Skorokhod embedding

Without some supplemental conditions on $T$, the problem of Skorokhod embedding is trivial. Suppose $W$ is a Brownian motion with respect to a filtration $\left\{ \mathcal{F}_{t}\right\}$ satisfying the usual conditions. Suppose $Y$ is a finite random variable and suppose $h$ is a real-valued function such that $h\left(W_{1}\right)$ has the same law as $Y$.

Show that the triple integral on the right-hand side of (15.10) is equal to the expression in $(15.11)$.

A sketch was given for the proof of Corollary 15.5. Provide a detailed proof.

Corollary 15.5: Let $W$ be a Brownian motion and let $\left\{ \mathcal{F}_{t}\right\}$ be the minimal augmented filtration for $W$. Let $Y$ be a random variable with $\mathbb {E}\left[Y\right]=0$ and $\operatorname {Var}\left[Y\right]<\infty$. There exists a stopping time $V$ with respect to $\left\{ \mathcal{F}_{t}\right\}$ such that $W_{V}$ has the same law as $Y$.

Here is another approach to proving Corollary 15.5. Let $Y, N, T$, and $\left\{ \mathcal{G}_{t}\right\}$ be as in the proof of Theorem 15.4 .

Theorem 15.4: Suppose $Y$ is a random variable with $\mathbb {E}\left[Y\right]=0$ and $\mathbb {E}\left[Y^{2}\right]<\infty$. There exists a Brownian motion $N$ and a stopping time $T$ with respect to the minimal augmented filtration $\left\{ \mathcal{G}_{t}\right\}$ of $N$ such that $N_{T}$ is equal in law to $Y$. Moreover $\mathbb {E}\left[T\right]=\mathbb {E}\left[Y^{2}\right]$.

Suppose $p \in (0,1 / 2)$ and $Y$ is a random variable such that $\mathbb {P}\left(Y=1\right)=\mathbb {P}\left(Y=-1\right)=p$ and $\mathbb {P}\left(Y=0\right)=1-2 p$. Let $W$ be a Brownian motion. Let $S_{x}=\inf \left\{ t>0: W_{t}=x\right\}$ and let $T=\inf \left\{ t>S_{x} \wedge S_{-x}: W_{t} \in \left\{ -1,0,1\right\} \right\}$. Determine $x$ such that $W_{T}$ and $Y$ have the same law.

Suppose $Y$ is a mean zero random variable and there exists a real number $K>0$ such that $\left|Y\right| \leq K$, a.s. Let $W$ be a Brownian motion and let $T$ be a stopping time with $\mathbb {E}\left[T\right]<\infty$ such that $W_{T}$ and $Y$ have the same law. (We do not necessarily assume that $T$ was constructed by the method of Section 15.2.) Let $S_{K}=\inf \left\{ t:\left|W_{t}\right| \geq K\right\}$. Prove that $T \leq S_{K}$, a.s.

Let $Y_{i}$ be a sequence of i.i.d. random variables with $\mathbb {P}\left(Y_{i}=1\right)=\mathbb {P}\left(Y_{i}=-1\right)=\frac{1}{2}$, and let $S_{n}=\sum_{i=1}^{n} Y_{i} . S_{n}$ is called a simple symmetric random walk. Let $T_{1}, T_{2}, \ldots$ and $U_{1}, U_{2}, \ldots$ be as in Section 15.3.

Let $S_{n}, T_{i}$, and $U_{i}$ be as in Exercise 15.7. Prove that

Let $S_{n}$ be a simple symmetric random walk; see Exercise 15.7. Let $Y$ be a bounded symmetric random variable that takes values only in $\mathbb {Z}$. ( $Y$ being symmetric means that $Y$ and $-Y$ have the same law.) Does there necessarily exist a stopping time $N$ such that $S_{N}$ and $Y$ have the same law? Why or why not?

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.

Using Theorem 13 of Chapter III, prove the invariance principle (Theorem 8) for the triangular array generated by a sequence of independent identically distributed random variables with zero mean and unit variance.

