Stochastic integral of Poisson process

Question

I want a way to calculate or approximate the stochastic integral $$ \int_{0}^{t} e^{\lambda (t - s)}dW_s $$ where $\lambda > 0$ is a real number, and $W_s$ is a Poisson process with intensity $\mu e^{\lambda s}$, where $\mu > 0$ is a real number. I was unable to find any information on how to calculate such integrals, which makes me think that it is not possible. Any help is appreciated, thanks.

Answer 1

Since the stochastic integral is, well, stochastic, by "calculating such integrals", I'm supposing you mean finding the transition probability density $p(x,t|x_0,0)$ of the process which is the solution to the SDE

$$X_t=X_0+\int_0^te^{\lambda(t-s)}dW_s.$$

I think that this problem is analytically intractable (it is likely that there is no closed-form expression for the transition density). However, we can calculate the moment-generating function $Z(a,t,x_0)\doteq\mathbb{E}[\exp(a X_t)|X_0=x_0]$, which is almost as useful as having the transition density. In particular, we can calculate all moments $\mathbb{E}[X_t^n]$ from it by taking derivatives with respect to $a$ and then integrating the result over $x_0$ using any initial distribution that we wish:

$$\mathbb{E}[X_t^n]=\int_{\mathbb{R}}\partial_a^nZ(0,t,x_0)p(dx_0).$$

Using the SDE for $X_t,$ namely

$$dX_t=-\lambda X_tdt+dW_t,$$

one easily finds an SDE for $Y_t=\exp(a X_t)$:

$$dY_t=-\lambda a X_tY_tdt+\big(e^a-1\big)Y_{t^-}dW_t.$$

Then, using $\mathbb{E}[dW_t]=\mu e^{\lambda t}dt$, we obtain the following PDE for $Z$:

$$\partial_t Z(a,t,x_0)=-\lambda a\partial_aZ(a,t,x_0)+\mu e^{\lambda t}\big(e^a-1\big)Z(a,t,x_0).$$

Integrating this PDE with initial condition $Z(a,0,x_0)=e^{ax_0}$ yields the somewhat intruiging and complicated expression

$$Z(a,t,x_0)=\begin{cases}\exp\Big[\frac{\mu}{\lambda}+a x_0e^{-\lambda t}-\frac{\mu e^{\lambda t}}{a\lambda}\big(a -e^{a}+e^{ae^{-\lambda t}}\big)\Big], & a\neq 0 \\ 1, & a=0\end{cases}.$$

In principle, the transition density could be recovered from the inverse Fourier transform of $Z(ik,t,x_0)$, which is an analytic continuation of $Z(\cdot,t,x_0)$, but I've had no luck so far calculating this inverse Fourier transform analytically.