SDE driven by Poisson Process

Question

Suppose that $(N_t)_{t\in \mathbb{R}^+}$ is a Poisson process with intensity $\lambda$>0 and that $a\in\mathbb{R}$ and $X$ being a stochastic process which solves the following SDE:$$dX_t=aX_t^-dN_t$$ Now I want to find an explicit representation for $X$ in terms of $X_0,a$ and $N_t$.Now I would like to try the change of variables $Z_t=log(aX_t)$ and use Ito's Lemma, however the material I have been given is very sparse and as far as I can see only applicable to SDEs driven by Brownian motions. So my question is if anyone could give a hint on how to solve this equation or any relevant material concerning similar SDEs.

Answer 1

We know $$\tag{1} \int_0^t aX_{s-}\,dN_s=\sum\limits_{0<s\leq t}aX_{s-}\Delta N_s\,. $$ The SDE $$\tag{2} dX_t=aX_{t-}\,dN_t $$ is nothing else than the integral equation $$\tag{3} X_t=1+\int_0^t aX_{s-}\,dN_s\,. $$ and the proposed solution of this SDE is $$\tag{4} X_t=X_0(1+a)^{N_t}\,. $$ Proof. The $X_t$ in (4) changes only by jumps of $N_t$ and only by the amount $$\tag{5} \Delta X_t=X_t-X_{t-}=X_0(1+a)^{N_{t-}+1}-X_{t-}=(1+a)X_{t-}-X_{t-}=aX_{t-}\,. $$ if and only if there is a jump of $N_t$ in $t\,.$ This can be written as $$ \Delta X_t=aX_{t-}\Delta N_t $$ which is the discrete version of the SDE (2). Due to the properties of $X_t$ (changes only by jumps) the integral equation (3) follows now from (1). $$\tag*{$\Box$} \quad $$