})\).

The filtration \(\mathcal{F}\) is contained in its right continuation.

Let \(\tau \) be an \((\mathcal{F}_t)_{t\in [0,T]}\) stopping time. We have \(\lim _n\mathbb {E}[A^n_\tau ]=\mathbb {E}[A_\tau ]\).

Let \(\tau \) be an \((\mathcal{F}_t)_{t\in [0,T]}\) stopping time. We have \(\limsup _n \mathcal{A}_\tau ^n = A_\tau \).

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\) and let \(J\) be a finite subset of \(T\). Let \(C\) be an \(\varepsilon \)-cover of \(J\) with \(C \subseteq J\). If \(\varepsilon {\lt} \inf _{s, t \in J; d_T(s, t){\gt}0} d_T(s, t)\) then

Suppose that the filtration is right-continuous. Let \(P, Q\) be two classes of stochastic processes such that \(P \subseteq Q_{\mathrm{loc}}\) and \(Q\) is stable. Let \(X\) be a stochastic process that satisfies \(P\) locally. Then \(X\) satisfies \(Q\) locally. In short, if \(P\) implies \(Q\) locally, then \(P\) locally implies \(Q\) locally.

For \(M, N\) local martingales, the process \(\langle M_t, N_t \rangle _E - \langle M, N \rangle _t\) is a local martingale.

For \(M\) a local martingale, the process \(\Vert M_t \Vert ^2 - \langle M \rangle _t\) is a local martingale.

The constant sequence \(\tau _n = \infty \) is a localizing sequence.

Let \((\sigma _n), (\tau _n)\) be localizing sequences. Then \((\sigma _n \wedge \tau _n)\) is a localizing sequence.

Assume \(T\) has a bottom element and that its closed intervals are compact, and that the filtration is right-continuous. A càdlàg process is locally of class D if and only if it is locally of class DL.

Assume \(T\) has a bottom element and that its closed intervals are compact, and that the filtration is right-continuous. A càdlàg process is locally of class D if and only if it has locally integrable supremum.

If the filtration is right-continuous and \(X\) is locally of class DL then it is locally of class D.

Assume \(T\) has a bottom element and that its closed intervals are compact, and that the filtration is right-continuous. If \(X\) is a càdlàg process, then it is locally of class DL if and only if it has locally integrable supremum.

If \(P, Q\) are stable classes of processes then \((P\cap Q)_{\mathrm{loc}} = P_{\mathrm{loc}}\cap Q_{\mathrm{loc}}\).

A stochastic process \(X\) is locally cadlag if and only if it is cadlag almost surely.

A stochastic process \(X\) has left limits locally if and only if it has left limits almost surely.

Suppose that the filtration is right-continuous. For any stable class of processes \(P\), we have \((P_{\mathrm{loc}})_{\mathrm{loc}} = P_{\mathrm{loc}}\).

If \(P \subseteq Q\) then \(P_{\mathrm{loc}} \subseteq Q_{\mathrm{loc}}\).

If \(P\) be a predicate on paths such that the constant path \(0\) satisfies \(P\) and \(X\) is a stochastic process satisfying \(P\) a.s. then, \(X\) satisfies \(P\) locally.

Let \(P\) be a stable class of processes and let \((\tau _n)_{n \in \mathbb {N}}\) be a pre-localizing sequence such that for all \(n \in \mathbb {N}\), \(X^{\tau _n}\mathbb {I}_{\tau _n {\gt} 0}\) is in \(P\). If the filtration is right-continuous, then \(X\) is locally in \(P\).

A stochastic process \(X\) is locally right continuous if and only if it is right continuous almost surely.

If a real-valued right-continuous function has bounded variation on a set, then it is a difference of right-continuous monotone functions there.

If a real-valued function has bounded variation on a set, then it is a difference of monotone functions there.

The quadratic variation \(\langle M \rangle \) of a local martingale \(M\) is locally integrable.

For \(i \ne j\), the balls \(B_{V_i}(t, (r_i-1)c)\) and \(B_{V_j}(t_j, (r_j-1)c)\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are disjoint.

