Suppose N1(t) and N2(t) are Poisson processes with intensities λ1 and λ2, respectively, both def

Step 1: Define the compensated Poisson processes:u000ALet u005C( M_1(t) u003D N_1(t) u002D u005Clambda_1 t u005C) and u005C(

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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.)

     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.)

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 11, Problem 4

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Step 1

These processes $ M_1(t) $ and $ M_2(t) $ are known as the compensated Poisson processes or martingales associated with $ N_1(t) $ and $ N_2(t) $, respectively. Show more…

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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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