Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb R^d)$ and $\lambda: \mathbb R^d\to\mathbb R^d$ and standard brownian motion $W_t$.
Now suppose I have some event $E\subset C^0$ for which I am interested in the probability that $X_t\in E$.
Define an operator $\mathcal W: C^1(\mathbb R^+,\mathbb R^d)\to C^1(\mathbb R^+,\mathbb R^d)$ by setting $$\mathcal Wf(t)=\int_0^t \left(\sigma(f(s))\right)^{-1} \left({f'(s)} - \lambda\right) ds.$$
So $\mathcal W$ is a naive attempt to reconstruct the original Brownian path $W_t$ from the path $X_t$.
If I can find some event $F\subset C^0$ such that $\mathcal Wf \in F$ for every $f\in E\cap C^1$ I might hope that $\mathbb P\left[X_t\in E\right]\leq\mathbb P\left[W_t\in F\right]$.
It's obviously not true in general (Take $F = C^1$) but I'd guess it was true for $F$ closed in the uniform topology.
So my question is:
- Is there a sufficient condition on $F$ that guarantees $\mathbb P\left[X_t\in E\right]\leq\mathbb P\left[W_t\in F\right]$?
- Can I relax the assumptions on my SDE?
