Uniqueness in Law from a Well-Posed Martingale Problem

From a Generator to a Probability Law

A diffusion operator records how a process changes when tested against smooth functions. The associated martingale problem turns that local description into a condition on the entire path law. When the problem has a solution and that solution is unique for every fixed starting point, the operator identifies a unique diffusion law from each such point. This conclusion is about distributions of paths; it does not say that two solutions driven by the same Brownian motion must coincide.

Fix a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\). For a scalar diffusion with drift \(b\) and diffusion coefficient \(\sigma\), define \(a(x)=\sigma(x)^2\) and \(\mathcal{L}f(x)=b(x)f'(x)+\tfrac12a(x)f''(x)\). Here \(b\) is the drift coefficient and \(\sigma\) is the diffusion coefficient. The natural test class used below is \(C_c^2(\mathbb{R})\), the twice continuously differentiable functions with compact support.

A martingale here is an integrable adapted process \(M_t\) satisfying \(\mathbb{E}[M_t\mid\mathcal{F}_s]=M_s\) for \(s\le t\). This is stronger than being a local martingale, which need only become a martingale after suitable stopping. The martingales in the definition are genuine martingales. Local martingales will arise when coordinate functions are recovered by stopping; they need not be true martingales without additional integrability.

Why Well-Posedness Gives Uniqueness in Law

Proof. Existence in the definition supplies at least one path law \(P_x\). Suppose \(P_x^{(1)}\) and \(P_x^{(2)}\) are laws of two weak solutions from \(x\), and suppose each solves the martingale problem for \(\mathcal{L}\). They are therefore two solutions of that same problem with the same initial state. Uniqueness in the definition gives \(P_x^{(1)}=P_x^{(2)}\). This is equality of probability measures on path space, hence equality of all finite-dimensional distributions and of the distributions of measurable path functionals. Applying the argument separately for each \(x\) proves the assertion for every deterministic starting state. \(\square\)

The link to an SDE comes from Itô’s formula. For example, suppose a weak solution satisfies \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t\), where \(B_t\) is Brownian motion on the filtered space, and the stochastic integral is an Itô integral. For each finite horizon \(T\), its integrand \(H_s=f'(X_s)\sigma(X_s)\) must be progressively measurable and satisfy \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\) for the integral to be square integrable; localization supplies the corresponding local statement when this integrability is only local. Itô’s formula, with the second-derivative correction displayed, gives \[ f(X_t)=f(X_0)+\int_0^t f'(X_s)b(X_s)\,ds +\int_0^t f'(X_s)\sigma(X_s)\,dB_s +\frac12\int_0^t f''(X_s)\sigma(X_s)^2\,ds. \] Thus the finite-variation terms combine to \(\int_0^t\mathcal{L}f(X_s)\,ds\). Under the stated square-integrability condition, the stochastic integral is a martingale: it is integrable by the Itô isometry and has conditional expectation at time \(s\) equal to its value at time \(s\). Consequently the path law solves the martingale problem. The earlier material on weak SDEs and martingale problems gives the converse construction under its hypotheses; the theorem here uses the implication from SDE laws to martingale-problem laws.

Well-posedness does not assert pathwise uniqueness. Pathwise uniqueness compares solutions on the same probability space driven by the same Brownian motion and asks whether their paths agree almost surely. Uniqueness in law compares distributions and does not require a shared probability space or shared noise. Nor does the result assert strong existence, which requires constructing a solution from a prescribed Brownian motion and initial value. All laws in this tutorial are under \(\mathbb{P}\); no risk-neutral measure \(\mathbb{Q}\) or change of measure is involved.

The Brownian Martingale Problem

For standard Brownian motion the operator is \(\mathcal{L}f=\tfrac12f''\). We now establish both existence and uniqueness for the martingale problem starting from any \(x\in\mathbb{R}\).

Proof. For existence, take a filtered probability space carrying standard Brownian motion \(B_t\) and set \(X_t=x+B_t\). For \(f\in C_c^2(\mathbb{R})\), Itô’s formula uses \(f'(X_s)\) as the stochastic-integral integrand, and gives \(f(X_t)=f(x)+\int_0^t f'(X_s)\,dB_s+\tfrac12\int_0^t f''(X_s)\,ds\). Both \(f'\) and \(f''\) are bounded. Thus \(f'(X_s)\) is progressively measurable and \(\mathbb{E}\int_0^T f'(X_s)^2\,ds\le T\|f'\|_\infty^2<\infty\). The integral is a true martingale by the Itô isometry and the martingale property of the Itô integral, so the required martingale problem holds.

For uniqueness, let \(X\) be any continuous coordinate process solving this problem from \(x\). Fix \(R>|x|\), and let \(\tau_R=\inf\{t\ge0:|X_t|\ge R\}\), with the infimum of the empty set taken as infinity. Choose compactly supported \(C^2\) functions that agree with \(y\) and \(y^2\), respectively, on \([-R,R]\). Up to \(\tau_R\), their derivatives give \(\mathcal{L}y=0\) and \(\mathcal{L}(y^2)=1\). Stopping their martingale identities shows that \(X_{t\wedge\tau_R}-x\) and \(X_{t\wedge\tau_R}^2-x^2-(t\wedge\tau_R)\) are martingales. Since paths are continuous, \(\tau_R\) tends to infinity as \(R\) tends to infinity on every path over any fixed bounded time interval. Hence \(M_t=X_t-x\) is a continuous local martingale and \(M_t^2-t\) is a local martingale.

For a continuous local martingale \(M\), the defining quadratic-variation property is that \(M_t^2-[M]_t\) is a local martingale. Subtracting this process from \(M_t^2-t\) shows that \([M]_t-t\) is a continuous finite-variation local martingale starting at zero, and such a process is identically zero. Therefore \([M]_t=t\). Lévy’s characterization states that a continuous local martingale starting at zero with quadratic variation \(t\) is Brownian motion relative to its filtration. Applying it to \(M\) shows that \(X_t-x\) is Brownian motion. Brownian motion has independent \(N(0,t-s)\) increments after time \(s\), so all finite-dimensional distributions of \(X\) are fixed. On continuous path space, the Borel sigma-field is generated by evaluations at rational times; those distributions determine the path law. Thus every solution has the law of \(x+B\), proving uniqueness. \(\square\)

The use of Lévy’s characterization is a standard result: the proof is not repeated here. The localization above is important because \(y\) and \(y^2\) are not compactly supported test functions. It also explains why the coordinate martingales obtained in the argument are initially local martingales, rather than being asserted to be true martingales without a moment bound.

Worked Consequences

Common Pitfall: Uniqueness Is State-by-State

Uniqueness from one starting point does not by itself identify laws from other starting points. The hypothesis “well posed from every state” is a family of separate assertions: existence and uniqueness are required for each \(x\). The Brownian proof verifies this for an arbitrary \(x\), rather than only for \(x=0\).

A related error is to conclude pathwise uniqueness from uniqueness of the martingale problem. The martingale problem identifies the distribution on path space. It does not compare two processes coupled to the same Brownian motion. For example, the theorem’s conclusion is that all Brownian solutions started at \(x\) have the law of \(x+B\); it is not a claim about arbitrary couplings of two solutions. Conversely, if weak SDE solutions from a fixed state all induce the same martingale-problem law, uniqueness in law follows even when no common driving Brownian motion has been specified.