Stochastic Processes

$\uparrow$ Suppose that $\left[X_{t}: t \in T\right]$ is a stochastic process on $(\Omega , \mathscr {F}, P)$ and $A \in \mathscr {F}$. Show that there is a countable subset $S$ of $T$ for which $P\left[A \| X_{t}, t \in T\right]= P\left[A \| X_{t}, t \in S\right]$ with probability 1. Replace $A$ by a random variable and prove a similar result.

Let $T$ be arbitrary and let $K(s, t)$ be a real function over $T \times T$. Suppose that $K$ is symmetric in the sense that $K(s, t)=K(t, s)$ and nonnegative-definite in the sense that $\sum_{i, j=1}^{k} K\left(t_{i}, t_{j}\right) x_{i} x_{j} \geq 0$ for $k \geq 1, t_{1}, \ldots , t_{k}$ in $T$, and $x_{1}, \ldots , x_{k}$ real. Show that there exists a process $\left[X_{t}: t \in T\right]$ for which $\left(X_{t_{1}}, \ldots , X_{t_{k}}\right)$ has the centered normal distribution with covariances $K\left(t_{i}, t_{j}\right), i, j=1, \ldots , k$.

*This topic may be omitted.

Let $L$ be a Borel set on the line, let $\mathscr {L}$ consist of the Borel subsets of $L$, and let $L^{T}$ consist of all maps from $T$ into $L$. Define the appropriate notion of cylinder, and let $\mathscr {L}^{T}$ be the $\sigma$-field generated by the cylinders. State a version of Theorem 36.1 for $\left(L^{T}, \mathscr {L}^{T}\right)$. Assume $T$ countable, and prove this theorem not by imitating the previous proof but by observing that $L^{T}$ is a subset of $R^{T}$ and lies in $\mathscr {R}^{T}$.

Suppose that the random variables $X_{1}, X_{2}, \ldots$ assume the values 0 and 1 and $P\left[X_{n}=1\right.$ i.o. $]=1$. Let $\mu$ be the distribution over ( 0,1 ] of $\sum_{n=1}^{\infty } X_{n} / 2^{n}$. Show that on the unit interval with the measure $\mu$, the digits of the nonterminating dyadic expansion form a stochastic process with the same finite-dimensional distributions as $X_{1}, X_{2}, \ldots$.

36.3 ↑ There is an infinite-dimensional version of Fubini's theorem. In the construction in Problem 36.3, let $L=I=(0,1), T=\left\{ 1,2, \ldots \right\}$, let $\mathscr {I}$ consist of the Borel subsets of $I$, and suppose that each $k$-dimensional distribution is the $k$-fold product of Lebesgue measure over the unit interval. Then $I^{T}$ is a countable product of copies of ( 0,1 ), its elements are sequences $x=\left(x_{1}, x_{2}, \ldots \right)$ of points of ( 0,1 ), and Kolmogorov's theorem ensures the existence on $(I^{T}, \mathscr {A}^{T})$ of a product probability measure $\pi : \pi \left[x: x_{i} \leq \alpha_{i}, i \leq n\right]=\alpha_{1} \cdots \alpha_{n}$ for $0 \leq \alpha_{i} \leq 1$. Let $I^{n}$ denote the $n$-dimensional unit cube.

Not all systems of finite-dimensional distributions can be realized by stochastic processes for which $\Omega$ is the unit interval. Show that there is on the unit interval with Lebesgue measure no process $\left[X_{t}: t \geq 0\right]$ for which the $X_{t}$ are independent and assume the values 0 and 1 with probability $\frac{1}{2}$ each. Compare Problem 1.1.

Here is an application of the existence theorem in which $T$ is not a subset of the line. Let $(N, \mathscr {N}, \nu )$ be a measure space, and take $T$ to consist of the $\mathscr {N}$-sets of finite $\nu$-measure. The problem is to construct a generalized Poisson process, a stochastic process $\left[X_{A}: A \in T\right]$ such that (i) $X_{A}$ has the Poisson distribution with mean $\nu (A)$ and (ii) $X_{A_{1}}, \ldots , X_{A_{n}}$ are independent if $A_{1}, \ldots , A_{n}$ are disjoint. Hint: To define the finite-dimensional distributions, generalize this construction: For $A, B$ in $T$, consider independent random variables $Y_{1}, Y_{2}, Y_{3}$ having Poisson distributions with means $\nu \left(A \cap B^{c}\right), \nu (A \cap B), \nu \left(A^{c} \cap B\right)$, take $\mu_{A, B}$ to be the distribution of $\left(Y_{1}+Y_{2}, Y_{2}+Y_{3}\right)$.

36.2 ↑ Show that $K(s, t)=\min \left\{ s, t\right\}$ is nonnegative-definite; use Problem 36.2 to prove the existence of a process with the finite-dimensional distributions prescribed for Brownian motion.

