I'm solving an exercise from a last year exam. Suppose we have an Poisson process $(N_t)$ with parameter $\lambda=\frac{1}{3}$ given with respect to a filtration $(\mathcal{F}_t)$.
The first question is:
1) for $t\ge 0$ show $N_t^3=\int_0^t (3N^2_{s-}+3N_{s-}+1)dN_s$ with the hint $(a+b)^3=a^3+3a^2b+3ab^2+b^3$.
and the second question
2) define the process $M_t:=N_t^3-\frac{t}{3}-\int_0^t(N^2_{s-}+N_{s-})ds$ and show that it is a local martingale. There is also hint, which says: we should use $1)$ and that the compensated Poisson process $X_t:=N_t-\lambda t$ is a martingale.
For the first one I tried quite everything. I used $Itô$ to solve it, but I never used the hint. So maybe my solution is wrong. How could you solve this with the hint? Here is what I've done
let $f(x)=x^3$, hence applying Itô yield,
$$N^3_t=\int_0^t3N^2_{s-}dN_s+\frac{1}{2}\int_0^t 6 N_{s-}d[N]_s+\sum_{0<s\le t}(f(N_s)-f(N_{s-}))$$ using $[N]=N$, we get $$N^3_t=\int_0^t3N^2_{s-}+3N_{s-}dN_s+\sum_{0<s\le t}(f(N_s)-f(N_{s-}))$$ hence the question reduces to show that $\sum_{0<s\le t}(f(N_s)-f(N_{s-}))=N_t$. But at this point I struggle. Why is this true? Clearly $f(N_s)-f(N_{s-})$ is either $0$, or $(n+1)^3-n^3=3(n^2+n+1)$, why should this, summing up, be $N_t$?
For the second question, I simply put the equation from $1)$ for $N^3_t$ in, but this does not help a lot. I also recognized that $\frac{t}{3}=\lambda t$, so I guess there is the point, where we have to use the result for the compensated poisson process.
However some help would be appreciated. Thanks in advance.
hulik
2.) For the second process you have $$ M_t = N_t^3 -\lambda t - \lambda \int\limits_0^t (3N^2_{s-}+3N_{s-})\mathrm ds = N_t^3 - \int\limits_0^t (3 N^2_{s-}+3N_{s-} +1)\mathrm d(\lambda \cdot ds) $$ $$ = \int\limits_0^t (3 N^2_{s-}+3N_{s-} +1)\mathrm d(N_s - \lambda \cdot ds) $$ which is an integral w.r.t. martingale.
1.) I think, it was a right way just to apply Ito to find $\mathrm dN^3_t$. I am not sure, whether you have the right Ito formula for the Poisson process. Forget about the stochastic framework and think about $N_t$ path-wise, as fortunately one can do this for Poisson processes. Then $$ N^3_t = N_0^3+\sum_{s\leq t}(N^3_s - N^3_{s-}) $$ where you can apply the hint to find that the latter sum is $$ \sum_{t\geq s:\;\mathrm dN_s = 1}(3N_{s-}^2+3N_{s-}+1) = \int_0^t (3N_{s-}^2+3N_{s-}+1)\mathrm dN_s $$ where you only used the fact that $N_t$ is a pure jump process, rather than the stochastic nature of it.