Brownian Motion. Properties of Sample Paths

Take the sets $T_{n} = \left\{ k 2^{-n+1}, k = 0, \ldots , 2^{n-1}\right\}$, $n \in \mathbb {N}$. Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ an array of independent standard Gaussian random variables $\left\{ X_{n, k}, k \in T_{n}, n \in \mathbb {N}\right\}$ is given.

Define inductively the processes $B_{n} = \left\{ B_{n}(t), t \in [0,1]\right\}$, $n \in \mathbb {N}$. Put $B_{1}(t) = t X_{1,1}$, $t \in [0,1]$. Thus on the set $T_{1}$ we have $B_{1}(0) = 0$ and $B_{1}(1) = X_{1,1}$, and for $t \in (0,1)$ we perform linear interpolation of the values of $B_{1}$ at the endpoints of the interval $[0,1]$. Assuming the process $B_{n}$ has been defined, we introduce the process $B_{n+1}$ as follows. Let

If $t = k 2^{-n} \in T_{n+1} \setminus T_{n}$, then put

Thus the process $B_{n+1}$ is defined on the set $T_{n+1}$. At the remaining points of the interval $[0,1]$ we define $B_{n+1}$ by linear interpolation through the nodes $(t, B_{n+1}(t))$, where $t \in T_{n+1}$. Thus the processes $B_{n}$ are continuous on $[0,1]$ for all $n \in \mathbb {N}$ and $\omega \in \Omega$.

The series

for a.e. $\omega$ converges uniformly on the interval $[0,1]$. By the Weierstrass theorem this series defines a.s. on $[0,1]$ a continuous function $B(t)$.

Find $\mathbb {E}\left[B_{n}(t)\right]$ and $\operatorname {cov}\left(B_{n}(s), B_{n}(t)\right)$ for $s, t \in [0,1]$ and $n \geqslant 0$. Prove that $B(t)$ is a Gaussian process with $\mathbb {E}\left[B(t)\right] = 0$ and $\operatorname {cov}(B(s), B(t)) = \min \left\{ s, t\right\}$ for $s, t \in [0,1]$.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process.

Let $\left\{ t_{n}\right\}_{n \geqslant 1}$ be a sequence of positive numbers such that $\sum_{n = 1}^{\infty } t_{n}^{-1 / 2} < \infty$. Prove that then $\left|W\left(t_{n}\right)\right| \rightarrow \infty$ a.s. as $n \rightarrow \infty$.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process.

Consider a sequence $\Pi_{n}, n \in \mathbb {N}$, of refining partitions $\left(t_{m}^{(n)}\right)$ of the interval $[0, t]$ by points of the special form $t_{m}^{(n)} = t m 2^{-n}, m = 0, \ldots , 2^{n}$. Then

Show that assertion (47) on a.s. convergence need not hold if one takes arbitrary refining partitions $\Pi_{n}, n \in \mathbb {N}$, of the interval $[0,1]$ by points $t_{m}^{(n)}, m = 0, \ldots , N_{n}(n \in \mathbb {N})$, for which

(but convergence in probability will nevertheless take place).

Assertion (47) (Exercise 3): let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process; consider a sequence $\Pi_{n}, n \in \mathbb {N}$, of refining partitions $\left(t_{m}^{(n)}\right)$ of the interval $[0, t]$ by points of the special form $t_{m}^{(n)} = t m 2^{-n}, m = 0, \ldots , 2^{n}$. Then

Prove that almost all sample paths of the Wiener process are Hölder functions with exponent $\gamma < 1 / 2$, i.e. on every time interval $[a, b] \subset [0, \infty )$

(A proof of this statement can be found, e.g., in [12], Theorem 3 on p. 127.) Can the constant $C_{\gamma }$ in the above result be taken independent of the interval $[a, b]$ ?

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process.

Prove that with probability one

(thus the Hölder condition of order $1 / 2$ fails at the point 0 ).

Using the reasoning carried out in the proof of Theorem 1, prove that the Wiener process $W = \left\{ W(t), 0 \leqslant t \leqslant 1\right\}$ has sample paths that a.s. do not satisfy at any point the Hölder condition with exponent $\gamma > 1 / 2$.

Theorem 1 (Paley-Wiener-Zygmund). With probability one the sample paths of Brownian motion $W = \left\{ W(t), t \geqslant 0\right\}$ are not differentiable at any point $t$ of the half-line $[0, \infty )$.

