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

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

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 \(m \le k\),

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.

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 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.

\(D\) is the class of all adapted processes \((S_t)_{0\leq t\leq T}\) such that the set \(\{ S_\tau \mid \tau \text{ is a stopping time}\} \) is uniformly integrable.

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 \([0,T]\).

Let \(V \in \mathcal{E}\) be a simple process and let \(X\) be a stochastic process. The elementary stochastic integral process \(V \cdot X : \mathbb {R}_+ \to \Omega \to E\) is defined by

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.

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

A stochastic process is a local martingale if it is locally a martingale in the sense of Definition 7.16. 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 martingale.

A stochastic process is a local submartingale if it is locally a submartingale in the sense of Definition 7.16. 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 submartingale.

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 \(\tau : \Omega \to T\) 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 \(\mathcal{F}\) a filtration indexed by \(T\) and \(t \in T\), we define \(\mathcal{F}_{t-} = \bigsqcup _{s {\lt} t} \mathcal{F}_s\) (if that supremum is nonempty: we set \(\mathcal{F}_{\bot -} = \mathcal{F}_\bot \)) and \(\mathcal{F}_{t+} = \bigsqcap _{s {\gt} t} \mathcal{F}_s\).

Note that \(\bigsqcap \) and \(\bigsqcup \) denote the infimum and supremum in the lattice of sigma-algebras on \(\Omega \).

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.).

Let \(P\) be a class of stochastic processes (or equivalently a predicate on stochastic processes). We say that a stochastic process \(X : \mathbb {R}_+ \to \Omega \to E\) 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 \(P_{\mathrm{loc}}\).

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.

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).

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 \(\{ (t, \infty ] \times A \mid t \in T, \: A \in \mathcal{F}_t\} \cup S\).

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 \(\omega \mapsto X(\cdot , \omega )\).

For any continuous local martingale \(M\), there exists a continuous process \([M]\) with \([M]_0 = 0\) such that \(M^2 - [M]\) is a local martingale. That process is a.s. unique and is called the quadratic variation of \(M\).

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

Let \(0 \le t_0 \le t_1 \le \ldots \le t_n\) in \(\mathbb {R}_+\). Let \((\eta _k)_{0 \le k \le n-1}\) be \(\mathcal{F}_{t_k}\)-measurable random variables. Then the simple process for that sequence is the process \(V : \mathbb {R}_+ \to \Omega \to E\) defined by

Let \(\mathcal{E}\) be the set of simple processes.

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 9.29 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

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.

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.

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 \(E\) 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 \(L' \in F^*\),

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

Let \(X:T\times \Omega \rightarrow \mathbb {R}^d\) a sub-martingale. Let \(\phi :\mathbb {R^d}\rightarrow \mathbb {R}\) convex 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.

Let \(M \in \mathcal{M}^2\). Then the set of simple processes is dense in \(L^2(M)\).

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

Let \(X:I\times \Omega \rightarrow \mathbb {R}\) be a sub-martingale. Let \(I\) be countable. 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

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

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

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

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 \(|t - s|\).

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 \(J \subseteq T\) 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 \(\beta \in (0, (q - d)/p)\),

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 \(X : T \to \Omega \to E\) 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 \(B \subseteq A\), \(N^{int}_\varepsilon (B) \le N^{int}_{\varepsilon /2}(A)\).

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 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

Every local submartingale \(X\) with \(X_0 = 0\) is locally of class D.

The standard Gaussian measure is a probability measure.

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

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

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 \(H\).

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.

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

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

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.

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

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

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

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

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.

Every martingale is a local 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.

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

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

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

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.

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.

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

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 \(V \cdot M \in \mathcal{M}^2\) and

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

Let \(X:\mathbb {R}\times \Omega \rightarrow \mathbb {R}\) be a cadlag martingale and \(\tau _0\) a stopping time. Then \((X_{t\wedge \tau _0})_{t\geq 0}\) is a martingale.

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,

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

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 \(\mu \).

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

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 \(\gamma \in (0, \sup _n (q_n - d)/p_n)\).