Noisy Brownian Noise

Question

Suppose that $W_t$ and $B_t$ are independent Brownian motions, and define the process $X_t\triangleq W_t + B_t$. What is the conditional expectation of $W_t$ given the $\sigma$-algebra generated by $X_t$?

Is this a special case of the Kalman filter? It seems that the usual assumptions are violated?

Answer 1

Note that (essentially by the symmetry in the problem) $$\mathbb{E}[W_t \mid X_t] = \mathbb{E}[B_t \mid X_t].$$ In particular, $$X_t = \mathbb{E}[X_t \mid X_t] = \mathbb{E}[W_t \mid X_t] + \mathbb{E}[B_t \mid X_t] = 2 \mathbb{E}[W_t \mid X_t]$$ and so $\mathbb{E}[W_t \mid X_t] = \frac{X_t}{2}$.