Derive the result of Exercise 5 from Theorem 19 of Chapter II and the property that for $q \in \mathbb {N}, 0 \leqslant s < t < \infty$

Recall that for a set $U \subset T(U$ may coincide with $T)$, where $(T, \delta )$ is a pseudometric space, an $\varepsilon$-net is a set $S_{\varepsilon } \subset T$ such that for every $t \in U$ there exists an element $s \in S_{\varepsilon }$ for which $\delta (s, t) \leqslant \varepsilon$. In other words, $U$ is covered by the closed balls

If $U$ has a finite $\varepsilon$-net, then we call a minimal $\varepsilon$-net any of its $\varepsilon$-nets $S_{\varepsilon }^{\min }(U)$ such that $\left(S_{\varepsilon }^{\min }(U)\right)$ is minimal, where $(V)$ denotes the number of elements of a finite set $V$. Put $N_{\delta }(\varepsilon , U) = \# \left(S_{\varepsilon }^{\min }(U)\right)$, emphasizing the dependence on the pseudometric $\delta$ as well.

Following [153], we introduce two conditions on the space ( $T, \delta$ ). Suppose there exist $d \in \mathbb {N}$ and $a > 0$ such that for each $U \subset T$ with diameter $D_{U}: = \sup \left\{ \delta (s, t): s, t \in U\right\} < \infty$ and any $\varepsilon > 0$

Suppose there exists $b > 0$ such that for any minimal $\varepsilon$-net $S_{\varepsilon }^{\min }(T)$

where $B_{5 \varepsilon }(t)$ is defined according to (II.53).

Theorem 19 of Chapter II (see [153; p. 134]). Let $X = \left\{ X(t), t \in T\right\}$ be a random function with values in a Polish space ( $\mathrm{S}, \rho$ ) for each $t \in T$, defined on some $(\Omega , \mathscr {F}, \mathbb {P})$ and a pseudometric space $(T, \delta )$ satisfying conditions (II.55) and (II.56). Suppose that for some $\gamma \in (0,1)$, $\alpha > d / \gamma$ and $\beta > 0$

Then $X$ has a modification $Y = \left\{ Y(t), t \in T\right\}$ with continuous sample paths such that for any $\lambda \in (0, \gamma -d / \alpha )$

and for each $t_{0} \in T$

Construct an example of a real-valued process $X = \left\{ X(t), t \geqslant 0\right\}$ of càdlàg type (but not continuous) and a closed set $F \subset \mathbb {R}$ such that $\tau_{\infty } < \tau_{F}$ a.s., where $\tau_{\infty }$ and $\tau_{F}$ are defined in the proof of Theorem 3.

Theorem 3. Let $X = \left\{ X(t), t \geqslant 0\right\}$ be a random function taking values (for each $t \geqslant 0$) in a metric space $(\mathrm{S}, \rho )$. Suppose the sample paths of $X$ are almost surely continuous. Then for every closed set $F \subset \mathrm{S}$ the time

is a Markov time with respect to the natural filtration $\mathbb {F}^{X}$ of the process $X$ (it is assumed that $\tau_{F}(\omega ) = \infty$ if $X(t, \omega ) \notin F$ for all $t \geqslant 0$).

In the proof of Theorem 3 one sets $\tau_{n} := \tau_{G_{n}} = \inf \left\{ t \geqslant 0: X(t, \omega ) \in G_{n}\right\}$, where the open sets $G_{n} = \left\{ x \in \mathrm{S}: \rho (x, F) < 1 / n\right\}$ form, as $n$ grows, shrinking neighborhoods of $F$. Clearly, $F \subset G_{n+1} \subset G_{n}$, whence $\tau_{n}(\omega ) \leqslant \tau_{n+1}(\omega ) \leqslant \tau_{F}(\omega )$ for all $n \in \mathbb {N}$, $\omega \in \Omega$. Therefore for each $\omega$ there exists

Let $X = \left\{ X_{n}, n \in \mathbb {N}\right\}$ be a sequence of real-valued random variables and let a set $B \in \mathscr {B}(\mathbb {R})$ be given. Let $\tau = \tau (\omega )$ be the first hitting time of the set $B$, defined by formula (I.49).

Formula (I.49):

($\tau (\omega ) := \infty$ if $X_{n}(\omega ) \notin B$ for all $n \in \mathbb {N}$).

We have already seen (see §3) that $\tau$ is a Markov time with respect to the natural filtration of the process $X$. Now introduce

(in the definition of the times $\tau$ and $\sigma$ it is assumed that if the set in curly braces is empty for some point $\omega$, then the value of the quantity under consideration at this point is taken to be $+\infty$ ). Prove that $\sigma$ is a Markov time with respect to the natural filtration of the process $X$. Construct an example of a sequence $X$ and a set $B$ for which the time $\gamma$ is a Markov time with respect to the natural filtration, and also an example of $X$ and $B$ for which $\gamma$ is not such a time.

