SDE with constant coefficients

So the simplest possible SDE would be $$$$\begin{cases} dX_t = a dt + b dW_t, \text{ with } a,b\in \mathbb{R} \\ X_0 = x_0 \end{cases}$$$$

describing a 1-dimensional stochastic process $$X_t$$ on a probability space $$(\Omega, \mathcal{F}, P)$$ and $$t \in [0,\infty)$$.

I know that in this case, the distribution of the solution is a Gaussian. So it is enough to characterize $$\mathbb{E}[X_t]$$ and $$\text{Var}(X_t)$$. Taking the expectation and variance of the equation, I get that $$X_t \sim \mathcal{N}(x_0 + at, b^2t)$$.

The problem I am interested in is a piecewise constant SDE, where $$a = \begin{cases} a_1, \text{ if } X_t >0\\ a_2, \text{ if }X_t \leq 0 \end{cases}$$

$$b = \begin{cases} b_1, \text{ if } X_t >0\\ b_2, \text{ if }X_t \leq 0 \end{cases}.$$

How would one "piece together" the two solutions on either side of $$0$$? It would be simple to solve the equation on either side, but I think there would have to be some compatibility equation at 0.

One problem I know is that a Brownian motion will hit $$0$$ infinitely many times in a time interval $$[0,\epsilon)$$, which means that each time $$X_t=0$$, there is a small interval where the process may not be well-defined. I was thinking to fix this by stopping the process everytime it hits $$0$$, and restarting it after it "escapes" some neighborhood of 0. Perhaps this could be reframed as a reflected diffusion.

Note: there are two relevant solution methods to similar problems that I've seen in the literature, one involving an analytic solution to the constant-coefficients problem (Karatzas and Shreve), and a numerical solution to the reflecting boundary problem (Skorokhod), not sure which is appropriate here.

• To answer your parenthetical question: Rather $X_t-X_0$ has the written distribution so that given $X_0=x_0$, $X_t\sim \mathcal{N}(x_0+at, b^2 t)$. – Nap D. Lover Feb 26 at 19:40
• Ah yes, edited! – 900edges Feb 26 at 19:46

You could replace the constant coefficients $$a$$ and $$b$$ with the coefficient functions in terms of indicator functions: $$a(x)=a_1 \cdot \mathbb{1}_{(0, \infty)}(x)+a_2 \cdot \mathbb{1}_{(-\infty, 0]}(x)$$ and a similar expression for $$b$$ but with the constants $$b_1, b_2$$. Then the SDE is $$dX_t=a(X_t)dt+b(X_t)dB_t$$
Here is a simulation of a sample-path via the Euler-Maruyama scheme implemented in R with initial point $$x_0=0$$, total time $$T=1$$, and parameters $$a_1=-0.05$$, $$a_2=0.1$$, $$b_1=0.1$$ and $$b_2=0.3$$ with $$n=1000$$ time-subintervals:
and here is one more simulation that crosses the line $$x=0$$ with all the same parameters except starting below zero at $$x_0=-1$$ and running the path for $$T=10$$. If I can derive any analytic information, I will update this post.
• @edges900 Indeed, unfortunately, at least at the moment, I cannot find much to say analytically. I added another sample path that starts below zero and eventually crosses $x=0$, just for reference. – Nap D. Lover Feb 26 at 20:28