# Are $L$-diffusions unique in law?

I've been trying to understand diffusions. We can show they exist by noting they solve particular SDEs, but are they unique? More precisely:

Fix a filtered probability space satisfying the usual conditions and locally bounded measurable functions $a_{i,j}$ and $b$ from $\mathbb{R}^d$ to $\mathbb{R}$, such that $(a_{i,j}(x))_{i,j}$ is symmetric non-negative definite for all $x\in\mathbb{R}^d$. Define the operator $L$ by

$$Lf(x)=\frac{1}{2}\sum_{i,j}a_{i,j}(x)\frac{\partial f}{\partial x_ix_j}+\sum_ib_i(x)\frac{\partial f}{\partial x_i}$$

Suppose that the continuous, adapted process $X$ is such that, for all $f\in C^2(\mathbb{R}^d)$,

$$f(X_t)-f(0)-\int_0^tLf(X_s)ds$$

is a local martingale, and $X_0=0$.

Then is $X$ unique in law?

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+1, I asked related question some time ago here – Ilya Mar 28 '12 at 16:26
Here are my thoughts: I don't think the law in general is unique. As far as I can tell this is a martingale problem, which is equivalent to solving an SDE with coefficient $(\sigma,b)$, where $a=\sigma\sigma^T$. But one can show that uniqueness in law of this martingale problem is equivalent to uniqueness in law of the corresponding SDE. But solutions to SDE's are not unique in law, although in many cases they are. Examples are when the coefficient is Lipschitz or when the Yamada-Watanabe conditions are satisfied. – Stefan Hansen Mar 28 '12 at 17:22
@StefanHansen Thanks. Is there a good book which proves the equivalence of uniqueness in law of the $L$-diffusion and the corresponding SDE? – Ben Derrett Mar 29 '12 at 10:09
I think Diffusions, Markov Processes and Martingales by Rogers and Williams is a good book. In Chapter V they introduce the martingale problem and later they prove equivalence of solutions to SDE's and solutions to the corresponding martingale problem. You could also take a look at Brownian Motion and Stochastic Calculus by Karatzas and Shreve around page 317. – Stefan Hansen Mar 29 '12 at 13:13

Here's a common counterexample to uniqueness that usually appears in the context of SDEs, e.g. Oksendal. The equivalence of SDEs and Martingale Problems is explained in this paper by Kurtz and the more technical book Ethier & Kurtz. Put $d=1$, $a\equiv 0$, $b(x)=2|x|^{\frac{1}{2}}$ and consider the (deterministic) process $$X_t= \begin{cases} 0 & \text{if }\hspace{2mm} 0\leq t \leq \tau \\ (t-\tau)^2 & \text{if }\hspace{2mm} t>\tau \end{cases}$$ for any $\tau>0$. Certainly $X$ is continuous and adapted and $L$ satisfies the required hypotheses. For any $f\in C^2(\mathbb{R})$ we have $f(X_t)-f(0)-\int_0^t2\sqrt{|X_s|}f'(X_s)ds=f(0)-f(0)=0$ when $t\leq\tau$. Moreover, $$f(X_t)-f(0)-\int_0^t2\sqrt{|X_s|}f'(X_s)ds=f((t-\tau)^2)-f(0)-\int_{\tau}^t2(t-\tau)f'((t-\tau)^2)ds=f((t-\tau)^2)-f(0)-\Big(f((t-\tau)^2)-f(0)\Big)=0$$ when $t>\tau$. Hence $f(X_t)-f(0)-\int_0^tLf(X_s)ds\equiv 0$ is a martingale. The fact that each $\tau>0$ leads to a different solution violates uniqueness. Note that requiring $a$ be positive definite or a Lipschitz $b$ coefficient would preclude this counterexample.