The Stroock-Varadhan Martingale Problem

From an Operator to a Law on Paths

A stochastic differential equation specifies dynamics through its drift and diffusion coefficients. The martingale problem gives a different way to specify dynamics: it asks whether a probability law on continuous paths makes certain compensated test functions into martingales. The operator determines which compensations are required, while the initial distribution specifies where the path starts. This formulation is useful because it describes the law of the process without requiring a Brownian motion or a particular probability space as part of the definition.

Let $d\geq 1$, and let $b:\mathbb{R}^d\to\mathbb{R}^d$ and $a:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be Borel measurable and locally bounded. Assume that each matrix $a(x)$ is symmetric and nonnegative definite. For $f\in C_c^2(\mathbb{R}^d)$, the space of twice continuously differentiable functions with compact support, define

Lf(x)=\sum_{i=1}^d b_i(x)\,\partial_i f(x) +\frac12\sum_{i,j=1}^d a_{ij}(x)\,\partial_{ij}f(x).

Here $b$ is the drift in the operator and $a$ is its second-order coefficient. In the diffusion notation $a=\sigma\sigma^\mathsf{T}$, the second-order term is $\tfrac12\sum_{i,j}(\sigma\sigma^\mathsf{T})_{ij}\partial_{ij}f$. The factor $\tfrac12$ is part of the generator convention. Since $f$ has compact support, its first and second derivatives vanish outside a compact set. Local boundedness of $b$ and $a$ therefore makes $Lf$ bounded. This ensures that the time integral in the definition below is finite on each bounded time interval.

Use the canonical continuous path space $\Omega=C([0,\infty),\mathbb{R}^d)$, with coordinate process $X_t(\omega)=\omega(t)$ and raw canonical filtration $\mathcal{F}_t^0=\sigma(X_s:0\leq s\leq t)$. A probability law $P$ on this space gives meaning to expectations and martingale properties of the coordinate process.

Definition: Given an operator $L$ as above and a probability measure $\mu$ on $\mathbb{R}^d$, the martingale problem for $(L,\mu)$ is to find a probability law $P$ on $\Omega$ such that $P(X_0\in A)=\mu(A)$ for every Borel set $A$, and, for every $f\in C_c^2(\mathbb{R}^d)$, the process $$ M_t^f=f(X_t)-f(X_0)-\int_0^t Lf(X_s)\,ds $$ is a martingale under $P$ with respect to $(\mathcal{F}_t^0)$. The first condition is the initial-law condition; the second is the martingale condition.

In particular, each $M_t^f$ is integrable: $f$ and $Lf$ are bounded, so $|M_t^f|\leq 2\lVert f\rVert_\infty+t\lVert Lf\rVert_\infty$. A solution is a probability law $P$ satisfying both conditions, not a particular sample path. The initial distribution may be a point mass $\delta_x$, in which case the problem is said to start at $x$.

Existence and Uniqueness Mean Properties of Laws

Definition: The martingale problem for $(L,\mu)$ has a solution if at least one probability law $P$ satisfies its defining conditions. It has uniqueness in law if any two such probability laws on canonical path space are equal. It is well posed for $\mu$ if it has a solution and that solution is unique. Saying the problem is well posed for every starting point means well posedness for $(L,\delta_x)$ for every $x\in\mathbb{R}^d$.

Uniqueness here concerns the entire distribution of the coordinate path, not equality of two processes constructed on the same probability space. It is therefore uniqueness in law, not pathwise uniqueness. If the initial law is not specified, the martingale condition alone does not select a single law: even the constant-path operator $L=0$ admits constant paths starting from different points.

