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Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve

Chapter 11

Introduction to Jump Processes - all with Video Answers

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Chapter Questions

Problem 1

. Let $M(t)$ be the compensated Poisson process of Theorem 11.2.4.
(i) Show that $M^2(t)$ is a submartingale.
(ii) Show that $M^2(t)-\lambda t$ is a martingale.

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Problem 2

Suppose we have observed a Poisson process up to time $s$, have seen that $N(s)=k$, and are interested in the value of $N(s+t)$ for small positive $t$. Show that
$$
\begin{aligned}
& \mathbb{P}\{N(s+t)=k \mid N(s)=k\}=1-\lambda t+O\left(t^2\right), \\
& \mathbb{P}\{N(s+t)=k+1 \mid N(s)=k\}=\lambda t+O\left(t^2\right), \\
& \mathbb{P}\{N(s+t) \geq k+2 \mid N(s)=k\}=O\left(t^2\right),
\end{aligned}
$$
where $O\left(t^2\right)$ is used to denote terms involving $t^2$ and higher powers of $t$.
526
11 Introduction to Jump Processes

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Problem 3

(Geometric Poisson process). Let $N(t)$ be a Poisson process with intensity $\lambda>0$, and let $S(0)>0$ and $\sigma>-1$ be given. Using Theorem 11.2.3 rather than the Itô-Doeblin formula for jump processes, show that
$$
S(t)=\exp \{N(t) \log (\sigma+1)-\lambda \sigma t\}=(\sigma+1)^{N(t)} e^{-\lambda \sigma t}
$$
is a martingale.

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Problem 4

Suppose $N_1(t)$ and $N_2(t)$ are Poisson processes with intensities $\lambda_1$ and $\lambda_2$, respectively, both defined on the same probability space $(\Omega, \mathcal{F}, P)$ and relative to the same filtration $\mathcal{F}(t), t \geq 0$. Show that almost surely $N_1(t)$ and $N_2(t)$ can have no simultaneous jump. (Hint: Define the compensated Poisson processes $M_1(t)=N_1(t)-\lambda_1 t$ and $M_2(t)=N_2(t)-\lambda_2 t$, which like $N_1$ and $N_2$ are independent. Use Itô's product rule for jump processes to compute $M_1(t) M_2(t)$ and take expectations.)

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Problem 5

. Suppose $N_1(t)$ and $N_2(t)$ are Poisson processes defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ relative to the same filtration $\mathcal{F}(t), t \geq 0$. Assume that almost surely $N_1(t)$ and $N_2(t)$ have no simultaneous jump. Show that, for each fixed $t$, the random variables $N_1(t)$ and $N_2(t)$ are independent. (Hint: Adapt the proof of Corollary 11.5.3.) (In fact, the whole path of $N_1$ is independent of the whole path of $N_2$, although you are not being asked to prove this stronger statement.)

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Problem 6

Let $W(t)$ be a Brownian motion and let $Q(t)$ be a compound Poisson process, both defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and relative to the same filtration $\mathcal{F}(t), t \geq 0$. Show that, for each $t$, the random variables $W(t)$ and $Q(t)$ are independent. (In fact, the whole path of $W$ is independent of the whole path of $Q$, although you are not being asked to prove this stronger statement.)

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Problem 7

Use Theorem 11.3.2 to prove that a compound Poisson process is Markov. In other words, show that, whenever we are given two times $0 \leq t \leq T$ and a function $h(x)$, there is another function $g(t, x)$ such that

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