For any log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\), for all \(k \ge \vert J \vert \), \(V_k = \emptyset \).

The sets \(V_i\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are a decreasing sequence of sets. That is, \(V_{i+1} \subseteq V_i\) for all \(i \in \mathbb {N}\).

The radius of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\) satisfies \(r_i \le n\) for all \(i \in \mathbb {N}\).

For every \(t\in [0,T]\) we have \(\mathcal{M}^n_t\stackrel{L^1}{\rightarrow }M_t\).

\(M_t\) admits a martingale cadlag modification.

\(\mathcal{M}^n\) is cadlag.

There are convex weights \(\lambda ^n_n,\cdots ,\lambda ^n_{N_n}\) such that \(\mathcal{M}^n_T\stackrel{L^1}{\rightarrow }M\), where \(\mathcal{M}^n:=\lambda ^n_nM^n+\cdots +\lambda ^n_{N_n}M^{N_n}.\)

\(M^n_t\) admits a modification which is a cadlag martingale.

The sequence \((M^n_T)_{n\in \mathbb {N}}\) is uniformly integrable (bounded in \(L^1\) norm).

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes. Then \(X\) and \(Y\) have same finite-dimensional distributions if and only if they have the same law.

Let \(X, Y : T \to \Omega \to E\) be two stochastic processes that are modifications of each other. Then for all \(t_1, \ldots , t_n \in T\), the random vector \((X_{t_1}, \ldots , X_{t_n})\) has the same distribution as the random vector \((Y_{t_1}, \ldots , Y_{t_n})\). That is, \(X\) and \(Y\) have same finite-dimensional distributions.

A càdlàg martingale is of class D if and only if it is uniformly integrable.

Every càdlàg martingale is of class DL.

If \(X\) is a martingale and \(Y\) is an adapted modification of \(X\), then \(Y\) is a martingale.

Let \(X\) be a martingale and \(V\) be a bounded simple process. Then the elementary stochastic integral \(V \bullet _B X\) is a martingale.

Every cadlag martingale is a local martingale.

Let \(X : T \rightarrow \Omega \rightarrow E\) a martingale with values in a normed space \(E\). Let \(\phi : E \rightarrow \mathbb {R}\) convex and continuous such that \(\phi (X_t)\in L^1(\Omega )\) for every \(t\in T\). Then \(\phi (X)\) is a sub-martingale.

An adapted integrable process \(X\) is a martingale if and only if for every bounded simple process \(V\) with values in \(\mathbb {R}\), \(\mathbb {E}[(V \bullet _{\mathbb {R}} X)_\infty ] = 0\).

Let \(X \in L^2(M)\) such that \(\langle X, V\rangle = 0\) for any simple process \(V\). Let \(A_t = \int _0^t X_s \: d\langle M \rangle _s\). Then \(A_t\) is a martingale.

Suppose that the filtration is sigma-finite. Then the martingale part of an adapted process \(X\) such that \(X_n\) is integrable for all \(n\) is a martingale.

Let \(X : \mathbb {N} \rightarrow \Omega \rightarrow \mathbb {R}\) be a non-negative sub-martingale. Then for every \(n \in \mathbb {N}\) and \(\lambda {\gt} 0\),

Let \((X_n)_{n \in \mathbb {N}}\) be a sequence of \(p\)-uniformly integrable stochastic processes and suppose \(X_n \to X\) in probability as \(n \to \infty \). Then, \(X\) is \(L^p\).

The simple processes \(\mathcal{E}_{T, F}\) form a module over the scalars \(\mathbb {R}\).

The quadratic variation \(\langle M \rangle \) of a local martingale \(M\) is non-decreasing.

The predictable part of a real-valued submartingale is an almost surely nondecreasing process.

Let \(X\) be a right-continuous \(\mathcal{F}\)-martingale on an approximable time index. Then, for any stopping times \(\sigma , \tau \) with \(\tau \) bounded, we have that \(X_{\sigma \wedge \tau } = P[X_{\tau } \mid \mathcal{F}_{\sigma }]\) almost surely.

