Let $W_t$ be a Wiener process and consider the stochastic differential equation $$dX_t = \sin(t)dW_t.$$
Is the solution to this SDE $X_t = W_t\sin(t)$?
2026-04-01 03:38:43.1775014723
Sine function and Wiener process
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No; if you apply Itô's lemma to $W_t \sin t$, you won't recover your proposed dynamics.
Another way to see this is by noting that a solution of this SDE would be a martingale, since $\sin$ is square integrable. But, $$E(X_t | \mathcal{F}_s) = W_s \sin t \neq X_s$$ showing that your proposed process cannot solve this SDE.