# Calculating the unconditional expectation of an Ito process

Suppose that: $$dX_t =\mu(t,X_t)dt+\sigma(t,X_t)dW_t,$$ where $$X_t$$ is a vector valued stochastic process, $$W_t$$ is a vector of Brownian motions, $$\mu$$ is a vector valued function and $$\sigma$$ is a matrix valued function.

I want to compute the unconditional expectation $$\mathbb{E}X_t$$ defined by: $$\mathbb{E}X_t := \lim_{s\rightarrow -\infty}{\mathbb{E}[X_t|X_s=0]}.$$ (Assume that this quantity is guaranteed to be well-defined and finite for the particular $$\mu$$ and $$\sigma$$ of interest.)

It seems that it is at least true that: $$\frac{d\mathbb{E}X_t}{dt}=\mathbb{E}\mu(t,X_t)?$$ (See: How to find mean of a stochastic differential equation . So, I take it no extra term can enter coming from some kind of instantaneous correlation between $$\sigma(t,X_t)$$ and $$dW_t$$.)

Are there any tricks to simplify evaluating $$\mathbb{E}\mu(t,X_t)$$? Applying Ito's lemma to get the law of motion for $$\mu(t,X_t)$$ is unlikely to help as that LOM will have a new drift term.

When $$\mu$$ and $$\sigma$$ are non-linear, is there any hope of deriving a solution for $$\mathbb{E}X_t$$ in terms of an ODE?

Does the problem simplify much if $$\mu$$ and $$\sigma$$ are not directly functions of $$t$$?

• I'm afraid there is no such technique in general, hence a case-by-case treatment. Jan 27 at 18:50