I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. To begin with, $(\zeta_{t})$ is assumed to revert toward zero following an Ornstein-Uhlenbeck process; $(\xi_{t})$ is assumed to follow a Brownian motion process The system is as follows: $$ d\zeta _{t}=(-\kappa \zeta _{t}-\varphi _{\zeta }-\phi _{\zeta }\xi _{t})dt+\sigma _{\zeta }dz_{\zeta } $$ $$ d\xi _{t}=(\mu _{\xi }-\varphi _{\xi }-\phi _{\xi }\zeta _{t})dt+\sigma _{\xi}dz_{\xi } $$
$dz_{\zeta}$ and $dz_{\xi}$ are correlated increments of standard Brownian motion process with $dz_{\zeta}dz_{\xi}=\rho_{\zeta\xi}dt$.
Can someone kindly help me? Or tell me how to solve analytically this system? Many thanks in advance
Hi fortunately your system of sde is a system of linear equation under the hypothesis that : $\phi _{\zeta },\sigma _{\zeta },\phi _{\xi },\sigma _{\xi}, etc...$ are not random functions (they might depend on time ) and are locally bounded.
From Karatzas and Shreve Brownian Motion and Stochastic Calculus we known(section 5.6) that there is a solution for a linear system of SDEs.
So let's be given the following system d dimensional system (written in a vectorial formulation identical to the reference) :
$$dX_t=[A(t)X_t+a(t)]dt+ \sigma_t.dW_t$$ -$a(t)$ is a d dimensional vector function of time (non random measurable (in t) and locally bounded).
-$A(t)$ is a $d \times d$ matrix (non random measurable (in t) and locally bounded).
-$W$ is a r dimensional Brownian motion.
-$\sigma(t)$ is $d \times r$ matrix (non random measurable (in t) and locally bounded).
Let's denote $\Phi$ the solution of the following ordinary matrix equation :
$$\Phi(t)'=A(t)\Phi(t)$$ with $\Phi(0)=Id$
Then : $$X(t) = \Phi(t).[X_0+\int_0^t \Phi(s)^{-1}.a(s).ds+\int_0^t \Phi(s)^{-1}.\sigma(s).dW_s]$$
Now what is left to to you is the calculation for which I have no personal interest but which seem tractable in your case.
PS: Even if the solution here is written in a framework where all component of W are independents, it's very easy to recast your framework in this setting with the proper matrix sigma (which is not diagonal).
Best regards