Free Martingale Problem Overview

• Free martingale problem is a formulation where the process law is defined by martingale identities rather than explicit stochastic equations.
• It replaces traditional pathwise constructions with generator-based, PDE, and duality formulations to ensure existence and uniqueness in various settings.
• The approach spans classical, degenerate, nonlinear, and free probability frameworks, providing versatile models for complex stochastic dynamics.

“Free martingale problem” denotes, in the literature considered here, a family of closely related constructions in which the law of a process is specified through martingale identities rather than through an a priori stochastic integral representation. In the classical Stroock–Varadhan sense, one starts from a generator and asks for a probability law on path space under which suitable test-function transforms are martingales; in abstract formulations, one specifies a class of test processes directly; in nonlinear and probability-free settings, the martingale property is imposed without a single reference probability measure; and in free probability the problem is formulated on a noncommutative state space of tracial states and solved by reconstructing a free Brownian driver from a weak PDE identity (Menozzi, 2014, Simoi et al., 2014, Criens et al., 2021, Guo et al., 2012, Vovk et al., 2017, Ueda, 26 Mar 2026).

1. Classical generator-specified formulation

In the standard framework, one fixes a state space $S$ and a linear operator

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$

thought of as the generator of a Markov process. The martingale problem asks for a probability measure $P_y$ on $C^0([0,T],S)$ such that $P_y(\{z:z(0)=y\})=1$ and, for every test function $A\in\mathcal D(L)$,

$M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$

is a martingale with respect to the natural filtration. This is explicitly described as a “free” characterization because it specifies the law of the process via the martingale property, without constructing it explicitly as a solution of an SDE or a Markov chain (Simoi et al., 2014).

In time-inhomogeneous settings the same idea is expressed through operators $(L_t)_{t\ge 0}$. For $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$, the canonical martingale is

$$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$

and well-posedness means existence and uniqueness of a probability $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$0 solving this condition for each initial state $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$1. In SDE language this is equivalent to weak existence and uniqueness in law for the diffusion generated by $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$2 (Menozzi, 2014).

A further abstraction replaces generators by a collection $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$3 of right-continuous, adapted processes on a fixed filtered measurable space. A probability measure $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$4 is then a solution to the martingale problem $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$5 if all processes in $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$6 are martingales, and a solution to the local martingale problem if all are local martingales. This formulation is explicitly not restricted to Markov processes, semimartingales, or to continuous or càdlàg paths (Criens et al., 2021).

2. Degenerate Kolmogorov equations and PDE-based well-posedness

A central classical realization of the viewpoint is Menozzi’s treatment of degenerate Kolmogorov operators of the form

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$7

where $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$8, the diffusion acts only on the first block, and the drift propagates randomness through a Hörmander-type chain structure. The associated SDE system is

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$9

Under Assumptions $P_y$0, $P_y$1, $P_y$2, and $P_y$3, the paper proves well-posedness of the associated martingale problem and hence uniqueness in law for the SDE (Menozzi, 2014).

The analytic core is a Calderón–Zygmund estimate for a frozen hypoelliptic proxy. After freezing the coefficients along the deterministic backward flow $P_y$4, Menozzi introduces a Gaussian kernel $P_y$5 whose covariance satisfies the multi-scale bound governed by

$P_y$6

The corresponding Green operator $P_y$7 satisfies

$P_y$8

for every $P_y$9 and small enough $C^0([0,T],S)$0 (Menozzi, 2014).

The PDE estimate is then turned into martingale-problem uniqueness by a perturbative argument. Writing the true operator as frozen operator plus remainder $C^0([0,T],S)$1, one shows

$C^0([0,T],S)$2

so that $C^0([0,T],S)$3 is invertible on $C^0([0,T],S)$4 for sufficiently small $C^0([0,T],S)$5 and sufficiently local freezing. The resulting backward equation yields a duality formula for expectations of $C^0([0,T],S)$6, which identifies any two solutions of the martingale problem. Combined with localization in space and chaining in time, this gives the main theorem: under Assumptions $C^0([0,T],S)$7, $C^0([0,T],S)$8, $C^0([0,T],S)$9, and $P_y(\{z:z(0)=y\})=1$0, the martingale problem associated with $P_y(\{z:z(0)=y\})=1$1 is well posed (Menozzi, 2014).

The same work explicitly states that it does not use the phrase “free martingale problem,” but that it provides a good illustration of what one might mean by a “free” or “abstract” martingale problem: the process is posed directly from the generator, uniqueness is proved through PDE duality, and no prior Itô construction, pathwise uniqueness, or Girsanov argument in the degenerate coordinates is required (Menozzi, 2014).

