Volterra Martingale Problem

• The Volterra martingale problem is a framework for stochastic Volterra equations characterized by memory effects and nonlocal dependencies.
• It replaces traditional generator-based methods with a coupled semimartingale and Volterra reconstruction approach to capture instantaneous dynamics.
• Existence, uniqueness, and stability are ensured under integrability and Lipschitz conditions on kernels and coefficients.

The Volterra martingale problem is a generalization of the classical martingale problem framework to stochastic Volterra equations (SVEs), which encompass integral and integro-differential equations with time-inhomogeneous or history-dependent coefficients. Unlike classical Markovian SDEs, SVEs feature nonlocal dependencies encoded by kernels, leading to intricate issues of existence, uniqueness, and weak solution formulation. The Volterra martingale problem replaces the standard generator-based martingale approach by a system that couples semimartingale “instantaneous” dynamics with a Volterra-type reconstruction constraint, thus capturing the evolution of non-Markovian systems including equations with jumps, general or singular kernels, and multi-dimensional noise.

1. Volterra Martingale Problem: Core Definitions

The Volterra martingale problem is formulated for stochastic Volterra equations of the general form: $$X_t = x_0(t) + \int_0^t K_\mu(s,t)\,\mu(s,X_s)\,ds + \int_0^t K_\sigma(s,t)\,\sigma(s,X_s)\,dB_s,$$ where $X$ is the state process with a continuous initial path $x_0$, $\mu, \sigma$ are possibly nonlinear drift and diffusion coefficients, and $K_\mu, K_\sigma$ are measurable kernels on the triangular domain $\Delta_T = \{(s,t) : 0 \leq s \leq t \leq T\}$. The Volterra martingale problem introduces an auxiliary process $Z$ which admits a semimartingale decomposition, and defines a system:

• For each $f$ in an appropriate test class (e.g., $C_0^2$), the process

$$\mathcal{M}^f_t := f(Z_t) - \int_0^t \mathcal{A}^f(s, X_s, Z_s)\,ds$$

is a local martingale, where $\mathcal{A}^f$ is a local Volterra generator.

• The Volterra reconstruction identity bridges the process $X$ and $Z$:

$$X_t = x_0(t) + \int_0^t K_\mu(s,t)\,dA_s + \int_0^t K_\sigma(s,t)\,dM_s,$$

with $Z = A + M$ the canonical finite variation and local martingale decomposition of $Z$ (Prömel et al., 2022).

In convolution-type SVEs or those with Lévy noise, the martingale problem is equivalently defined via an operator $A$ acting on the semimartingale characteristics and test functions, with a constraint ensuring that $X$ and $Z$ are coupled via the Volterra structure (Jaber et al., 2019). For more intricate dynamics, such as backward doubly stochastic Volterra integral equations (BDSVIEs), yet another variant of the martingale problem, the “symmetrical martingale solution,” appears (Wen et al., 2019).

2. Main Results: Existence, Uniqueness, and A Priori Bounds

Existence of weak solutions to the Volterra martingale problem is established under minimal integrability and growth conditions on kernels and coefficients:

• If $K_\mu(\cdot, t) \in L^1$, $K_\sigma(\cdot, t) \in L^2$ for each $t$, and
• $|\mu(t, x)| + |\sigma(t, x)| \leq C(1 + |x|)$ (linear growth),

then the SVE admits a weak solution, equivalently a solution to the associated local martingale problem (Prömel et al., 2022). Additional kernel regularity (e.g., uniform Hölder-type continuity in $t$, absolute continuity or convolution structure for $K_\sigma$) enables tightness arguments yielding compactness and convergence of solution approximations (Prömel et al., 2022). In the presence of jumps, similar existence and stability theorems hold under appropriate p-integrability and Sobolev–Slobodeckij regularity conditions (Jaber et al., 2019).

Uniqueness is typically associated with global Lipschitz assumptions on the coefficients and suitable regularity of kernels; pathwise uniqueness implies uniqueness in law and strong existence via abstract results of Kurtz (Jaber et al., 2019).

A priori moment and regularity estimates, such as

$\E\left[\|X\|_{W^{\eta, p}(0,T)}^p\right] < \infty$

for the solution $X$, are established by estimates on the kernels and coefficients, leading to tightness results useful in weak convergence and stability (Jaber et al., 2019).

