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 \(X : T \to \Omega \to \mathbb {R}\) be an adapted stochastic process which is right-continuous in probability and such that one of the two conditions of Theorem 11.2 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 \(m \le k\),

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 \(n \to \infty \).

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 adapted 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 adapted 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 any continuous local martingales \(M\) and \(N\), there exists a continuous process \([M,N]\) with \([M,N]_0 = 0\) such that \(MN - [M,N]\) is a local martingale. That process is a.s. unique and is called the covariation of \(M\) and \(N\).

It can be defined by \([M, N]_t = \frac{1}{4}\left([M+N]_t - [M-N]_t \right)\) .

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 \(B \in \mathcal{F}_s\).

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

An important example is \(G = E\), \(F = \mathbb {R}\) and \(B(x, r) = r \cdot x\) the scalar multiplication.

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 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 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 \(x, y \in c\), \(d_E(x, y) {\gt} \varepsilon \).

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 \(V \mapsto V \cdot M\) 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 \(t \ge 0\), \(\mathbb {E}\left[ \int _0^t X_s^2 \: d[M]_s \right] {\lt} \infty \).

Let \(M \in \mathcal{M}^2\) be a continuous square integrable martingale. We define

in which \(\mathcal{P}\) is the predictable \(\sigma \)-algebra and \(d[M]\) is the measure induced by the quadratic variation of \(M\). The norm on that Hilbert space is \(\Vert X \Vert ^2 = \mathbb {E}\left[ \int _0^{\infty } X_t^2 \: d[M]_t \right]\) .

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 \(P_{\mathrm{loc}}\).

We say that a stochastic process is integrable if for all \(t\), \(X_t\) is integrable. A process has locally integrable supremum if \((\sup _{s \le t} \Vert X_s \Vert )_t\) is locally integrable.

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 any continuous local martingale \(M\), there exists a continuous process \([M]\) with \([M]_0 = 0\) such that \(\Vert M \Vert ^2 - [M]\) is a local martingale. That process is a.s. unique and is called the quadratic variation of \(M\). \([M]\) is defined as the predictable part of the Doob-Meyer decomposition of the local sub-martingale \(\Vert M \Vert ^2\) .

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

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 \(\bigsqcap \) 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 random variables with values in a normed space \(F\) 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 \(V : T \to \Omega \to F\) defined by

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

Let \(\mathcal{M}^2\) be the set of square integrable continuous martingales with respect to a filtration \(\mathcal{F}\) indexed by \(\mathbb {R}_+\),

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 \(x \mapsto \sum _{i=1}^d x_i \cdot e_i\).

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 13.19 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 \(\mathcal{F}_\tau = \bigcup _{t \in T} \{ A \in \mathcal{F} \mid A \cap \{ \tau \le t\} \in \mathcal{F}_t\} .\)

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).

If 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 \(1 \leq i,j \leq n\).

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

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

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 \(\bot \) and that its closed intervals are compact. If \(X\) is a càdlàg and adapted process, and if the filtration is right-continuous, then any process of class DL 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 continuous square integrable martingales. Then

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.

\(V \bullet (W \bullet X) = (V \times W) \bullet X\) for simple processes \(V, W\) and stochastic process \(X\).

TODO: that’s for \(B\) the scalar multiplication. What is the right statement for general \(B\)?

The elementary stochastic integral is linear in both arguments.

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

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

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 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 \(\bot \) and that its closed intervals are compact. If \(X\) is a càdlàg and adapted process, and if the filtration is right-continuous, then any process locally of class DL 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.

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

Let \(X\) be a stochastic process such that for all \(t \in T\), \(X^*_t := \sup _{s \le t} \| X_t\| \) is integrable. 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 \(t\), \(\int _0^t X_s \: d[M]_s = 0\).

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 \(\int _0^t X_s \: d[M]_s = 0\) for all \(t \ge 0\) a.s.. Then \(X = 0\) \((\mathbb {P} \times d[M])\)-almost everywhere.

Let \(X, Y \in L^2(M)\) such that \(\int _0^t X_s \: d[M]_s = \int _0^t Y_s \: d[M]_s\) 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 pseudometric space. If \(X\) is a càdlàg process then it maps compact sets 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 \(\bot \) and that its closed intervals are compact. If \(X\) is a real-valued càdlàg and adapted process, and if the filtration is right-continuous, 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 martingale if and only if it is of class DL.

Every local submartingale is locally of class D.

A nonnegative local submartingale is a 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.

The class D is stable.

The class DL 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 \(g\).

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.

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.

A cadlag adapted process is locally of class D if and only if it is locally of class DL.

A cadlag adapted 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.

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.

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 \(V \bullet X\) is a martingale.

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

Let \(X : T \to \Omega \to \mathbb {R}\) be a martingale. 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 \(T \setminus S\).

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.

Let \(X=(X_t)_{t\in \mathcal{D}}\) be a martingale indexed by the dyadics. Then almost surely, for every \(t\geq 0\) the limit

exists and is finite.

Let \(X=(X_t)_{t\in \mathcal{D}}\) be a martingale indexed by the dyadics. Then almost surely, for every \(t\geq 0\) the limit

exists and is finite.

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

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

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

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

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.

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.

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 \([B]_t = t\) .

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 \(\mathcal{F}_{t+} = \bigsqcap _{u {\gt} t} \mathcal{F}_u\).

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 7.36.

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 \(\mathcal{F}_{t+} = \bigsqcap _{u {\gt} t} \mathcal{F}_u\).

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 \(\mathcal{F}_{t+} = \bigsqcap _{u {\gt} t} \mathcal{F}_u\).

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 7.23.

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 \(V \in \mathcal{E}\) and \(M \in \mathcal{M}^2\), then \(V \bullet M \in \mathcal{M}^2\) 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 càdlàg submartingale is of class D if and only if it is uniformly integrable.

Every nonnegative càdlàg 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 \(V \bullet X\) is a submartingale.

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

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.

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

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, \((V \bullet X)_t\) tends to its limit process \((V \bullet X)_\infty \) 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 \(n \to \infty \).

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

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 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 \(\| X_t\| \le \| Y_s\| \) 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 one of the two following conditions hold:

1. \(X\) 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.

2. For every \(t \in T\) the set \(\{ (\mathbb {1}_A \bullet X)_t \mid A \text{ elementary predictable}\} \) is bounded in probability.

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 with the inner product defined by

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 \(X\) 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.