3. Abstract, duality-based, and dynamical-systems formulations

In deterministic fast–slow systems, De Simoi and Liverani use the martingale problem as a limit-identification device. For the averaged dynamics on $P_y(\{z:z(0)=y\})=1$2, the limiting generator is

$P_y(\{z:z(0)=y\})=1$3

where

$P_y(\{z:z(0)=y\})=1$4

and the limiting path $P_y(\{z:z(0)=y\})=1$5 solves

$P_y(\{z:z(0)=y\})=1$6

For fluctuations,

$P_y(\{z:z(0)=y\})=1$7

the limiting time-dependent generator is

$P_y(\{z:z(0)=y\})=1$8

The method shows that every subsequential limit solves the corresponding martingale problem, and uniqueness identifies the limit as the deterministic ODE in the averaging regime and as the linear diffusion in the fluctuation regime (Simoi et al., 2014).

The same paper describes the method as free of explicit stochastic structure: one defines the limiting process by specifying the generator $P_y(\{z:z(0)=y\})=1$9 and checking a martingale property for observables, without realizing the limiting process pathwise ahead of time. In this sense the martingale problem is model-free at the level of probability: once one has a candidate generator, existence and uniqueness for the corresponding martingale problem, and approximate martingale relations for the prelimit processes, convergence in distribution follows (Simoi et al., 2014).

A different generalization is the abstract local-martingale-problem framework of Conforti, De Marco, and collaborators. There the martingale problem is formulated on an arbitrary filtered measurable space, and convergence criteria are proved using weak–strong convergence for joint laws of a control variable and a path variable. This permits treatment of non-Markovian problems, semimartingale-characteristics problems, SPDEs in mild form, Volterra SDEs, and processes with fixed times of discontinuity (Criens et al., 2021).

Duality provides another route to well-posedness. For Polish spaces $A\in\mathcal D(L)$0 and $A\in\mathcal D(L)$1, a bounded continuous duality function $A\in\mathcal D(L)$2, and a dual process $A\in\mathcal D(L)$3 solving a $A\in\mathcal D(L)$4-martingale problem, Hammer, Hutzenthaler, and Kliem show that existence and uniqueness of a $A\in\mathcal D(L)$5-martingale problem follow if the duality relation

$A\in\mathcal D(L)$6

defines a Markov semigroup through probability kernels $A\in\mathcal D(L)$7. Under separation of measures by $A\in\mathcal D(L)$8, the resulting martingale problem is well posed (Depperschmidt et al., 2019).

4. Nonlinear and probability-free martingale problems

A nonlinear version replaces linear expectation by a nonlinear expectation space $A\in\mathcal D(L)$9. In this setting, for a nonlinear generator

$M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$0

belonging to the admissible class $M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$1, the martingale problem asks that for each $M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$2,

$M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$3

be an $M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$4-martingale. The associated PDE is fully nonlinear,

$M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$5

and the paper proves an existence theorem for the nonlinear martingale problem as well as the existence of weak solutions to $M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$6-SDEs under Hölder continuity of the coefficients (Guo et al., 2012).

The same work emphasizes the difference from the classical Stroock–Varadhan picture: the generator is no longer linear, the expectation is nonlinear and dominated by a sublinear expectation $M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$7, and the associated change-of-measure argument is a generalized Girsanov transformation that no longer requires the two probability measures to be absolutely continuous. The process is thus specified by a nonlinear generator and a nonlinear expectation rather than by a single reference probability measure (Guo et al., 2012).

A more radical reformulation removes probability measures entirely. In the probability-free theory of continuous martingales, the sample space is

$M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$8

there is no probability measure $M_t^A:=A(z(t))-A(z(0))-\int_0^t LA(z(s))\,ds$9, and the primitive objects are simple trading strategies and their capital processes. A nonnegative supermartingale is the smallest $(L_t)_{t\ge 0}$0-closed class containing all nonnegative simple capital processes, and a continuous martingale is the smallest $(L_t)_{t\ge 0}$1-closed class containing all simple capital processes. Within this framework one obtains probability-free versions of the Dubins–Schwarz theorem, the Girsanov theorem, the Itô integral, Itô’s formula, and a semimartingale decomposition (Vovk et al., 2017).

This probability-free construction is described as addressing a “free martingale problem” in the sense that martingale and semimartingale structure are defined directly on path space via trading and limiting operations, before any stochastic model is chosen. Under any compatible probability measure making the basic price paths continuous local martingales, the probability-free martingales embed into the classical theory (Vovk et al., 2017).