Let \(X\) be a discrete time martingale with respect to the filtration \(\mathcal{F}\) and let \(\tau , \sigma \) be stopping times. Then, if \(\tau \) is bounded, we have that almost surely, \(X_{\tau \wedge \sigma } = P[X_{\tau } \mid \mathcal{F}_{\sigma }]\).

Let \(X\) be a discrete time submartingale with respect to the filtration \(\mathcal{F}\) taking values in a real Banach space \(E\). Assume \(E\) is an order-closed partial order, an ordered monoid and an ordered module. Let \(\tau , \sigma \) be stopping times. Then, if \(\tau \) is bounded, we have that almost surely, \(X_{\tau \wedge \sigma } \le P[X_{\tau } \mid \mathcal{F}_{\sigma }]\).

Let \(X\) be a discrete time supermartingale with respect to the filtration \(\mathcal{F}\) taking values in a real Banach space \(E\). Assume \(E\) is an order-closed partial order, an ordered monoid and an ordered module. Let \(\tau , \sigma \) be stopping times. Then, if \(\tau \) is bounded, we have that almost surely, \(X_{\tau \wedge \sigma } \ge P[X_{\tau } \mid \mathcal{F}_{\sigma }]\).

Let \(X\) be a right-continuous \(\mathcal{F}\)-submartingale on an approximable time index. Then, for any stopping times \(\sigma , \tau \) with \(\tau \) bounded, we have that \(X_{\sigma \wedge \tau } \le P[X_{\tau } \mid \mathcal{F}_{\sigma }]\) almost surely.

\(P_\varepsilon (A) V(B_{\varepsilon /2}) \le V(A + B_{\varepsilon /2})\).

\(P_{2\varepsilon }(A) \le N^{ext}_\varepsilon (A)\).

Let \((T,d_T)\) be a metric space. Let \(J \subseteq T\) be finite, \(a {\gt} 1\), \(c{\gt}0\) and \(n \in \{ 1, 2, ...\} \) such that \(|J| \le a^n\). Then, there is \(K \subseteq J^2\) such that for any function \(f : T \to E\) with \((E,d_E)\) a metric space,

For \(I = \{ t_1, \ldots , t_n\} \) a finite subset of \(\mathbb {R}_+\), let \(C \in \mathbb {R}^{n \times n}\) be the matrix \(C_{ij} = \min (t_i, t_j)\) for \(1 \leq i,j \leq n\). Then \(C\) is positive semidefinite.

A gram matrix is positive semidefinite.

\((A^n_t)_{t\in \mathcal{D}_n^T}\) is an increasing process.

A predictable process is progressively measurable.

The covariation \(\langle M, N \rangle \) of local martingales \(M\) and \(N\) is a predictable process.

Sets of the form \((s, t] \times A\) for any \(A \in \mathcal{F}_s\) is measurable with respect to the predictable \(\sigma \)-algebra.

Let \(X : \mathbb {N} \to \Omega \to E\) be a stochastic process and let \(\mathcal{F}\) be a filtration indexed by \(\mathbb {N}\). Then \(X\) is predictable if and only if \(X_0\) is \(\mathcal{F}_0\)-measurable and for all \(n \in \mathbb {N}\), \(X_{n+1}\) is \(\mathcal{F}_n\)-measurable.

The predictable part of a process is predictable.

The quadratic variation \(\langle M \rangle \) of a local martingale \(M\) is a predictable process.

A simple process is predictable.

For any integer \(n \ge 0\), \(A_{n+1} = A_n + \mathbb {E}[X_{n+1} - X_n \mid \mathcal{F}_n]\).

We have \(A_0 = 0\).

An elementary predictable set is measurable with respect to the predictable \(\sigma \)-algebra.

Let \(B\) be a standard Brownian motion. Then the quadratic variation of \(B\) is given by \(\langle B \rangle _t = t\) .

\(\langle M \rangle _0 = 0\) .

