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The notation $\mu(X_t,t)$ and $\sigma(X_t,t)$ is indeed a special case of the more general notation $\mu_t$ and $\sigma_t$. The latter may be any stochastic processes (with the conditions given in the defintion). Note firstly that if $\sigma_t=0$, the SDE reduces to an ODE and we already know that not all ODEs are solvable (explicitly and uniquely). ...

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