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

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

-
What's the question? –  Graviton Dec 1 '10 at 7:19
This could be interesting, so please extend your question (a little more flesh is needed!) –  vonjd Dec 1 '10 at 9:09
Ok, I will try, Suppose a deterministic dynamical system defined using the ordinary differential equation dx/dt=ax+b Then what happens if the system is shaken by a white noise process, for instance if the parameter a stops being a constant and assumes a form as a=c+g where g is a parameter and c is a white noise process. Then the sde should be written in a form dx=((c+g)x+b)dt+σ()dz What is the algebra from ode to sde? And what if the functional form of σ() ? Any reference or suggestions? thanks!! –  newb Dec 1 '10 at 9:42

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

According your equation, lets rewrite it:

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

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

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

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

-
Did you mean to switch from $a$ in the first equation to $c$ in the second? –  Mike Spivey Dec 1 '10 at 22:33
I mean switch from $ax\,dt$ to $(c+g)x\,dt$ then I just group coefficient at $dt$ and at $g\,dt$ –  rystsov Dec 1 '10 at 22:53
Now I see that the OP makes the same variable switch, which I should have seen before. My apologies. –  Mike Spivey Dec 2 '10 at 1:03
Thank for the reference and your answer! –  newb Dec 2 '10 at 7:00

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 $dB_t = \xi_t\sqrt{dt}$.

-