Suppose \(T\) is a topological space with the topology being the order topology. For \(t \in T\), the right continuation of \(\mathcal{F}\) at \(t\) is given by

If \(T\) is a densely ordered linear order with no maximal element, then forall \(t \in T\) we have \(\mathcal{F}_{t+} = \mathop{\Large \sqcap }_{u {\gt} t} \mathcal{F}_u\).

If \(T\) is a linear order and there exists \(u {\gt} t\) such that \((t, u) = \emptyset \), then \(\mathcal{F}_{t+} = \mathcal{F}_t\).

Fake lemma for the dependency graph. Import this to depend on Definition 7.38.

If \(\mathcal{F}\) is right-continuous, then for all \(t \in T\), any set \(A \subseteq \Omega \) which is \(\mathcal{F}_t\)-measurable is also \(\mathcal{F}_{t+}\)-measurable.

The right continuation of \(\mathcal{F}\) is right-continuous.

If \(t \in T\) is maximal, \(\mathcal{F}_{t+} = \mathcal{F}_t\).

Suppose \(T\) is a topological space with the topology being the order topology. Assume that \(t \in T\) is isolated on the right, meaning that there exists a neighbourhood \(\mathcal{V}\) of \(t\) such that \(\mathcal{V} \cap \{ u \mid u {\gt} t\} = \emptyset \). Then \(\mathcal{F}_{t+} = \mathcal{F}_t\).

Suppose \(T\) is a topological space with the topology being the order topology. Assume that \(t \in T\) is not isolated on the right, meaning that for all neighbourhood \(\mathcal{V}\) of \(t\), \(\mathcal{V} \cap \{ u | u {\gt} t\} \ne \emptyset \). Then \(\mathcal{F}_{t+} = \mathop{\Large \sqcap }_{u {\gt} t} \mathcal{F}_u\).

Assume that \(T\) is a densely ordered linear order, meaning that for all \(s {\lt} t\), there exists \(u\) such that \(s {\lt} u {\lt} t\). If \(t\) is not maximal, then \(\mathcal{F}_{t+} = \mathop{\Large \sqcap }_{u {\gt} t} \mathcal{F}_u\).

The right continuation of the right continuation of \(\mathcal{F}\) is equal to the right continuation of \(\mathcal{F}\).

Assume that \(T\) is a linear order with successor. This means that for any \(t\), there is an element \(succ(t) \ge t\) such that for any \(u {\gt} t\), \(u \ge succ(t)\), and if \(succ(t) \le t\), then \(t\) is maximal. Then the right continuation of \(\mathcal{F}\) is equal to \(\mathcal{F}\).

Fake lemma for the dependency graph. Import this to depend on Definition 7.25.

Let \(X : T \to E\). Let \((\varepsilon _n)_{n \in \mathbb {N}}\) be a sequence of positive numbers, \(C_n\) a finite \(\varepsilon _n\)-cover of \(J \subseteq T\) with \(C_n \subseteq J\). For \(m \le k\),

Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\). Let \(C_n\) a finite \((\varepsilon _0 2^{-n})\)-cover of \(T\) for \(\varepsilon _0 \le \mathrm{diam}(T)\) with \(C_n \subseteq T\), and with minimal cardinality. Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Then for \(m \le k\),

For \(V, W\) two simple processes with values in \(E\) and \(F\) respectively, the process defined by \(B(V, W)_t(\omega ) = B(V_t(\omega ), W_t(\omega ))\) is a simple process.

Note: in Lean, the above is not a lemma but a definition of a simple process with a lemma that its coercion to a function is given by above.

Any simple process in \(\mathcal{E}_{T, E}\) is in \(L^2(\mu , P)\) .

(Lean remark: we mean that the uncurried version of its coercion to a function satisfies MemLp)

For \(V \in \mathcal{E}\) and \(M \in \mathcal{M}^2\), then \(V \bullet M \in \mathcal{M}^2\) (by Lemma 13.23) and

The class of cadlag martingales is stable. That is, if \(M\) is a cadlag martingale and \(\tau \) is a stopping time, then the stopped process cadlag \(M^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is also a martingale.