Let $\tau$ be a Markov time with respect to a filtration of $\sigma$-algebras $(\mathscr {F}_{t})_{t \in T}$. Put

The system of sets $\mathscr {F}_{\tau }$ is (as is easy to check) a $\sigma$-algebra, called the $\sigma$-algebra of events observable up to the random time $\tau$ (inclusive).

Let $X = \left\{ X(t), t \in T\right\} , T \subset \mathbb {R}$, be a real-valued stochastic process and let $\mathbb {F}_{T}$ be its natural filtration. Let $\tau = \tau (\omega )$ be a Markov time with respect to this filtration; if $\tau (\omega ) = \infty$, we set $X(\tau (\omega ), \omega ) = 0$. Can one assert that $X(\tau (\omega ), \omega )$ is an $\mathscr {F}_{\tau } \mid \mathscr {B}(\mathbb {R})$-measurable random variable? What can be said in the case $T = \mathbb {N}$ ?

Set $\tau_{a} = \inf \left\{ t > 0: W(t) = a\right\}$, where $W = \left\{ W(t), t \geqslant 0\right\}$ is a Wiener process, $a \in \mathbb {R}$. Prove that $\tau_{a} \stackrel{\mathscr {D}}{ = } a^{2} \tau_{1}$, i.e. the distributions of the random variables $\tau_{a}$ and $a^{2} \tau_{1}$ coincide: $\operatorname {Law}\left(\tau_{a}\right) = \operatorname {Law}\left(a^{2} \tau_{1}\right)$.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process.

Let $U$ be the largest zero of the process $W$ on the interval $[0, t]$. Prove that the distribution of the random variable $U$ obeys the arcsine law:

Prove Wald's second identity: let $\operatorname {Var}\left[\xi_{1}\right] < \infty$ in addition to the conditions of Theorem 11; then

Theorem 11 (Wald's first identity). Let $\xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random variables, $S_{n} = \sum_{k = 1}^{n} \xi_{k}$, and let $\tau$ be a Markov time with respect to the filtration of $\sigma$-algebras $\mathscr {F}_{n} = \sigma \left\{ \xi_{1}, \ldots , \xi_{n}\right\} , n \in \mathbb {N}$, where $\mathbb {E}\left[\left|\xi_{1}\right|\right] < \infty$ and $\mathbb {E}\left[\tau \right] < \infty$. Then

where $S_{\tau (\omega )}(\omega ) = \sum_{k = 1}^{\tau (\omega )} \xi_{k}(\omega )$ for $\omega \in \left\{ \tau < \infty \right\}$, and we set $S_{\tau (\omega )}(\omega ) = 0$ for $\omega \in \left\{ \tau = \infty \right\}$.

Let $X = X(\omega )$ be a centered random variable on $(\Omega , \mathscr {F}, \mathbb {P})$ with $\mathbb {E}\left[X^{2}\right] < \infty$. Take, on $\left(\Omega^{\prime }, \mathscr {F}^{\prime }, \mathbb {P}^{\prime }\right)$, a Brownian motion $W = \left\{ W\left(t, \omega^{\prime }\right), t \geqslant 0\right\}$. Extend $X$ and $W$ to $(\Omega , \mathscr {F}, \mathbb {P}) \otimes \left(\Omega^{\prime }, \mathscr {F}^{\prime }, \mathbb {P}^{\prime }\right)$ according to (I.46).

Formula (I.46). Let $\left(\Omega^{\prime }, \mathscr {F}^{\prime }, \mathbb {P}^{\prime }\right)$ be some other probability space. Define a new probability space $(\widetilde{\Omega }, \widetilde{\mathscr {F}}, \widetilde{\mathbb {P}})$ by setting $\widetilde{\Omega } = \Omega \times \Omega^{\prime }$, $\widetilde{\mathscr {F}} = \mathscr {F} \otimes \mathscr {F}^{\prime }$, $\widetilde{\mathbb {P}} = \mathbb {P} \otimes \mathbb {P}^{\prime }$. If $X = \left\{ X_{t}(\omega ), t \in T\right\}$ is a stochastic process on $(\Omega , \mathscr {F}, \mathbb {P})$, then it can naturally be regarded as defined also on $(\widetilde{\Omega }, \widetilde{\mathscr {F}}, \widetilde{\mathbb {P}})$. Indeed, for $\widetilde{\omega } = \left(\omega , \omega^{\prime }\right)$ set

