It is well known that for an Stochastic differential equation (on the real line) of the form: $dX_t = \mu(X_t)dt + \sigma(X_t)dW$ where $W$ is the standard Wiener process, the transition probability densities of the process can be related to a PDE of the form $\frac{\partial u}{\partial t} = \frac{\partial^2}{\partial x^2}(a^2(x)u)-\frac{\partial}{\partial x}(v(x)u)$. Using standard PDE theory it can be proved that weak solutions of the PDE exist in suitably weighted Lesbesgue spaces for essentially bounded coefficients $a^2(x)$ and $v(x)$ with suitable growth conditions (and ellipticity). However almost all books/papers on SDEs I see that $\mu$ and $\sigma$ are taken to be at least continuous. Are there any counterexamples where solutions for PDEs exist but not for the corresponding SDE? Or even if solutions for the SDE exist is there a counterexample such that the sample paths are not continuous any more?
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1$\begingroup$ The phrase "existence for the solution of the SDE" is a bit ambiguous. There are several notions of solution. Strong solutions (i.e. where the process X is a measurable function of W) may fail to exist when $\mu$ is discontinuous. See Tsirelson's example: tau.ac.il/~tsirel/Research/mydrift/main.html $\endgroup$– ofer zeitouniCommented Dec 4, 2015 at 15:11
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4$\begingroup$ Under boundedness of $\mu$ (and continuity of $\sigma$), weak solutions to the SDE exist by Girsanov's theorem (and time change). $\endgroup$– ofer zeitouniCommented Dec 4, 2015 at 15:13
1 Answer
One of the latest conditions on $\mu,\sigma$ are from "A Numerical Method for SDEs with Discontinuous Drift":
This result states that the SDE (1) ($dX_t = \mu(X_t)dt + \sigma(X_t)dW$) admits a unique strong solution $X$ if the drift coefficient $\mu$ has finitely many discontinuity points and is piecewise Lipschitz continuous and the diffusion coefficient $\sigma$ is globally Lipschitz continuous and non-degenerate at the discontinuity points of $\mu$.
Also, as mentioned in the comments, one can even do with just bounded, measurable $\mu$ (cf "A Numerical Method for SDEs with Discontinuous Drift"):
In the case where the diffusion coefficient $\sigma$ is bounded, Lipschitz, and (partly) uniformly elliptic, and the drift coefficient $\mu$ is only bounded and measurable, the pioneering work by Zvonkin [15] and Veretennikov [13, 14] yields existence and uniqueness of the solution.
It seems to be the latest one based on overview here "Existence, uniqueness and approximation of solutions of SDEs with superlinear coefficients in the presence of discontinuities of the drift coefficient'
see here What work has been done on SDE with diffusion coefficients of bounded variation in $\mathbb R^d$? too for more references.
A nice result is also in Revuz-Yor book "Continuous Martingales and Brownian Motion" in (1.14) Corollary:
If $\sigma$ is a bounded function on the line such that $|\sigma(x,t)|\geq \epsilon>0$ and $\mu$ a bounded function on $R_{+}\times R$ there is existence and uniquenss in law for the SDE.
