How to find the expectation of joint non-linear SDEs?

Question

I'm working with the following pair of joint stochastic differential equations \begin{align} X_t &= (\alpha X_t - Y_t)dt + \sigma\left(X_t - \frac{Y_t}{a}\right)dW_t^{(1)} \\ Y_t &= (b(X_t)X_t + \beta Y_t)dt + \nu \sqrt{Y_t\left(X_t - \frac{Y_t}{a}\right)}dW_t^{(2)} \\ \end{align} In the case that $b(X_t) \equiv b \in \mathbb{R}$ is a constant, we can explicitly recover the expectations of both $X$ and $Y$ by solving a system of ordinary differential equations. My question is this, if the function $b(X_t)$ is non-constant (and hence the term $b(X_t)X_t$ becomes non-linear), are there any other situations in which we can still find the expectations of the system?

Answer 1

Setting $f_t:=\mathbb{E}\left[X_t\right],g_t:=\mathbb{E}\left[Y_t\right],h_t:=\mathbb{E}\left[b(X_t)X_t\right]$, we still have that $$\dot{f}_t=af_t-g_t$$ such that $$f_t=-e^{at}\int_0^te^{-as}g(s)ds.$$ Similarly, $$g_t=e^{\beta t}\int_0^te^{-\beta s}h(s)ds$$ so it depends what $h$ is. If it is $x\mapsto \frac{1}{x}$, then this would be an example for a case, where one obtains an explicit solution.