5
$\begingroup$

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

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.

$\endgroup$
2
  • 1
    $\begingroup$ To answer your parenthetical question: Rather $X_t-X_0$ has the written distribution so that given $X_0=x_0$, $X_t\sim \mathcal{N}(x_0+at, b^2 t)$. $\endgroup$ – Nap D. Lover Feb 26 at 19:40
  • $\begingroup$ Ah yes, edited! $\endgroup$ – 900edges Feb 26 at 19:46
2
$\begingroup$

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: samplepath

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.

enter image description here

$\endgroup$
3
  • 2
    $\begingroup$ Thanks for the response! However I am attempting an analytic solution and looking for some insight on the compatibility conditions needed at the boundary. What happens when the simulation goes below 0? $\endgroup$ – 900edges Feb 26 at 20:20
  • $\begingroup$ @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. $\endgroup$ – Nap D. Lover Feb 26 at 20:28
  • 2
    $\begingroup$ That image makes sense intuitively-- I'm interested in what happens to the transition density as it passes through 0. Interestingly it looks like it passes right through, but I imagine that depends on the parameters. Thanks again $\endgroup$ – 900edges Feb 26 at 20:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.