There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts:
$$E^x\left[u(W_0)\right]=u(x)$$
What is the notation $E^x$? what kind of expectation is it?
2026-04-02 17:40:17.1775151617
Notation in stochastic integrals
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$E^x$ is the expectation under the path measure of $W(\cdot)$ started at $W(0) = x$.
In plain English, $E^x$ is just ordinary expectation with the understanding that the starting point of BM is a variable, $x$.