The class of cadlag submartingales is stable. That is, if \(M\) is a cadlag submartingale and \(\tau \) is a stopping time, then the stopped process \(M^{\tau }\mathbb {I}_{\tau {\gt} 0}\) is also a cadlag submartingale.

Let \(X\) be a stochastic process and \(τ\) be a stopping time taking finitely many values. Then \(\mathbb {1}_{[0, τ]}\) is a simple process and

\(T = \mathbb {R}_+\) is an approximable time index. In particular, for any stopping time \(\tau \) on \(\overline{\mathbb {R}_+}\), defining \(\tau _n = 2^{-n} \lceil 2^n \tau \rceil \), we have that \((\tau _n)\) is a discrete approximation sequence of \(\tau \).

\(T = \mathbb {N}\) is an approximable time index.

Let \(\tau , \sigma \) be stopping times such that \(\tau \le \sigma \). Then, \(\mathcal{F}_\tau \subseteq \mathcal{F}_\sigma \).

A nonnegative right-continuous submartingale is of class D if and only if it is uniformly integrable.

Every nonnegative right-continuous submartingale is of class DL.

If \(X\) is a submartingale and \(Y\) is an adapted modification of \(X\), then \(Y\) is a submartingale.

Let \(X\) be a submartingale and \(V\) be a nonnegative bounded simple process. Then the elementary stochastic integral \(V \bullet _{\mathbb {R}} X\) is a submartingale.

Let \(X : T \to \Omega \to \mathbb {R}\) be a submartingale which is right-continuous in probability. Then \(X\) has a cadlag modification.

Let \(X : T \to \Omega \to \mathbb {R}\) be a submartingale. Then \(X\) has a modification \(Y\) such that for all \(t \in T\), \(Y\) has left and right limits at \(t\) and such that there is a countable set \(S \subseteq T\) for which \(Y\) is right-continuous on \(T \setminus S\).

Let \(X\) be a submartingale. Then for all bounded stopping times \(\tau \), the stopped value \(X_\tau \) is integrable.

Let \(X : T \to \Omega \to \mathbb {R}\) be a submartingale. Then for every \(t \in T\), the set \(\{ \mathbb {E}[(\mathbb {1}_A \bullet X)_t] \mid A \text{ elementary predictable}\} \) is bounded.

Let \(X : T \to \Omega \to \mathbb {R}\) be a submartingale and \(t \in T\). Let \(t_n\) be a decreasing sequence in \(T\) converging to \(t\), and such that \(X_{t_n}\) tends to a limit \(X_{t+}\) almost surely. Then \(X_t \le P[X_{t+} \mid \mathcal{F}_t]\) almost surely.

Every cadlag submartingale is locally of class D.

Every cadlag submartingale for a right-continuous filtration has locally integrable supremum.

Let \(X : \mathbb {N} \to \Omega \to \mathbb {R}\) be a sub-martingale and \(\tau \) a stopping time with respect to the filtration \(\mathcal{F}\). Then, the stopped process \(X^{\tau }\) is a sub-martingale with respect to the filtration \(\mathcal{F}\).

Let \(X:\mathbb {R}_+ \to \Omega \to \mathbb {R}\) be a cadlag submartingale and \(\tau \) a stopping time. Then the stopped process \(X^\tau \) is a submartingale.

Let \(X : T \to \Omega \to \mathbb {R}\) be a submartingale and let \(t_n\) be a decreasing sequence in \(T\) which is bounded below. Then the family \(\{ X_{t_n}\} _{n \in \mathbb {N}}\) is uniformly integrable.

An adapted integrable process \(X\) is a submartingale if and only if for every bounded simple process \(V\) with values in \(\mathbb {R}_+\), \(\mathbb {E}[(V \bullet _{\mathbb {R}} X)_\infty ] \ge 0\).

Note that in Lean \(V\) comes with the data of a sum representation: by “values in \(\mathbb {R}_+\)” we mean that the random variables in that particular sum representing \(V\) are nonnegative.

