My process is the SDE $dX_t=X_tdt + dB_t$, for $t>0$, and $X_0=1$, where $B_t$ is Brownian Motion. I was told I should solve using $Y_t=X_te^{-t}$.
How does it become $dY_t=e^{-t}dX_t-X_te^{-t}dt=e^{-t}(X_tdt+dB_t)-X_te^{-t}dt=e^{-t}dB_t$ ?
2026-03-31 05:20:10.1774934410
Stochastic Differential Equation Explanation
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can you take a look at https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process ? Your question is intimately related to the so-called "Ornstein-Uhlenbeck process"