Introduce $\tau \left(\omega , \omega^{\prime }\right) = \inf \left\{ t \geqslant 1: W\left(t, \omega^{\prime }\right) = X(\omega )\right\}$. Then $W(\tau ) = X$. Prove that in this case $\widetilde{\mathrm{E}} \tau = \infty$, where $\widetilde{\mathrm{E}}$ denotes averaging with respect to the measure $\mathbb {P} \otimes \mathbb {P}^{\prime }$.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process, $a < 0 < b$, and

Define the random variables $\xi_{n, m} = W(m / n)-W((m-1) / n), m, n \in \mathbb {N}$, and set

Prove that the following property holds: $\tau_{a, b}^{(n)} / n \rightarrow \tau_{a, b}$ a.s. as $n \rightarrow \infty$.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process, $a < 0 < b$, and

Define the random variables $\xi_{n, m} = W(m / n)-W((m-1) / n), m, n \in \mathbb {N}$, and set

Prove that the family of random variables $\left\{ W\left(\tau_{a, b}^{(n)} / n\right), n \in \mathbb {N}\right\}$ is uniformly integrable.

Let a centered nondegenerate random variable $X$ with distribution function $F(x) = \mathbb {P}\left(X \leqslant x\right)$ and a Wiener process $W = \left\{ W(t), t \geqslant 0\right\}$ be given on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$. For $y < 0 < z$ set $\tau_{y, z} = \inf \left\{ t \geqslant 0: W(t) \in \left\{ y, z\right\} \right\}$, and also $\tau_{0, b} = 0$ for $b > 0$. Let

On some probability space $\left(\Omega^{\prime }, \mathscr {F}^{\prime }, \mathbb {P}^{\prime }\right)$ construct a random vector $(Y, Z)$ with values in $\mathbb {R}^{2}$ such that

Take the probability space $(\widetilde{\Omega }, \widetilde{\mathscr {F}}, \widetilde{\mathbb {P}}) = (\Omega , \mathscr {F}, \mathbb {P}) \otimes \left(\Omega^{\prime }, \mathscr {F}^{\prime }, \mathbb {P}^{\prime }\right)$ and for $\widetilde{\omega } = \left(\omega , \omega^{\prime }\right) \in \widetilde{\Omega }, t \geqslant 0$ set

Show that on $(\widetilde{\Omega }, \widetilde{\mathscr {F}}, \widetilde{\mathbb {P}})$

is a random variable, and $\widetilde{W} = \left\{ \widetilde{W}(t), t \geqslant 0\right\}$ is a Brownian motion.

Let $\mathbb {P}$ and $\mathbb {Q}$ be probability measures on a metric space $(\mathrm{S}, \mathscr {B}(\mathrm{S}))$. Suppose that for every continuous bounded function $f: \mathrm{S} \rightarrow \mathbb {R}_{+}$

Then $\mathbb {P} = \mathbb {Q}$ on $\mathscr {B}(\mathrm{S})$.

Let $W = \left\{ W(t), t \geqslant 0\right\}$ be a Wiener process, $a < 0 < b$, and $\tau_{a, b} = \inf \left\{ t \geqslant 0: W(t) \in \left\{ a, b\right\} \right\}$.

Using Wald's second identity (Exercise 14), prove that

Definition 7. A process $W = \left\{ W(t), t \geqslant 0\right\}$ is called a real Brownian motion with respect to a filtration $(\mathscr {F})_{t \geqslant 0}$ if

1. $W(t)$ is an $\mathscr {F}_{t}$-measurable variable for each $t \geqslant 0$;

2. $\left(W(t)-W(s)\right) \; \perp \! \! \! \perp \; \mathscr {F}_{s}$ for $0 \leqslant s < t$ (i.e. $W(t)-W(s)$ is independent of $\mathscr {F}_{s}$);

3. $W_{0} = 0$ a.s. and $W(t)-W(s) \sim \mathrm{N}(0, t-s)$ for $0 \leqslant s < t$;

4. the sample paths of the process $W$ are continuous a.s.

Let $\mathscr {N}$ be the class of events of zero probability of the (complete) probability space $(\Omega , \mathscr {F}, \mathbb {P})$. Without loss of generality we shall assume that in Definition 7 each $\sigma$-algebra $\mathscr {F}_{t}(t \geqslant 0)$ contains all events from $\mathscr {N}$.

