用伊藤公式计算泊松过程积分

fungarwai

Hui-Hsiung Kuo
Introduction to Stochastic Integration
Chapter 7 Exercise 7
用伊藤公式计算 \(\int_0^t \widetilde{N}(s-)^3 d \widetilde{N}(s)\)
\(F(M_t)=F(M_a)+\int_a^t F'(M_{s-})dM_s+\frac{1}{2}\int_a^t F''(M_s)d[M]_s^c+\sum_{a<s\le t}(F(M_s)-F(M_{s-})-F'(M_{s-})\Delta M_s)\)

fungarwai

己解決（好似係）

\(\int_0^t {\widetilde N}(s-)^{m-1} d {\widetilde N}(s)=\frac{1}{m}{\widetilde N}(t)^m - \frac{1}{m}\sum_{k = 2}^m C_m^k\int_0^t {\widetilde N}(s-)^{m-k} d {\widetilde N}(t) - \frac{\lambda }{m}\sum_{k = 2}^m C_m^k\int_0^t {\widetilde N}(s)^{m - k}ds\)

\(\int_0^t {\widetilde N}(s-)^3 d {\widetilde N}(s)=\frac{1}{4}{\widetilde N}(t)^4 - \frac{1}{2}{\widetilde N}(t)^3 + \frac{1}{4}{\widetilde N}(t)^2 - \frac{3}{2}\lambda \int_0^t {\widetilde N}(t)^2 ds  + \frac{1}{2}\lambda \int_0^t {\widetilde N}(s)ds\)