5. Free-probabilistic formulation for unitary Brownian motion

In free probability, the phrase is used literally. Ueda formulates and solves a “free martingale problem” for the unitary Brownian motion and the matrix liberation process. The basic state space is the universal $(L_t)_{t\ge 0}$2-algebra

$(L_t)_{t\ge 0}$3

generated by selfadjoint variables $(L_t)_{t\ge 0}$4 and unitary process coordinates $(L_t)_{t\ge 0}$5. A continuous tracial state $(L_t)_{t\ge 0}$6 on this algebra determines a filtered noncommutative probability space through its GNS representation, with conditional expectations $(L_t)_{t\ge 0}$7 onto the von Neumann subalgebras generated by the coordinates up to time $(L_t)_{t\ge 0}$8 (Ueda, 26 Mar 2026).

The generator-like objects are derivations

$(L_t)_{t\ge 0}$9

and for an adapted selfadjoint process $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$0 one defines

$$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$1

If the large-deviation rate function $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$2 is finite, there exists a unique adapted $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$3 such that

$$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$4

and, for every $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$5 and $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$6,

$$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$7

This identity is interpreted as a noncommutative weak PDE and as the martingale-problem formulation of the process (Ueda, 26 Mar 2026).

The reconstruction theorem is the distinctive feature. Setting $$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$8 and

$$\varphi\in C_0^{1,2}([0,\infty)\times \mathbb R^{nd})$$9

one proves that each $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$0 is a martingale in $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$1. Then one defines

$$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$2

and shows that $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$3 is an $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$4-dimensional free Brownian motion. Finally the canonical unitaries satisfy the free SDE

$$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$5

In this way the weak PDE or martingale formulation is shown to be equivalent to the existence of a genuine free stochastic process driven by free Brownian motion (Ueda, 26 Mar 2026).

The same program is carried over to the matrix liberation process. The rate function again becomes a quadratic $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$6-energy of a drift, and the associated weak PDE is solved by embedding the liberation coordinates into the unitary setting and transferring the free SDE construction (Ueda, 26 Mar 2026).

6. Related optimization and field-theoretic interpretations

A distinct, but related, use of the expression appears in martingale optimal transport. Given marginal laws $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$7 on $$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$8, the martingale transport problem minimizes

$$M_t^\varphi :=\varphi(t,X_t)-\varphi(s,X_s)-\int_s^t(\partial_u+L_u)\varphi(u,X_u)\,du,$$9

over couplings $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$00 with marginals $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$01 and martingale constraint $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$02. Equivalently,

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$03

for all bounded measurable $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$04. Existence of admissible martingale couplings is equivalent to convex order $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$05, and the paper interprets this as what one might call a “free martingale problem”: the marginals are fixed, but the martingale coupling is otherwise free and must be selected by an optimization principle (Beiglböck et al., 2012).

The dynamic analogue is the martingale Benamou–Brenier problem. For $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$06 in convex order, one considers continuous martingales

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$07

and maximizes

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$08

The problem has a unique-in-law optimizer $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$09, called stretched Brownian motion, and the corresponding interpolation

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$10

is time-consistent. This is a free-martingale viewpoint in which the generator is not fixed in advance; instead, one optimizes over all martingale laws with prescribed marginals (Backhoff-Veraguas et al., 2017).

Infinite-dimensional Gaussian fields provide another neighboring perspective. The Gaussian free field and fractional Gaussian free fields are characterized through resampling dynamics that converge to the stochastic heat equation and fractional stochastic heat equation. For each test function $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$11, the limiting coordinate process satisfies a martingale problem of the form

$$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$12

for the GFF, or with $$L:\mathcal D(L)\subset C^0(S,\mathbb R)\to C^0(S,\mathbb R),$$13 in the fractional case, with Gaussian quadratic variation determined by the noise coefficient. The field is then identified as the unique stationary solution of the corresponding martingale problem (Aru et al., 2024).

Taken together, these developments show that the “free martingale problem” can refer to at least three technical patterns present in current research: a generator-first characterization of a law without prior pathwise construction; a probability-free or nonlinear reformulation in which the martingale structure precedes the choice of a model; and a noncommutative free-probabilistic problem in which tracial states, derivations, and free Brownian motion replace classical path-space objects. The common thread is that martingale identities, rather than explicit stochastic equations, are treated as the primary data from which stochastic dynamics are recovered (Simoi et al., 2014, Vovk et al., 2017, Guo et al., 2012, Ueda, 26 Mar 2026).