Higher Moments of a Time-Varying Mean-Reverting Jump Diffusion

# How can one calculate third and fourth moments of a jump-diffusion process with time-varying parameters?


# How can one calculate third and fourth moments of a jump-diffusion process with time-varying parameters?












Suppose that $x_t$ is a random process that satisfies the mean-reversion jump-diffusion process governed by the stochastic differential equation

$$dx_t=\alpha(t)(\beta(t)-x_t)\,dt+\sigma(t)\,dW_t+J_t\,d\pi_t\tag1$$

where $\alpha$, $\beta$, and $\sigma$ are deterministic smooth functions of $t$, $J_t$ is a random process, $W_t$ is a Wiener process, and $\pi_t$ is a Poisson process with time-varying deterministic arrival frequency $h(t)$. It is assumed herein that $W_t$, $J_t$, and $\pi_t$ are independent random processes.

Solution to $(1)$ can be written

$$x_T=A(t,T)+\int_t^T e^{-\int_{t'}^T \alpha(t'')\,dt''}[\sigma(t')dW_{t'}+J_{t'}d\pi_{t'}] \tag2$$

where $A(t,T)=\beta(T)+e^{-\int_t^T\alpha(t')\,dt'}(x_t-\beta(t))-\int_t^T e^{\int_{t'}^T\alpha(t'')\,dt''}\beta'(t')\,dt'$ is purely determinisitic.

Now, suppose we wish to calculate the expected value, $\mathbb{E_t}\{x_T\}$, of $x_T$ as given by $(2)$. Inasmuch as $\mathbb{E_t}\{dW_t\}=0$ and $J_t$ and $\pi_t$ are independent, we find that

$$\begin{align} \mathbb{E_t}\{x_T\}&=A(t,T)+\int_t^T e^{-\int_{t'}^T \alpha(t'')\,dt''}\mathbb{E_t}\{\sigma(t')dW_{t'}+J_{t'}d\pi_{t'}\}\\\\ &=A(t,T)+\int_t^T e^{-\int_{t'}^T \alpha(t'')\,dt''} h(t')\mathbb{E_t}\{J_{t'}\}\,dt'\tag3 \end{align}$$

Similarly, we can evaluate the second moment, $\mathbb{E_t}\{x^2_T\}$, of $x_T$. Proceeding and exploiting the independence of the random processes along with knowledge of the mean and variance of both $dW_t$ and $d\pi_t$, we find that

$$\mathbb{E_t}\{x^2_T\}=\left(\mathbb{E_t}\{x_T\}\right)^2+\int_t^T e^{-2\int_{t'}^T \alpha(t'')}\left[\sigma^2(t')+h(t')\mathbb{E_t}\{J^2_{t'}\}\right]\,dt'$$

To evaluate higher order moments, one would need to evaluate the expectation of the stochastic multiple integrals given by

$$ \prod_{i=1}^N \int_t^T e^{-\int_{t_i}^T \alpha(\tau)\,d\tau}\left[\sigma(t_i)dW_{t_i}+J_{t_i}\,d\pi_{t_i}\right]\,dt_i\tag4$$

for $N>2$.

How can one evaluate $(4)$ for $N=3$ and $N=4$, respectively, by direct computation of the expectations?

I realize that one may use the characteristic function of the MRJD process to compute moments. Instead, I am asking how to take expectations of powers of $x_T$ directly using the stochastic multiple integrals.