Let \(K\) be the pair set of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\). Then for any function \(f : T \to E\) with \((E,d_E)\) a metric space,

For \(Y\) a stochastic process, let \(Y^*_t = \sup _{s \le t} \Vert Y_s \Vert \). Let \(X\) be a stochastic process and let \(\tau = \inf \{ t \mid \Vert X_t \Vert \ge n\} \) for some \(n \in \mathbb {R}\). Then

For \(V\) a simple process and \(X\) a stochastic process, \((V \bullet _B X)_t\) tends to its limit process \((V \bullet _B X)_\infty \) as \(t\) goes to infinity.

Let \(\tau \) be a stopping time bounded by \(t \in T\) and \((\tau _n)\) be a discrete approximation sequence of \(\tau \). Then, for any right continuous martingale \(X\), \(X_{\tau } \in L^1\) and \(X_{\tau _n \wedge t} \to X_{\tau }\) in \(L^1\) as \(n \to \infty \).

Let \(u : \beta \to \alpha \), for \(\alpha \) a densely ordered, conditionally complete linear order equipped with the order topology. Let \(S\) be a dense subset of \(\alpha \) and \(f\) a filter of \(\beta \). If for all \(a {\lt} b\) in \(S\), the number of upcrossings of the interval \([a, b]\) by \(u\) along \(f\) is finite, then \(u\) tends to a limit along \(f\).

Given a right continuous process \(X\) and a discrete approximation sequence \((\tau _n)\) of the stopping time \(\tau \), we have that

\(\vert df \vert = df_+ + df_-\) .

Let \((X_t)_{t \in T}\) be a family of \(p\)-uniformly integrable stochastic processes. Then the family of limits in almost everywhere convergence of sequences of \(X\) is \(p\)-uniformly integrable.

Let \((X_t)_{t \in T}\) be a family of \(p\)-uniformly integrable stochastic processes. Then the family of limits in probability of sequences of \(X\) is \(p\)-uniformly integrable.

Let \(\tau \) be a stopping time bounded by \(t \in T\) and \((\tau _n)\) be a discrete approximation sequence of \(\tau \). Then, for any martingale \(X\), the sequence of stopped values \((X_{\tau _n \wedge t})\) is uniformly integrable.

Let \(X\) be a martingale on a discrete index set and let \((\tau _k)_{k \in \mathbb {N}}\) be a sequence of stopping times that are uniformly bounded by \(n\). Then, the family of stopped values \(\{ X_{\tau _k}\} _{k \in \mathbb {N}}\) is uniformly integrable.

Let \(X\) be a martingale and let \((\tau _k)_{k \in \mathbb {N}}\) be a sequence of stopping times that are uniformly bounded by \(n\). Then, the family of stopped values \(\{ X_{\tau _k}\} _{k \in \mathbb {N}}\) is uniformly integrable if for each \(k\), \(\tau _k\) takes value in a countable set.

Let \(X\) be a submartingale on a discrete index set and let \((\tau _k)_{k \in \mathbb {N}}\) be a sequence of stopping times that are uniformly bounded by \(p\). Then, the family of stopped values \(\{ X_{\tau _k}\} _{k \in \mathbb {N}}\) is uniformly integrable.

Let \((X_t)_{t \in T}\) and \((Y_t)_{t \in T}\) be two families of uniformly integrable random variables. Then the family \((X_t + Y_t)_{t \in T}\) is uniformly integrable.

If \((X_t)_{t \in T}\) is uniformly integrable and \(\phi : S \to T\), then \((X_{\phi (s)})_{s \in S}\) is uniformly integrable.

Let \((X_s)_{s \in S}\) be a family of random variables and \((Y_t)_{t \in T}\) be a family of uniformly integrable random variables. If for all \(s\), there exists \(t\) such that \(\| X_s\| \le \| Y_t\| \) almost surely, then \(X\) is uniformly integrable.

Let \((X_t)_{t \in T}\) be a family of random variables and \(Y\) be a real random variable in \(L^p\). If for all \(t\), \(\| X_t\| \le Y\) almost surely, then \(X\) is uniformly integrable.

