provide in the following chapter the proof of the existence and uniqueness of solution of SDEs with jumps under global Lipschitz conditions.

Existence and uniqueness of solution of the jump-diffusion Itô’s stochastic differential equations
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In this chapter, we give the general formulation of the compensated stochastic differential equation (CSDE) with jumps which will be helpful to prove the existence and the uniqueness solutions of the stochastic differential equation with jump.

General formulation
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Along this work, $||.||$ denote the Frobenius matrix norm, $(\Omega, \mathcal{F}_t, \mathbb{P})$ denote a complete probability space. For all $x, y\in\mathbb{R}^n$, $\langle x,y\rangle=x_1y_1+\cdots, x_ny_n$ is the inner product. For all $a, b\in\mathbb{R}$,

$ a\vee b :=\max(a,b)$.

Throughout this work, we consider a jump diffusion Itô’s stochastic differential (SDEs) of the form $$\begin{aligned}
dX(t)=f(X(t^-))dt+g(X(t^-))dW(t)+h(X(t^-))dN(t),\hspace{0.5cm} X(0^-)=X_0,
\label{ch2jdi1}\end{aligned}$$ where $X_0$ is the initial condition, $X(t^-)=\lim\limits_{s\longrightarrow t^-}X(s)$, $W(t)$ is an $m$-dimensional Brownian motion and $N(t)$ is a $1$-dimensional poisson process with intensity $\lambda>0$. We assume $W_t$ and $N_t$ to be both $\mathcal{F}_t$-measurable. $f : \mathbb{R}^n\longrightarrow \mathbb{R}^n$, $g : \mathbb{R}^n \longrightarrow\mathbb{R}^{n\times m}$ and $h : \mathbb{R}^n\longrightarrow \mathbb{R}^n$.

Our aim in this chapter is to prove the existence and the uniqueness of solution of equation in the strong sense.

**\[Strong solution\]**

A 