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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 $X_t = W_t\sin(t)$?

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  • $\begingroup$ All you need to do is check if it is a solution or not, because uniqueness comes for free from the Lipschitz nature of the coefficient. Is there some trouble while applying the Ito rule? $\endgroup$ Commented Feb 1, 2022 at 13:31
  • $\begingroup$ The increments of $X_t$ are distributed as $dX_t\sim N(0,\sin^2(t)\,dt)$ or $2\,dX_t\sim N(0,2(1-\cos(2t))\,dt)$. This means that there exists a second Brownian motion $\widetilde W$ so that $$ X_t=X_0+\frac12\widetilde W_{2t-\sin(2t)}. $$ $\endgroup$ Commented Feb 1, 2022 at 17:09
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    $\begingroup$ It's actually an overload to call this an SDE. This equation means nothing but $X_{t}=X_{0}+\int_{0}^{t}\operatorname{sin}(s)dW_{s}$. $\endgroup$
    – Tobsn
    Commented Feb 1, 2022 at 17:22

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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.

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  • $\begingroup$ @Sinem does this answer your question? $\endgroup$ Commented Apr 26, 2022 at 21:18

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