Question

. 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.

    . 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.

Stochastic Calculus for Finance II : Continuous-Time Models

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 11, Problem 1

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Step 1

A compensated Poisson process $M(t)$ can be defined as $M(t) = N(t) - \lambda t$, where $N(t)$ is a standard Poisson process with rate $\lambda > 0$. The term $\lambda t$ is the compensator, making $M(t)$ a martingale. Show more…

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. 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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