Let \((X_t)_{t \in T}\) be a family of uniformly integrable random variables. It is uniformly integrable if and only if \((\| X_t\| )_{t \in T}\) is.

If \((X_t)_{t \in T}\) is a family of uniformly integrable random variables, then so is \((\| X_t\| )_{t \in T}\).

Let \(X : T \to \Omega \to \mathbb {R}\) be a stochastic process on a finite time domain \(T\). Then for all \(a {\lt} b\) in \(\mathbb {R}\) and \(t \in T\), there exists an elementary predictable set \(A\) such that the number of upcrossings \(U_t[a, b]\) of the interval \([a, b]\) before time \(t\) satisfies

A sequence of functions converges in \(L^1\) if and only if it converges in probability and is uniformly integrable.

If \(0 {\lt} \varepsilon \) then \(\frac{V(A)}{V(B_\varepsilon )} \le N^{int}_\varepsilon (A)\).

If \(0 {\lt} \varepsilon \) then \(V(A) \le N^{ext}_\varepsilon (A) V(B_\varepsilon )\).

Let \(A \subseteq E\) and \(C \subseteq E\) be a finite \(\varepsilon \)-cover of \(A\). Denote by \(V(A)\) the volume of \(A\). Then \(V(A) \le \vert C \vert V(B_\varepsilon )\), in which \(B_\varepsilon \) is the closed ball of radius \(\varepsilon \) in \(E\).

The characteristic function of a multivariate Gaussian measure \(\mathcal{N}(m, \Sigma )\) is given by

The Borel sigma-algebra on \(C(\mathbb {R}_+, \mathbb {R})\) coming from the compact-open topology is equal to the smallest sigma-algebra for which the evaluation maps \(f \mapsto f(t)\) are measurable for every \(t \in \mathbb {R}_+\).

Suppose that \(T\) has bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). Let \(\beta \in (0, (q - d)/p)\). Then for every countable subset \(T' \subseteq T\) with positive diameter,

Let \(X: \mathbb {R}_+ \to \Omega \to \mathbb {R}\) be a right-continuous non-negative sub-martingale. For every \(T \in \mathbb {R}_+\) and \(\lambda {\gt}0\) we have

Let \(X:\mathbb {R} \rightarrow \Omega \rightarrow \mathbb {R}\) be a right-continuous non-negative sub-martingale. For every \(T, \lambda {\gt}0\) and \(p{\gt}1\) we have

That is, for \(\Vert \cdot \Vert _p\) the \(L^p\) norm, \(\left\Vert \sup _{t\in [0,T]} X_t \right\Vert _p \leq \frac{p}{p-1} \left\Vert X_T \right\Vert _p \: .\)

Let \(S = (S_t )_{0\leq t\leq T}\) be a cadlag submartingale of class \(D\). Then, \(S\) can be written in a unique way in the form \(S = M + A\) where \(M\) is a cadlag martingale and \(A\) is a predictable increasing process starting at \(0\).

Let \(\mu \) be a finite measure on \(F\) such that \(\mu \times \mu \) is invariant under the rotation of angle \(-\frac{\pi }{4}\). Then there exists \(C {\gt} 0\) such that the function \(x \mapsto \exp (C \Vert x \Vert ^2)\) is integrable with respect to \(\mu \).

Let \(X : T \to \Omega \to \mathbb {R}\) be an adapted stochastic process such that \(X\) is integrable and for every \(t \in T\) the set \(\{ \mathbb {E}[(\mathbb {1}_A \bullet X)_t] \mid A \text{ elementary predictable}\} \) is bounded. Then \(X\) has a modification \(Y\) which has left and right limits everywhere and such that there is a countable set \(S \subseteq T\) for which \(Y\) is right-continuous on \(T \setminus S\).

In a separable Hilbert space, if two finite measures have same characteristic function, they are equal.

In a separable Banach space, if two finite measures have same characteristic function, they are equal.