Proposition: For a law $P$ with the stated integrability, the process $M^f$ is a martingale if and only if, for every $0\leq s\leq t$ and every bounded $\mathcal{F}_s^0$-measurable random variable $G$, $$ \mathbb{E}^{P}\!\left[G\left(f(X_t)-f(X_s)-\int_s^t Lf(X_r)\,dr\right)\right]=0. $$

Proof. By definition, an integrable adapted process $M^f$ is a martingale exactly when $\mathbb{E}^{P}[M_t^f\mid\mathcal{F}_s^0]=M_s^f$ almost surely for every $s\leq t$. The defining property of conditional expectation says this is equivalent to $\mathbb{E}^{P}[G(M_t^f-M_s^f)]=0$ for every bounded $\mathcal{F}_s^0$-measurable $G$. Subtracting the two expressions for $M_t^f$ and $M_s^f$ gives $M_t^f-M_s^f=f(X_t)-f(X_s)-\int_s^t Lf(X_r)\,dr$, which is the displayed identity. The equivalence holds in both directions, and all expectations exist by the boundedness established above. $\square$

This criterion makes explicit what the martingale condition tests: once the path up to time $s$ is observed, the expected increment of $f(X)$ is the expected accumulated increment prescribed by $Lf$. It also avoids treating the expression $M^f$ as though its martingale property followed merely from its form; the conditional expectation identity must hold under the proposed law.

Proposition: Suppose $P_1$ solves the martingale problem for $(L,\mu_1)$, $P_2$ solves it for $(L,\mu_2)$, and $0\leq\lambda\leq1$. Then $P_\lambda=\lambda P_1+(1-\lambda)P_2$ solves the martingale problem for $(L,\lambda\mu_1+(1-\lambda)\mu_2)$.

Proof. The initial law under $P_\lambda$ is the claimed mixture, since for every Borel set $A$, $P_\lambda(X_0\in A)=\lambda\mu_1(A)+(1-\lambda)\mu_2(A)$. Fix $f\in C_c^2$, $s\leq t$, and bounded $\mathcal{F}_s^0$-measurable $G$. The random variable $G(M_t^f-M_s^f)$ is integrable under both laws. Linearity of expectation gives $$ \mathbb{E}^{P_\lambda}[G(M_t^f-M_s^f)] =\lambda\mathbb{E}^{P_1}[G(M_t^f-M_s^f)] +(1-\lambda)\mathbb{E}^{P_2}[G(M_t^f-M_s^f)]=0, $$ because each law solves its martingale problem and the conditional-expectation criterion applies. Thus $M^f$ is a martingale under $P_\lambda$ for every test function. When $\lambda=0$ or $1$, the same argument reduces to the corresponding given solution, so the endpoint cases are included. $\square$

Worked Examples

Worked Example: The Zero Operator and Constant Paths

Let $d=1$, $L f=0$, and $\mu=\delta_3$. Consider the law $P$ concentrated on the single path $\omega(t)=3$ for every $t\geq0$. Its initial value is $X_0=3$, so its initial distribution is $\delta_3$. For any $f\in C_c^2(\mathbb{R})$, $f(X_t)=f(3)$ and $\int_0^t Lf(X_s)\,ds=0$. Hence $M_t^f=f(3)-f(3)=0$ for all $t$. The zero process is a martingale, since it is integrable and its conditional expectation at every time is zero.

Check: at $t=0$, the expression is $f(3)-f(3)-0=0$; at any later $t$, the path and the expression have not changed. This verifies that the proposed law is a solution. This check establishes existence for this particular initial law; it does not by itself establish uniqueness.

Worked Example: A Deterministic Drift

Take $d=1$, $b(x)=2$, $a(x)=0$, so that $Lf(x)=2f'(x)$. Let the initial law be $\delta_{-1}$, and consider the deterministic path $X_t=-1+2t$. For $f\in C_c^2(\mathbb{R})$, the ordinary chain rule gives $$ f(X_t)-f(X_0)=\int_0^t 2f'(-1+2s)\,ds =\int_0^t Lf(X_s)\,ds. $$ Therefore $M_t^f=0$ at every time, and the initial distribution is $\delta_{-1}$. This law solves the martingale problem. The generator has no second-order term here, so the verification uses the ordinary chain rule, not a Brownian correction.

