\int_0^tH(s)dX(s) : =\lim_{n\longrightarrow +\infty}\int_0^tH^{n}(s)dX(s),\end{aligned}$$ where $(H^{n})$ is a sequence of simple process converging to $H$.

Let $f\in\textbf{L}_{ucp}$ and $(\overline{N_t})$ be a compensated poisson process. The following holds

1.  $\mathbb{E}\left(\int_0^tf(s)d\overline{N}(s)\right)=0$.

2.  $\mathbb{E}\left(\int_0^tf(s)d\overline{N}(s)\right)^2=\lambda\int_0^t\mathbb{E}(f(s))^2ds$.

[@Oksa]

### Itô formula for jump process

**\[Itô jump-diffusion process\]**

The Itô jump-diffusion process is any process of the form $$\begin{aligned}
   X_t=X_0+\int_0^ta(X_s)ds+\int_0^tb(X_s)dW_s+\int_0^tc(X_s)dN_s.
   \label{ch1ito1}
   \end{aligned}$$ The coefficient $a\in\mathbb{L}^1([0,T], \mathbb{R}^n)$ is called the drift coefficient, $b\in \mathbb{L}^2([0,T], \mathbb{R}^{n\times m})$ is called the diffusion coefficient and $c(X_s)\in\mathbb{L}^2([0,T], \mathbb{R}^n)$ is called the jump coefficient. $W$ is a $m$-dimentional Brownian motion and $N$ a one dimentional poisson process.

[@Oksa pp 6]**\[Itô formula for jump process\]**

If $(X_t)$ is a jump-diffusion process of the form and $f :[0,\infty)\longrightarrow \mathbb{R}^n$ any function twice derivable, then $f(X_t)$ is a jump-diffusion process and satisfies the following equation $$\begin{aligned}
 f(X_t)=f(X_0)+\int_0^t\dfrac{\partial f}{\partial x}(X_s)a_sds+\dfrac{1}{2}\int_0^t\dfrac{\partial^2f}{\partial x^2}(X_s)b_s^2ds +\int_0^t\dfrac{\partial f}{\partial x}b_sdW_s+\int_0^t(f(X_{s^-}+c( X_s))-f(X_{s^-}))dN_s.
 \end{aligned}$$

\[ch1Itoproduct\] **\[Itô’s lemma for product\]**

If $X_t$ and $Y_t$ are two Itô’s jump-diffusion process, Then $X_tY_t$ is an Itô jump-diffusion process and $$\begin{aligned}
d(X_tY_t)=Y_tdX_t+X_tdY_t+dX_tdY_t.\end{aligned}$$ $dX_1dX_2$ is called the Itô’s corrective term and it is computed according to the relations $$\begin{aligned}
dt.dt=dN_tdt=dtdN_t=dW_tdN_t=dN_tdW_t=0,\hspace{0.5cm}
dN_tdN_t=dN_t.\end{aligned}$$

[@Oksa]

After being familiar with some basic notions in probability theory and stochastic process, we are now ready to 