Suppose that \(T\) is a finite set with bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). For all \(\delta \ge 4\mathrm{diam}(T)\),

Suppose that \(T\) is a finite set with bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). For all \(\delta \in (0, 4\mathrm{diam}(T)]\),

The space \(\mathcal{M}^2\) is a Hilbert space.

If \(X : T \to \Omega \to E\) is a progressively measurable process with respect to a right-continuous filtration and \(B\) is a Borel-measurable subset of \(E\), then the hitting time of \(X\) in \(B\) is a stopping time.

Under the assumptions of Theorem 5.77, for \(E\) a complete space, there exists a modification \(Y\) of \(X\) (i.e., a process \(Y\) with \(\mathbb {P}(Y_t \ne X_t) = 0\) for all \(t\)) such that the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\).

Let \(X\) and \(Y\) be two continuous semi-martingales. Then we have almost surely

For a Gaussian measure, there exists \(C {\gt} 0\) such that the function \(x \mapsto \exp (C \Vert x \Vert ^2)\) is integrable.

A finite measure \(\mu \) on \(F\) is Gaussian if and only if for every continuous linear form \(L \in F^*\), the characteristic function of \(\mu \) at \(L\) is

in which \(\mathbb {V}_\mu [L]\) is the variance of \(L\) with respect to \(\mu \).

Let \(M\) be a continuous local martingale with \(M_0 = 0\). If \(M\) is also a finite variation process, then \(M_t = 0\) for all \(t\).

Let \(X^1, \ldots , X^d\) be continuous semi-martingales and let \(f : \mathbb {R}^d \to \mathbb {R}\) be a twice continuously differentiable function. Then, writing \(X = (X^1, \ldots , X^d)\), the process \(f(X)\) is a semi-martingale and we have

Let \(\mathcal{X}\) be a Polish space, equipped with the Borel \(\sigma \)-algebra, and let \(T\) be an index set. Let \(P\) be a projective family of finite measures on \(\mathcal{X}\). Then the projective limit \(\mu \) of \(P\) exists, is unique, and is a finite measure on \(\mathcal{X}^T\). Moreover, if \(P_I\) is a probability measure for every finite set \(I \subseteq T\), then \(\mu \) is a probability measure.

An adapted process \(X\) is a cadlag local submartingale iff \(X = M + A\) where \(M\) is a cadlag local martingale and \(A\) is a predictable, cadlag, locally integrable and increasing process starting at \(0\). The processes \(M\) and \(A\) are uniquely determined by \(X\) a.s.

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and the same exponent \(d\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition with exponents \((p, q)\) with \(q {\gt} d\). Then \(X\) has a modification \(Y\) such that almost surely the paths of \(Y\) are locally Hölder continuous of all orders \(\gamma \in (0, (q - d)/p)\).

Let \(T\) be a metric space with a cover \((T_n)\) with bounded covering numbers with constants \(c_n\) and the same exponent \(d\). Let \((p_n, q_n)_{n \in \mathbb {N}}\) be a sequence of pairs of positive numbers such that \(q_n {\gt} d\) for all \(n \in \mathbb {N}\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition with exponents \((p_n, q_n)\) for all \(n \in \mathbb {N}\). Then \(X\) has a modification \(Y\) such that almost surely the paths of \(Y\) are locally Hölder continuous of all orders \(\gamma \in (0, \sup _n (q_n - d)/p_n)\).

Let \(X : T \to \Omega \to \mathbb {R}\) be a martingale with respect to a right-continuous filtration. Then \(X\) has a cadlag modification.

Let \(X : T \to \Omega \to \mathbb {R}\) be a submartingale with respect to a right-continuous filtration. Then \(X\) has a cadlag modification if and only if \(t \mapsto \mathbb {E}[X_t]\) is right-continuous.

Let \(X\) be an uniformly integrable cadlag martingale with respect to the filtration \(\mathcal{F}\). Then there exists a limit process \(X_\infty \) measurable with respect to \(\mathcal{F}_\infty \) such that \(X_t\) converges to \(X_\infty \) almost surely as \(t\) goes to infinity. Furthermore, \(X_t = P[X_\infty \mid \mathcal{F}_t]\) almost surely.