Math Problems Vault

Problem 12.7.1

Let $X$ be a continuous-time Markov chain with finite state space $S$ and generator $\mathbf{G}$ . Let $\mathbf{\eta } = \left\{ \eta (i): i \in S\right\}$ be a root of the equation $\mathbf{G} \mathbf{\eta }^{\prime } = \mathbf{0}$ . Show that $\eta (X(t))$ constitutes a martingale with respect to $\mathcal{F}_{t} = \sigma (\left\{ X(u): u \leq t\right\} )$ .

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Problem 12.7.2

Let $N$ be a Poisson process with intensity $\lambda$ and $N(0) = 0$ , and let $T_{a} = \min \left\{ t: N(t) = a\right\}$ , where $a$ is a positive integer. Assuming that $\mathbb {E}\left[\exp \left(\psi T_{a}\right)\right] < \infty$ for sufficiently small positive $\psi$ , use the optional stopping theorem to show that $\operatorname {Var}\left[T_{a}\right] = a \lambda^{-2}$ . Show further that $T_{a}$ has characteristic function $\phi_{a}(s) = [\lambda /(\lambda -i s)]^{a}$ .

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Problem 12.7.3

Let $S_{m} = \sum_{r = 1}^{m} X_{r}, m \leq n$ , where the $X_{r}$ are independent and identically distributed with finite mean. Denote by $U_{1}, U_{2}, \ldots , U_{n}$ the order statistics of $n$ independent variables which are uniformly distributed on $(0, t)$ , and set $U_{n+1} = t$ . Show that $R_{m} = S_{m} / U_{m+1}, 0 \leq m \leq n$ , is a backward martingale with respect to a suitable sequence of $\sigma$ -fields, and deduce that

\mathbb {P}\left(R_{m} \geq 1 \text{ for some } m \leq n \mid S_{n} = y\right) \leq \min \left\{ y / t, 1\right\}

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Problem 12.7.4

(a)(b)

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(a)

Let $W$ be a standard Wiener process. Show that the following are martingales:

(i) $W(t)$ ,

(ii) $W(t)^{2}-t$ ,

(iii) $W(t)^{3}-3 t W(t)$ ,

(iv) $W(t)^{4}-6 t W(t)^{2}+3 t^{2}$ .

(b)

Let $T$ be the earliest time at which $W$ exits the interval $(-a, a)$ . Show that $\mathbb {E}\left[T^{n}\right] < \infty$ for $n \geq 1$ . Use the above martingales to show that $\mathbb {E}\left[T^{2}\right] = \frac{5}{3} a^{4}$ . [You may use appropriate optional stopping theorems and the dominated convergence theorem.]

Problem 12.7.5

Let $X$ and $Y$ be independent standard Wiener processes. Show that

M(t) = Y(t) X(t)^{2}-\int _{0}^{t} Y(u) d u

defines a martingale with respect to the natural filtration $\mathcal{F}_{t} = \sigma \left(\left\{ X_{v}, Y_{v}: v \leq t\right\} \right)$ .

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Problem 12.7.6

Brownian motion in a disc. Let $Z(t) = z+X(t)+i Y(t)$ , where $X$ and $Y$ are independent standard Wiener processes and $\left|z\right| < 1$ . Show that the first passage time $T$ to the unit circle has expected value $\mathbb {E}\left[T\right] = \frac{1}{2}\left(1-|z|^{2}\right)$ .

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Problem 12.7.7

(a)(b)

Let $X$ and $Y$ be independent Poisson processes with rate $\lambda$ , and let $Q(t) = X(t)-Y(t)$ . For positive integers $m, n$ , find:

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(a)

the probability that $Q$ hits $n$ before $-m$ ,

(b)

the expected time for $Q$ to hit either $-m$ or $n$ .

Problem 12.7.8

(a)(b)(c)

Let $N$ be a Poisson process with rate $\lambda$ . Show the following.

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(a)

The predictable quadratic variation of $N$ over the interval $[0, t]$ equals $\lambda t$ , which is to say that

\sum _{r = 0}^{n-1} \mathbb {E}\left[[N((r+1) t / n)-N(r t / n)]^{2} \mid \mathcal{F}_{r t / n}\right] \rightarrow \lambda t \quad \text{ as } n \rightarrow \infty

where $\mathcal{F}_{u} = \sigma (\left\{ N(s): s \leq u\right\} )$ .

(b)

The optional quadratic variation of $N(s)-\lambda s$ over [0,t] equals $N(t)$ . (See Exercise (12.2.4).)

(c)

$M(t) = (N(t)-\lambda t)^{2}-N(t)$ defines a martingale.