How can I obtain from a differential equation a stochastic version?

Question

Suppose $\frac{dx}{dt}=ax+b$ and then assume that $a=c+g$ where $g$ is a Wiener process.

Answer 1

I think that 'Stochastic Differential Equations' by Bernt Oksendal is a good book about SDE.

According your equation, lets rewrite it:

$$dx = ax\,dt +b\,dt$$

and add noise

$$dx = (cx+b)\,dt + xg\,dt$$

If $g$ is a Wiener process, then $g\,dt$ is a Brownian motion, so your SDE equation is

$$dX_t(\omega) = (cX_t(\omega)+b)\,dt + X_t(\omega)dB_t(\omega)$$

Answer 2

You should distinguish white noise and Brownian motion. Let $\xi_t$ be $\mathcal{N}(0,1)$ white noise, then if $$ dy_t = \xi_t dt $$ then $y_t\equiv$const (from the Law of Large Numbers).

In fact it's better to obtain ODE from SDE, vice versa can be done in the following way: you write perturbed equation $$ dX_t = (aX_t + b)dt+\sigma(t,X_t)dB_t $$ where $B_t$ is a Brownian motion ( = Wiener process). If you put $\sigma\equiv 0$ then $$ dx_t = (ax_t+b)dt $$ Also, for arbitrary $\sigma$ it's true for the expected value of the perturbed system $m_t = \mathrm{E}[x_t]$ we have $$ dm_t = (am_t+b)dt. $$ You can choose any $\sigma$ you like - for example constant, linear in $X_t$, linear in time and so on. I suggest to put $\sigma$ independent of time to leave the system autonomous.

P.S. You can read in Oksendal that for the white noise $\xi_t$ it's not interesting to consider process with increments $\xi_t dt$ because of its triviality. In fact it's more "correct" to use agreement $dB_t = \xi_t\sqrt{dt}$.