3. Symmetrical Martingale Solutions in Backward DS Volterra Equations

BDSVIEs require a refined notion: the symmetrical martingale solution (SM-solution). Considering the backward (in time) Itô integral and the coupled filtration generated by independent forward and backward Brownian motions, a pair $(Y, Z)$ is an SM-solution if:

• The equation

$$Y(t) = v(t) + \int_t^T f(t, s, Y(s), Z(t, s), Z(s, t))\,ds + \int_t^T g(t, s, Y(s), Z(t, s), Z(s, t))\,d\overleftarrow{B}(s) - \int_t^T Z(t,s)\,dW(s)$$

holds,

• The kernel $Z$ on the domain where $s < t$ is specified via a forward/backward martingale representation with a symmetry constraint $Z(t, s) = Z(s, t)^T$.

Under Lipschitz/growth conditions (labeled (H3)), including a contraction property in the backward noise parameter (with contraction coefficient $\alpha < 1$), the BDSVIE admits a unique SM-solution, together with explicit a priori estimates (Wen et al., 2019). The proof proceeds via Picard iteration and the Banach fixed point theorem on an appropriate Hilbert space.

4. Connection to Classical Martingale Problems

The Stroock–Varadhan martingale problem for SDEs characterizes Markovian diffusions via a generator and local martingale property on functionals of the process. In the Volterra setting, key differences emerge:

• Non-Markovianity: The law of $X_t$ depends on the full past through the kernels.
• The introduction of an auxiliary semimartingale $Z$ to capture the instantaneous drift/diffusion dynamics, with $X$ reconstructed via integral Volterra constraints.
• An additional identification condition coupling $X$ and $Z$, absent in the Markovian case, encapsulates memory effects and allows for singular or time-inhomogeneous kernels (Prömel et al., 2022, Jaber et al., 2019).

In the limit when $K_\mu = K_\sigma = 1$, the Volterra martingale problem collapses to the classical martingale problem. The SM-solution extends previous M- and S-solution paradigms, unifying existence and uniqueness statements for a broader class of non-Markovian equations (Wen et al., 2019).

5. Approximation, Stability, and Scaling Limits

The Volterra martingale problem framework naturally leads to generic approximation and stability results:

• If sequences of kernels, initial conditions, and coefficient operators converge in suitable $L^p$ or locally uniform senses, solutions to the corresponding martingale problems converge weakly to those of the limit problem (Jaber et al., 2019).
• Applications include scaling limits for non-linear Hawkes processes: Under suitable rescaling of parameters and intensity, solutions to discrete systems converge in law to the SVE characterized by the Volterra martingale problem (Jaber et al., 2019).
• Approximation by Markovian semimartingales: Singular or rough Volterra kernels can be approximated by exponentially weighted sums, with limit points in the Skorokhod or $L^p$ topology again solving the original non-Markovian problem.

These results extend the domain of classical Itô theory to encompass a wide range of practical stochastic systems with memory and path-dependence.

6. Special Cases, Applications, and Open Problems

When certain coefficients vanish or are independent of history (e.g., $g \equiv 0$ or $f, g$ linear in the BDSVIE setting), the Volterra martingale problem reduces to classical or semi-classical cases with explicit solution structures and stability properties (Wen et al., 2019). The Volterra martingale approach supports the rigorous analysis of models in finance (insider models, rough volatility), risk management, and stochastic control, and allows for extensions to mean-field and systems with jumps or quasilinear SPDEs via Feynman–Kac formulas (Wen et al., 2019).

Open problems remain in the areas of path regularity, Malliavin differentiability, and numerical analysis of SVEs and BDSVIEs, as well as extension to more general classes of kernels and multi-dimensional noise (Wen et al., 2019).

7. Summary Table: Variants of the Volterra Martingale Problem

Variant	Key Features	References
Standard SVE Martingale Problem	Auxiliary semimartingale $Z$, generator $\mathcal{A}$, Volterra reconstruction constraint	(Prömel et al., 2022)
Convolution/Jump Case	Lévy noise, operator $A$ with jump component, Sobolev–Slobodeckij regularity	(Jaber et al., 2019)
BDSVIE/SM-Solution	Forward/backward Brownian motion, symmetric kernel, two-parameter martingale representation	(Wen et al., 2019)

These frameworks systematically extend the martingale problem paradigm to non-Markovian Volterra equations, offering a unified route for establishing existence, uniqueness, regularity, and convergence of weak solutions under broad classes of kernel and coefficient data.