stochastic process $X=\{X_t\}_{t\in[0,T]}$ is called strong solution of Itô jump-diffusion differential equation if :

1.  $X_t$ is $\mathcal{F}_t-$measurable $\forall\; t\in[0,T]$.

2.  $\mathbb{P}\left[\int_0^t|f(X_s,s)|ds<\infty\right]=\mathbb{P}\left[\int_0^t|g(X_s,s)|^2ds<\infty\right]
    =\mathbb{P}\left[\int_0^t|h(X_s,s)|^2ds<\infty\right]=1$.

3.  $X$ satisfies equation almost surely.

\[ch2assump\] Troughout this chapter, we make the following hypothesis :

There exist positive constants $L$ and $K$ such that for all $x, y\in\mathbb{R}^n$,

1.  $\mathbb{E}||X(0)||^2<+\infty$ and $X(0)$ is independent of the Weiner process $W(t)$ and of the poisson process $N(t)$.

2.  $f$, $g$ and $h$ satisfy the global Lipschitz condtion : $$\begin{aligned}
    ||f(x)-f(y)||^2\vee ||g(x)-g(y)||^2\vee ||h(x)-h(y)||^2\leq L||x-y||^2\end{aligned}$$

3.  $f$, $g$ and $h$ satisfy the linear growth condition : $$\begin{aligned}
    ||f(x)||^2\vee ||g(x)||^2\vee ||h(x)||^2\leq K(1+||x||^2).\end{aligned}$$

The globaly Lipschitz condition implies the linear growth condition. So it is enough to make the assumptions only on the global Lipschitz condition.

A soluion $\{X_t\}$ of is said to be pathwise unique if any other solution $\overline{X}$ is a stochastilly indistinguishable from it, that is $\mathbb{P}\{X(t)=\overline{X}(t)\}=1$, $\forall\; t\in [0,T]$ almost surely.

In order to prove the existence and uniqueness solution of , it is useful to write in its compensated form.

### Compensated stochastic differential equation

(CSDE)

From the relation $\overline{N}(t)= N(t)-\lambda t$, we have $d\overline{N}(t)=dN_t-\lambda dt$. Substituting this latter relation in leads to : $$\begin{aligned}
dX(t)=f_{\lambda}(X(t^-))dt+g(X(t^-))dW(t)+h(X(t-))d\overline{N}(t),
\label{ch2jdi2}\end{aligned}$$ 