Quadratic covariation of two Compound Poisson Process.

Question

This is not a homework problem but a curiosity question and hence the question may not be well built but I want the community to help me in understanding this concept and hence answer it. If you think the question can be framed better please do tell me. I would really really appreciate any help in the form of comments/answers.

I want to find out the Quadratic covariation of two semimartingales $Z_1,Z_2$ where both are compound poison processes and they are correlated by say a factor of $\rho_1$.

If $Z_1(t)=\sum^{N_1(t)}Y_i$ and $Z_2(t)=\sum^{N_2(t)}U_j$ are two compound poisson process then what will be the quadratic covariation of $Z_1,Z_2$ , i.e $[Z_1,Z_2](t)$ ?

In this $N_1$ and $Y_i$ are independent and $N_2$ and $U_j$ are independent and with $Y_i$ is an i.i.d and also $U_j$ is an i.i.d. We can assume that $E(Y_i)=\mu_1$ and $E(U_j)=\mu_2$. We can also assume that the jumps $N_1,N_2$ are correlated by $\rho_2$ and $Y_i,U_j$ are corrleated by a factor of $\rho_3$ (if needed).

One of the ways in which I thought was to calculate $[Z_1,Z_2](t)=Z_1(t)Z_2(t)−Z_1(0)Z_2(0)−\int^t_0Z_1(s)dZ_2(s)−\int^t_0 Z_2(s)dZ_1(s)$ where we will have $dZ_1(s)=Y_iN_1(s)$ and $dZ_2(s)=U_jN_2(s)$.

I am open to other methods to calculate the same as I was unable to proceed much in above. Also if you can find a reference please mention the link below.

Answer 1

So.

We have that $ Z_1(t) = \sum^{N_1(t)}Y_i $ and $ Z_2(t) = \sum^{N_2(t)}U_j $ are both semimartingale and compound poisson processes with correlation coefficient $ \rho_1 $ and i.i.d. jump sizes (from $ Y_i $ & $ U_j $ being i.i.d.) where $ \left( N_1 \perp \!\!\! \perp Y_i \right) \land \left( N_2 \perp \!\!\! \perp U_j \right) $. As $ Z_1 $ and $ Z_2 $ are both semimartingale, we know that quadratic covariance exists between them.

It must be assumed that $ E(Y_i) = \mu_1 $ and $ E(U_j) = \mu_2 $ are finite for the two processes to be correlated; that is, the random variables $ Y_i $ and $ U_j $ must be normally distributed.

This leads me to the more reckless assumption that we are working with Gaussian distributions here... but luckily, we can see that the form of the Gaussian distribution, $ f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $, is amenable to a (continuous and thus unique) semimartingale decomposition in that we can use $Y_i=M_1(i) + X_1(i)$ and $U_j=M_2(j) + X_2(j)$ to write $ y(i) = \frac{1}{\sigma_1\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{Y_i-\mu_1}{\sigma_1}\right)^2} = \frac{1}{\sigma_1\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{M_1(i)+X_1(i)-\mu_1}{\sigma_1}\right)^2} $ and $ u(j) = \frac{1}{\sigma_2\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{U_j-\mu_2}{\sigma_2}\right)^2} = \frac{1}{\sigma_2\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{M_2(j)+X_2(j)-\mu_2}{\sigma_2}\right)^2} $. Working with this could be an option.

I also note that the Itô integral $ [Z,Z]_t = \int_0^t{\sigma^2(s)ds} $ may be used to find the quadratic covariance between two Itô processes (because all semimartingale processes are Itô processes) via the polarization formula $ \left[Z_1,Z_2\right]_t = \frac{1}{2}\left[Z_1+Z_2,Z_1+Z_2\right]_t - \left[Z_1,Z_1\right]_t - \left[Z_2,Z_2\right] $. Use of the Itô integral obtains $ \left[Z_1,Z_1\right]_t = \int_0^t{\sigma_1^2(s)ds} $ and $ \left[Z_2,Z_2\right]_t = \int_0^t{\sigma_2^2(s)ds} $. Since sums of Itô processes are Itô processes, we also get $ \left[Z_1+Z_2,Z_1+Z_2\right]_t = \int_0^t{\left(\sigma_1+\sigma_2\right)^2(s)ds} $ so that we have

$$ \left[Z_1,Z_2\right]_t = \frac{1}{2}\int_0^t\left(\sigma_1+\sigma_2\right)^2(s)ds - \int_0^t\sigma_1^2(s)ds - \int_0^t\sigma_2^2(s)ds $$ which is more or less similar but different to what you're working with?

This may contain errors, but the basic ideas are:

1. Consider working with Gaussian formulae.
2. Treat the processes as Itô processes.

Answer 2

The following is a simple model for correlated compound processes. Let $ N=\{N_t,t\ge 0\} $ be a Poisson process with $ \mathsf{E}[N_t]=\lambda t $ and $ Y=\{ ( Y_k^{(1)} ,Y^{(2)}_k), k\ge 1 \} $ be a sequence of i.i.d. 2-d random variables, with $$ \mathsf{E}[Y^{(i)}_k Y^{(j)}_k]=m_{ij},\qquad i,j=1,2,\quad k\ge 1 $$ Also, $ Y $ and $ N $ are mutual independent. Now define $X=\{X_t=(X^{(1)}_t,X^{(2)}_t), t\in\mathbb{R}_+\}$ as following: $$ X^{(i)}_t=\sum_{k=1}^{N_t}Y_k^{(i)}, \qquad i=1,2. $$ Then $X^{(1)}=\{ X^{(1)}_t,t\in\mathbb{R}_+\}$, $X^{(2)}=\{ X^{(2)}_t,t\in\mathbb{R}_+\} $ are correlated compound Poisson Processes. Using the general results of compound Poisson Processes we can get the quadratic covariation $[X^{(i)},X^{(j)}] $ as following: $$ [X^{(i)},X^{(j)}]_t=\sum_{s\le t} \Delta X^{(i)}_s \Delta X^{(j)}_s = \sum_{k=1}^{N_t}Y_k^{(i)} Y_k^{(j)},\qquad i,j=1,2 $$ And the predictable quadratic covariations are $$ \langle X^{(i)},X^{(j)}\rangle_t=m_{ij}\lambda t,\qquad i,j=1,2 $$ About the general results of stochastic calculus and compound Poisson processes please refer to the following books:

J. Jacod and A. N. Shiryayev, Limit Theory for Stochastic Processes, 2ed. Springer, 2003.

A. E. Kyprianou, Fluctuations of Lévy Processes with Applications, 2nd Ed., Springer, 2014.

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd Ed, Cambridge University Press, 2009.