where $$\begin{aligned}
f_{\lambda}(x) :=f(x)+\lambda h(x).
\label{ch2jdi4}\end{aligned}$$ can be rewriten into its integral form $$\begin{aligned}
X(t)=X_0+\int_0^tf_{\lambda}(X(s^-))ds+\int_0^tg(X(s^-))dW(s)+\int_0^th(X(s^-))d\overline{N}(s),
\label{ch2jdi3}\end{aligned}$$

\[ch2lemma1\] If assumptions \[ch2assump\] are satisfied, then the function $f_{\lambda}$ satisfies the global Lipschitz and the linear growth conditions with constants $L_{\lambda}=(1+\lambda)^2L$ and $K_{\lambda}=(1+\lambda)^2K$ respectively.

1.  Using the global lipschitz condition satisfied by $f$ and $h$, it follows from that: $$\begin{aligned}
    ||f_{\lambda}(x)-f_{\lambda}(y)||^2&=&||f(x)-f(y)+\lambda (h(x)-h(y))||^2\\
    &\leq& \left(\sqrt{L}||x-y||+\lambda\sqrt{L}||x-y||\right)^2=(1+\lambda)^2L||x-y||^2.\end{aligned}$$

2.  Along the same lines as above, we obtain the linear growth condition satisfies by $f_{\lambda}$.

Well-posedness problem
----------------------

Based on Lemma \[ch2lemma1\] and using the fact that equations and are equivalent, the existence and uniqueness of solution of equation is equivalent to the existence and uniqueness of solution of equation .

\[ch2th1\] If Assumptions \[ch2assump\] are fulfilled, then there exist a unique strong solution of equation .

In order to prove Theorem \[ch2th1\], we need the following lemma.

\[ch2lemma2\] let $X^0(t)=X_0(t), \hspace{0.5cm} \forall t\in [0,T]$ and $$\begin{aligned}
X^{n+1}(t)=X_0+\int_0^tf(X^n(s^-))ds+\int_0^tg(X^n(s^-))dW(s)+\int_0^th(X^n(s^-))d\overline{N}(s)ds,\end{aligned}$$ then $$\begin{aligned}
\mathbb{E}||X^{n+1}(t)-X^n(t)||^2 \leq \dfrac{(Mt)^{n+1}}{(n+1)!},
\label{lemI1}\end{aligned}$$

where $M$ is a positive constant depending on $ \lambda, K, L, X_0$.

: By induction

1.  For $n=0$ $$\begin{aligned}
    ||X^1(t)-X^0(t)||^2&=&\left\|\int_0^tf_{\lambda}(X_0(s^-))ds+\int_0^tg(X_0(s^-))dW(s)+\int_0^th(X_0(s^-))d\overline{N}(s)\right\|^2\nonumber\\
    &\leq& 3\left\|\int_0^tf_{\lambda}(X_0(s^-))ds\right\|^2+3\left\|\int_0^tg(X_0(s^-))dW(s)\right\|^2\nonumber\\
    &+&3\left\|\int_0^th(X_0(s^-))d\overline{N}(s)\right\|^2.
    \label{E1}\end{aligned}$$ Using Cauchy-Schwartz inequality and the linear growth condition, it follows that: $$\begin{aligned}
    \mathbb{E}(I_1(t)):=\mathbb{E}\left(3\left\|\int_0^tf_{\lambda}(X_0(s^-))ds\right\|^2\right)&\leq & 3T\int_0^t\mathbb{E}||f_{\lambda}(X_0(s^-))||^2ds\nonumber\\
    &\leq & 3TK_{\lambda}\int_0^t(1+E||X_0||^2)ds.
    \label{EI1}\end{aligned}$$ 