Check: the path's derivative is $2$, and the integrand on the right is $2f'(X_s)$, exactly the derivative of $f(X_s)$. At $t=0$, both sides of the identity are zero.

Worked Example: Brownian Motion and the Heat Operator

Let $B_t$ be standard Brownian motion on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})$, with $B_0=0$, and take $L f=\tfrac12 f''$ and $\mu=\delta_0$. For any $f\in C_c^2(\mathbb{R})$, Ito's formula, with the function $g(t,x)=f(x)$, gives $g_t=0$, $g_x=f'(x)$, and $g_{xx}=f''(x)$. Its second-order correction is $\tfrac12g_{xx}(t,B_t)\,dt=\tfrac12f''(B_t)\,dt$; using $dB_t\,dB_t=dt$, the formula is $$ f(B_t)-f(B_0)=\int_0^t f'(B_s)\,dB_s+\frac12\int_0^t f''(B_s)\,ds. $$ The integrand $H_s=f'(B_s)$ is adapted and progressively measurable, and $\mathbb{E}\int_0^T H_s^2\,ds\leq T\lVert f'\rVert_\infty^2<\infty$ for every finite $T$. Thus $H\in\mathcal{L}^2$, and the Ito integral is a true martingale: it is square-integrable by the Ito isometry, and its conditional expectation increment from $s$ to $t$ is zero. Rearranging the formula shows that $M_t^f=\int_0^t f'(B_s)\,dB_s$, so the Brownian path law solves the martingale problem for $(L,\delta_0)$.

Check: the compensating integral in the definition is $\int_0^t Lf(B_s)\,ds=\tfrac12\int_0^t f''(B_s)\,ds$, exactly the correction term in Ito's formula. The initial value is $B_0=0$, as required by $\delta_0$.

What the Formulation Does—and Does Not—Assert

The martingale problem is a specification of a law through a family of tests. The condition must hold for every $f\in C_c^2$, not just for one convenient function. Compact support is useful because it controls the derivatives where the operator acts; the local boundedness assumption ensures $Lf$ is bounded. If coefficients are not locally bounded, one must instead impose appropriate integrability or localization conditions before writing the same definition.

Existence and uniqueness are separate questions. A proposed law can be checked against the initial condition and the test-function martingales to prove existence, as in the examples. Uniqueness asks whether any other law on continuous path space can satisfy all those same tests. The definition alone does not answer that question. Nor does uniqueness in law identify two sample paths on a common probability space: that is a different assertion, called pathwise uniqueness.

A common mistake is to use $Lf$ without the factor $\tfrac12$ in a diffusion generator. For one-dimensional Brownian motion, the incorrect choice $Lf=f''$ would produce the proposed compensation $\int_0^t f''(B_s)\,ds$. Ito's formula instead gives $\tfrac12\int_0^t f''(B_s)\,ds$. The discrepancy is a factor of two, not a change in notation. The correction follows from the second-order term in Ito's formula and the convention $dB_t\,dB_t=dt$.

The martingale problem is valuable because it can be formulated directly for probability laws and operators, without choosing a driving Brownian motion in its definition. The examples show how a candidate law is verified; determining whether all solutions have the same law is the separate uniqueness question.

Check Your Understanding

Use the definition and propositions above to answer the following questions.

For $Lf=b f'+\tfrac12a f''$ in one dimension, which assumptions in this tutorial ensure that $Lf$ is bounded for every $f\in C_c^2(\mathbb{R})$?
State the conditional-expectation identity that is equivalent to the martingale condition between times $s$ and $t$.
If two laws solve martingale problems for the same operator but have different initial distributions, what initial distribution does their mixture have?
For Brownian motion, identify the test-function martingale when $Lf=\tfrac12 f''$, and state why its stochastic integral is a true martingale on a finite time interval.
Explain why uniqueness in law does not assert pathwise uniqueness.

The Stroock-Varadhan Martingale Problem

What You'll Learn