Let $\left\{ \left(Y_{n}, Z_{n}\right)\right\}_{n \geqslant 1}$ be a sequence of independent random vectors with values in $\mathbb {R}^{2}$, and moreover $\left\{ \left(Y_{n}, Z_{n}\right)\right\}_{n \geqslant 1} \; \perp \! \! \! \perp \; W$, where $W = \left\{ W(t), t \geqslant 0\right\}$ is a Brownian motion (with respect to the filtration $\left(\mathscr {F}_{t}^{W}\right)_{t \geqslant 0}$ ). For $t \geqslant 0$ set $\mathscr {H}_{t} = \sigma \left\{ \mathscr {F}_{t}^{W}, \left\{ \left(Y_{n}, Z_{n}\right)\right\}_{n \geqslant 1}\right\}$, i.e. consider the $\sigma$-algebra generated by the variables $W(s)$ for $s \in [0, t]$ and $\left(Y_{n}, Z_{n}\right)_{n \geqslant 1}$. We shall assume it to be augmented by the class $\mathscr {N}$ of null events.

Show that $W$ is a Brownian motion with respect to the filtration $\left(\mathscr {H}_{t}\right)_{t \geqslant 0}$ introduced above.

Prove that the function appearing on the right-hand side of (79) is a covariance function if and only if $H \in (0,1]$.

Formula (79):

Definition 9. A process $X = \left\{ X(t), t \geqslant 0\right\}$ is called a process with stationary increments if for each $n \geqslant 1$, any $0 \leqslant t_{1} < \cdots < t_{n}$ and arbitrary $h > 0$

where $\stackrel{\mathscr {D}}{ = }$ denotes equality in distribution.

Let $X = \left\{ X(t), t \geqslant 0\right\}$ be a Gaussian process such that for all $0 \leqslant s \leqslant t < \infty$

where $c \in \mathbb {R}$, and $f: \mathbb {R}_{+} \rightarrow \mathbb {R}_{+}$ is a function. Then $X$ is a process with stationary increments.

Definition 8. Fractal (or fractional) Brownian motion with exponent $H \in (0,1]$ is a centered Gaussian process $B^{(H)} = \left\{ B^{(H)}(t), t \geqslant 0\right\}$ with covariance function

Show that for fractional Brownian motion with exponent $H \in (0,1]$

Let $(T, \rho )$ be a metric space with some fixed point $\theta$.

Definition 10. A (real) Lévy Brownian function on the set $T$ is a real Gaussian process $B^{L} = \left\{ B^{L}(t), t \in T\right\}$ with zero mean and covariance function

A Gaussian random function $X = \left\{ X(t), t \in T\right\}$ is a Lévy Brownian function on $T$ if and only if $X(\theta ) = 0$ a.s. and

(here the function introduced in (82) is assumed to be nonnegative definite).

Definition 12. A Wiener-Chentsov random field is a real Gaussian function $X = \left\{ X(t), t \in \mathbb {R}_{+}^{d}\right\}$ such that

where $t = \left(t_{1}, \ldots , t_{d}\right), s = \left(s_{1}, \ldots , s_{d}\right) \in \mathbb {R}_{+}^{d}$.

By the increments of a random field $X = \left\{ X_{t}, t \in \mathbb {R}^{d}\right\}$ on the class of rectangles (parallelepipeds) $B = (a, b] = \left(a_{1}, b_{1}\right] \times \cdots \times \left(a_{d}, b_{d}\right] \subset \mathbb {R}^{d}$ we mean the random functions of the form

where the sum is taken over all vectors $\varepsilon = \left(\varepsilon_{1}, \ldots , \varepsilon_{d}\right)$ having components 0 or 1, and $\left\| \varepsilon \right\| = \sum_{k = 1}^{d} \varepsilon_{k}$.

Let $X$ be a Wiener-Chentsov field. Prove that $\left\{ X(B), B \in \Pi \right\}$, where $\Pi$ is the collection of "rectangles" $B = (a, b] \subset \mathbb {R}_{+}^{d}$, is a Gaussian random function. Find its mean and its covariance function. Prove that for any $n \geqslant 2$ and disjoint rectangles $B_{1}, \ldots , B_{n} \in \Pi$ the variables $X\left(B_{1}\right), \ldots , X\left(B_{n}\right)$ are independent. Give a definition of the Wiener-Chentsov field in terms of the variables $X(B), B \in \Pi$.

Definition 13. The Ornstein-Uhlenbeck process with parameters $\alpha , \beta > 0$ is the process given by the formula

where $W = \left\{ W(t), t \geqslant 0\right\}$ is a standard Brownian motion.

Prove that $V = \left\{ V_{t}, t \in \mathbb {R}\right\}$ is a Gaussian process and find its covariance function.