I have a question about stochastic differential equations with additive noise. My question is: Is the solution of a SDE with additive noise almost surely equal to the solution of the corresponding deterministic equation plus the noise?
Mathematically formulated, my question is the following:
Let $X_t$ be the solution to the SDE $$ \mathrm{d}X_t = b(t,X_t) \mathrm{d}t + \mathrm{d} B_t,\; X_0=\xi, $$ where $B_t$ denotes a Brownian motion and with suitable assumptions on $b$ to assure existence and uniqueness of a solution. Furthermore, let $Y_t$ be the solution to the integral equation $$ \mathrm{d}Y_t = b(t,Y_t)\mathrm{d}t,\; Y_0 =\xi. $$ Is it true that it holds $X_t = Y_t +B_t$ almost surely?
Thank you in advance for your input!
Best, Luke
No, that’s not true. The effect of the noise feeds back into the state ($X_t$), which in turn affects the drift term ($b(t,X_t)$). You can only separate the two effects if $b$ does not depend on $X_t$.
As a blatant illustration, consider a particle in a double-well potential:
$$b(X_t) = -X_t(X_t - 1)(X_t + 1)$$
If the initial condition is at the bottom of one of the wells (e.g., $ξ=−1$), the particle will never leave that well in the deterministic case. By contrast, the particle can hop between the wells if moved by noise.