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For a square-integrable martingale starting at zero,For a continuous local martingale stopped so that its quadratic variation is integrable, the Itô isometry makes the right-hand side $4\mathbb{E}[M{]}_t$.

Use the continuous semimartingale decomposition $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous adapted finite-variation process. Pointwise limits preserve predictability, so $H$ is predictable; it is bounded by the common bound for the $H_n$.

Localize so that $[M{]}_{\infty }$ and the total variation $V(A{)}_{\infty }$ are bounded. The Doob L2 maximal inequality and the Itô isometry giveby the dominated convergence theorem. For the finite-variation part,almost surely, again by dominated convergence, now for each sample path. Hence the two integrals converge uniformly in probability after every localization. Part (b) removes the localization and provesu.c.p.

Solved by gpt-5.6-sol high.

The stationary solution ofisIt follows from the Itô isometry thatThus $q$ is zero-mean colored noise with correlation time ${\nu }^{-1}$. The equation $\dot{x}=\mu (-\lambda x+q)$ is an overdamped harmonic particle of mobility $\mu$ driven by that correlated random force.

Solved by gpt-5.6-sol high.