Theorem 13 of Chapter III (Skorokhod). Let $X_{1}, X_{2}, \ldots$ be a sequence of centered independent random variables defined on some probability space. Then there exists a probability space on which one can find a sequence of random variables $\left\{ T_{k}\right\}_{k \geqslant 1}$ and a Brownian motion $W = \left\{ W(t), t \geqslant 0\right\}$ such that

where the nonnegative variables $T_{k}-T_{k-1}, k \in \mathbb {N}\left(T_{0} \equiv 0\right)$, are independent. If $\mathbb {E}\left[X_{k}^{2}\right] < \infty$, then in addition to the indicated properties of the sequence $\left\{ T_{k}\right\}_{k \geqslant 1}$ one can ensure the relation $\mathbb {E}\left[T_{k}-T_{k-1}\right] = \mathbb {E}\left[X_{k}^{2}\right], k \in \mathbb {N}$.

Let $(\Omega , \mathscr {F}, \mathbb {P})$ be a probability space and let $X_{n, i}, i = 1, \ldots , m_{n}, n \geqslant 1$, be series of (for each $n$) independent real random variables defined on this space such that $\mathbb {E}\left[X_{n, i}\right] = 0$, $i = 1, \ldots , m_{n}$, and $\sum_{i = 1}^{m_{n}} \sigma_{n, i}^{2} = 1$, where $\sigma_{n, i}^{2} = \mathbb {E}\left[X_{n, i}^{2}\right]$. Denote $t_{n, 0} = 0, t_{n, j} = \sum_{i = 1}^{j} \sigma_{n, i}^{2}$ (then $t_{n, m_{n}} = 1$), $S_{n, 0} = 0, S_{n, j} = \sum_{i = 1}^{j} X_{n, i}$, $j = 1, \ldots , m_{n}$. Now, for each $\omega \in \Omega$, define the function $S_{n}(t, \omega ), t \in [0,1]$, as the continuous polygonal line with vertices $\left(t_{n, j}, S_{n, j}\right), j = 0, \ldots , m_{n}$, i.e., let

Here and below we assume that all $\sigma_{n, i} > 0$; in other words, we exclude from consideration the variables $X_{n, i} = 0$ a.s. Let $\mathbb {P}_{n}$ be the probability distribution of $S_{n}$ on $\mathscr {B}(C[0,1])$.

Theorem 8 (Prokhorov). Suppose that for the variables $X_{n, i}$ described above, where $n \geqslant 1$ and $i = 1, \ldots , m_{n}$, the Lindeberg condition holds:

Then $\mathbb {P}_{n} \Rightarrow \mathbb {W}$ ($n \rightarrow \infty$), where $\mathbb {W}$ is the Wiener measure. In other words, $S_{n} \stackrel{\mathscr {D}}{\rightarrow } W$ as $n \rightarrow \infty$, where $W = \left\{ W(t), t \in [0,1]\right\}$ is the Wiener process on the interval $[0,1]$.

In the proof of Theorem 14.2 we assume that $\left(B_{t}\right)_{t \geqslant 0}$ and $(U, W)$ are independent. Show that $\mathscr {F}_{t}: = \sigma \left(B_{s}, s \leqslant t ; U, W\right)$ is an admissible filtration for a Brownian motion, cf. Definition 5.1.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$ and $\tau$ some a.s. finite stopping time such that $\left|B_{t \wedge \tau }\right|, t \geqslant 0$, is bounded. Show that $\mathbb {E}\left[\tau^{2}\right] \leqslant 4 \mathbb {E}\left[B_{\tau }^{4}\right]$. Conclude from this that we have in Theorem 14.2, and Corollary 14.3, "for free" the bounds $\mathbb {E}\left[\tau^{2}\right] \leqslant 4 \mathbb {E}\left[X^{4}\right]$, and $\mathbb {E}\left[\left(\tau_{n} - \tau_{n - 1}\right)^{2}\right] \leqslant 4 \mathbb {E}\left[X_{1}^{4}\right]$, respectively. (These latter bounds are understood in $[0, \infty ]$, though).