Conditional Expectation in Diffusion Process

Question

Consider a $d$-dimensional diffusion process $\mathbf{X}=(\mathbf{X}_t)_{t\in [0,T]}=([X^1_t,...,X^d_t])_{t\in [0,T]}$ that is the unique strong solution of the following SDE:

$$\left\{\begin{matrix} d X^1_t = \lambda_1(\mathbf{X}_t)dt + d W^1_t\\ d X^2_t = \lambda_2(\mathbf{X}_t)dt + dW^2_t\\ \cdots \\ d X^d_t = \lambda_d(\mathbf{X}_t)dt + dW^d_t \end{matrix}\right.$$

where $W^1, W^2, ..., W^d$ are independent Wiener processes.

Denote $\mathcal{F}_t^i:=\sigma(X_s^i; s\leq t)$ and $\mathcal{F}_t^C:=\sigma(\mathcal{F}_t^i; i\in C)$ as the filtration generated by $X^i_t$ and $\mathbf{X}^C_t$, respectively. The question is then: How to estimate the conditional expectations:

$$t\mapsto \mathbb{E}[X_t^i|\mathcal{F}_t^C], \quad t\mapsto \mathbb{E}[\lambda_i(\mathbf{X}_t)|\mathcal{F}_t^C]$$

for any $i=1,2,...,d$ and $C\subseteq [d]:=\{1,2,...,d\}$.

Answer 1

Here’s my attempt at getting started. The following answer concerns the easier case of conditioning on $\sigma(\{W^i\}_{i \in C})$.

Without loss of generality, take $C = \{1, \dots k\}$.

Now for $1 \leq i \leq k$, conditional on $\mathcal F_t^C$, $X^i$ follows the dynamics

$$X^i_t= X^i_0 + \int_0^t \lambda_i (\mathbf X_s) \, ds + w_t,$$

where now $W^i = w^i \in C[0, t]$ is deterministic given $\mathcal F^C_t$.

As such, by taking conditional expectations, we have the equation

$$\mathbb E[X^i_t| \mathcal F^C_t] = X^i_0 + \int_0^t\mathbb E[\lambda_i (\mathbf X_s) | \mathcal F^C_s] \, ds+ w^i_t.$$

We can of course write similarly, for $i > k$,

$$\mathbb E[X^i_t| \mathcal F^C_t] = X^i_0 + \int_0^t\mathbb E[\lambda_i (\mathbf X_s) | \mathcal F^C_s] \, ds + W^i_t,$$

or in SDE form,

$$d \mathbb E[X^i_t| \mathcal F^C_t] = \mathbb E[\lambda_i (\mathbf X_t) | \mathcal F^C_t] \, dt + dW^i_t.$$

I am not sure how to proceed from here. We may be able to get something much more if $\lambda$ is assumed smooth enough to apply Ito’s formula.