Let $X(t)$ be independent, standard normal variables, one for each dyadic rational $t$ (Theorem 20.4; the unit interval can be used as the probability space). Let $W(0)=0$ and $W(n)=\sum_{k=1}^{n} X(k)$. Suppose that $W(t)$ is already defined for dyadic rationals of rank $\eta$, and put

Prove by induction that the $W(t)$ for dyadic $t$ have the finite-dimensional distributions prescribed for Brownian motion. Now construct a Brownian motion with continuous paths by the argument leading to Theorem 37.1. This avoids an appeal to Kolmogorov's existence theorem.

↑ For each $n$ define new variables $W_{n}(t)$ by setting $W_{n}\left(k / 2^{n}\right)=W\left(k / 2^{n}\right)$ for dyadics of order $n$ and interpolating linearly in between. Set $\delta_{n}= \sup_{t \leq n}\left|W_{n+1}(t)-W_{n}(t)\right|$, and show that

The construction in the preceding problem makes it clear that the difference here is normal with variance $1 / 2^{n+2}$. Find positive $x_{n}$ such that $\sum x_{n}$ and $\sum \mathbb {P}\left(\delta_{n} \geq x_{n}\right)$ both converge, and conclude that outside a set of probability 0, $W_{n}(t, \omega )$ converges uniformly over bounded intervals. Replace $W(t, \omega )$ by $\lim_{n} W_{n}(t, \omega )$. This gives another construction of a Brownian motion with continuous paths.

36.6 ↑ Let $T=[0, \infty )$, and let $P$ be a probability measure on $(R^{T}, \mathscr {R}^{T})$ having the finite-dimensional distributions prescribed for Brownian motion. Let $C$ consist of the continuous elements of $R^{T}$.

Suppose that $\left[W_{t}: t \geq 0\right]$ is some stochastic process having independent, stationary increments satisfying $\mathbb {E}\left[W_{t}\right]=0$ and $\mathbb {E}\left[W_{t}^{2}\right]=t$. Show that if the finite-dimensional distributions are preserved by the transformation (37.11), then they must be those of Brownian motion.

Show that $\bigcap_{t>0} \sigma \left[W_{s}: s \geq t\right]$ contains only sets of probability 0 and 1. Do the same for $\bigcap_{\epsilon >0} \sigma \left[W_{t}: 0<t<\epsilon \right]$; give examples of sets in this $\sigma$-field.

Show by a direct argument that $W(\cdot , \omega )$ is with probability $\mathbf{1}$ of unbounded variation on $[0,1]$ : Let $Y_{n}=\sum_{i=1}^{2^{n}}\left|W\left(i 2^{-n}\right)-W\left((i-1) 2^{-n}\right)\right|$. Show that $Y_{n}$ has mean $2^{n / 2} \mathbb {E}\left[\left|W_{1}\right|\right]$ and variance at most $\operatorname {Var}\left[\left|W_{1}\right|\right]$. Conclude that $\sum \mathbb {P}\left(Y_{n}<n\right)<\infty$.

Show that the Poisson process as defined by ( 23.5 ) is measurable.

Show that for $T=[0, \infty )$ the coordinate-variable process $\left[Z_{t}: t \in T\right]$ on $\left(R^{T}, \mathscr {R}^{T}\right)$ is not measurable.

Extend Theorem 37.4 to the set $[t: W(t, \omega )=\alpha ]$.

Let $\tau_{\alpha }$ be the first time the Brownian motion hits $\alpha >0: \tau_{\alpha }=\inf \left[t: W_{t} \geq \alpha \right]$. Show that the distribution of $\tau_{\alpha }$ has over $(0, \infty )$ the density

Show that $\mathbb {E}\left[\tau_{\alpha }\right]=\infty$. Show that $\tau_{\alpha }$ has the same distribution as $\alpha^{2} / N^{2}$, where $N$ is a standard normal variable.

$\uparrow$ Suppose that $X_{1}, X_{2}, \ldots$ are independent and each has the distribution (37.46).

$37.11 \uparrow$ Let $\rho (s, t)$ be the probability that a Brownian path has at least one zero in $(s, t)$. From (37.46) and the Markov property deduce

Hint: Condition with respect to $W_{s}$.

$\uparrow$

$\uparrow$ Show that $\rho (s, t)=\rho \left(t^{-1}, s^{-1}\right)=\rho (c s, c t)$. Check this by (37.47) and also by the fact that the transformations (37.11) and (37.12) preserve the properties of Brownian motion.

Deduce by the reflection principle that $(M_{t}, W_{t})$ has density

on the set where $y \geq 0$ and $y \geq x$. Now deduce from Theorem 37.8 the corresponding limit theorem for symmetric random walk.

Show by means of the transformation (37.12) that for positive $a$ and $b$ the probability is 1 that the process is within the boundary $-a t<W_{1}<b t$ for all sufficiently large $t$. Show that $a /(a+b)$ is the probability that it last touches above rather than below.

The martingale calculation used for (37.39) also works for slanting boundaries. For positive $a, b, r$, let $\tau$ be the smallest $t$ such that either $W_{t}=-a+r t$ or $W_{t}=b+r t$, and let $p(a, b, r)$ be the probability that the exit is through the upper barrier-that $W_{\tau }=b+r \tau$.