Expectation of Gaussian White Noise and Functional of Noise

Question

If I have a Gaussian white-noise process $$\langle \xi(t)\rangle=0\,,$$ $$\langle \xi(t) \xi(s)\rangle=\lambda \delta(t-s)\,,$$ where $\langle \cdot \rangle=$ denotes the expectation value, what is the expectation value of the noise and a functional of the noise $$\langle\xi(t) F(t)\rangle,$$ where $\dot{F}=g(t,\xi(t))$ and $g(t,\xi(t))$ is a function linear in $\xi(t)$?

Answer 1

Let us take a finite time interval $[0,T]$ for clarity. The function $F$ you write is actually the solution to an SDE (with $F(t, \xi) = X_t)$: $$dX_t = b_t d W_t,$$ assuming for simplicity that $g(t, \xi(t)) = b_t \xi_t$ (an additional affine term will not change the discussion - and for the same reason we consider $X_0=0$).

Note: The SDE cannot be solved for all $b,$ so you need that the process is either adapted or in some way smooth.

At this point take functions $f,h \in L^2([0,T])$ and compute: $$ \mathbb{E} [ \xi(f) X(h)]= \mathbb{E} \int_0^T f_s dW_s \int_0^T h_s\int_0^sb_r dW_r ds \\ = \int_0^T h_s\int_0^s f_r b_r d r $$ and if you formally insert $f=h=\delta_t$ one finds: $$ \mathbb{E} [\xi(t) X_t] = b_t. $$