Stochastic Perron’s Method

• Stochastic Perron’s Method is a probabilistic extension of classical Perron’s method that uses martingale criteria to define viscosity solutions for parabolic PDEs.
• It constructs stochastic sub- and super-solutions along sample paths of SDEs, circumventing the need for smooth value functions.
• The method ensures uniqueness via a viscosity comparison principle, supporting practical verification in stochastic control, game theory, and related applications.

Stochastic Perron's Method is a probabilistic extension of the classical Perron method, providing a robust framework for constructing viscosity solutions to (typically linear) parabolic partial differential equations associated with stochastic differential equations, without requiring smoothness of the value function or the expected payoff. Its central insight is the replacement of analytic sub- and supersolutions with stochastic counterparts defined in terms of martingale and supermartingale properties along solutions of an SDE, enabling comparison with the expected payoff and establishing uniqueness via viscosity comparison principles (Bayraktar et al., 2011).

1. Classical Perron’s Method and its Stochastic Analogue

The classical Perron method is an analytic technique in which the solution to a PDE is constructed as the supremum of subsolutions (or infimum of supersolutions), with regularity and boundary/terminal condition constraints. However, in stochastic or control-theoretic settings, the value function associated with an SDE-driven optimization problem is rarely smooth and may even fail to have a classical solution. The stochastic analogue introduced by Bayraktar and Sîrbu (Bayraktar et al., 2011) transfers the mechanism of Perron’s method from analytic to probabilistic objects by defining sub- and supersolutions in terms of their behavior along sample paths of SDEs. These are:

• Stochastic sub-solutions: Lower semicontinuous, bounded functions $u$ with $u(T,x)\leq g(x)$ such that, for every (possibly non-unique) weak solution $X^{s,x}$ of the SDE,

$\left(u(t, X^{s,x}_t)\right)_{s \leq t \leq T}$

is a submartingale.

• Stochastic super-solutions: Upper semicontinuous, bounded functions $u$ with $u(T,x)\geq g(x)$ such that the analogous process is a supermartingale.

This martingale-based definition fundamentally leverages the probabilistic representation of solutions via the Feynman–Kac formula, even in settings without Markovianity or uniqueness in law.

2. Construction of Viscosity Solutions for Parabolic PDEs

For the linear parabolic Cauchy problem

$$\begin{cases} -u_t(t,x) - L_t u(t,x) = 0, & (t,x) \in [0,T)\times\mathbb{R}^d,\ u(T, x) = g(x), & x \in \mathbb{R}^d, \end{cases}$$

where

$$L_t u(x) = \langle b(t,x), \nabla u(t,x) \rangle + \frac{1}{2} \operatorname{Tr}\left(\sigma(t,x)\sigma^T(t,x) u_{xx}(t,x)\right),$$

the stochastic Perron procedure defines

$$v^-(t,x) := \sup_{u\in \mathcal{U}^-} u(t,x), \qquad v^+(t,x) := \inf_{u\in \mathcal{U}^+} u(t,x),$$

where $\mathcal{U}^-$ and $\mathcal{U}^+$ are the above classes of stochastic sub- and super-solutions.

The stochastic Perron theorem (Theorem 2.5 in (Bayraktar et al., 2011)) shows that, assuming the payoff function $g$ is bounded and semicontinuous, $v^-$ is a viscosity supersolution and $v^+$ is a viscosity subsolution for the PDE, with terminal conditions derived from the respective semicontinuity. These envelopes are constructed directly via martingale properties, and their viscosity character follows from stochastic–analytic switching arguments adapted to the lack of right-continuity.

3. Viscosity Comparison Principle and Uniqueness

A viscosity comparison principle, denoted $CP(T,g)$, is assumed for the PDE: if $u$ (bounded, USC viscosity subsolution) and $v$ (bounded, LSC viscosity supersolution) satisfy $u(T,x)\leq g(x)\leq v(T,x)$, then $u\leq v$ everywhere. If $CP(T,g)$ holds and $g$ is continuous and bounded, then uniqueness follows:

$v^-(t,x) = v^+(t,x) = v^*(t,x) = v(t,x),$

where $v$ is the unique continuous viscosity solution and $v^*$ denotes the continuous envelope.

A crucial aspect is that this unique solution $v$ coincides with the expected payoff—i.e., the probabilistic value function—thereby cementing the equivalence between the probabilistically defined martingale solution and the analytic solution emerging from viscosity techniques. Moreover, $v\big(t,X^{s,x}_t\big)$ is a martingale for any weak solution $X^{s,x}$ of the SDE, extending the standard verification property known from Itô's Lemma to non-smooth settings.

4. Verification Without Smoothness and the Martingale Property

Traditional verification arguments rely on the availability of a classical solution $u$ and the application of Itô's formula:

$$u(t, X^{s,x}_t) = u(s,x) + \int_s^t \left(u_t + L_t u\right)(r, X^{s,x}_r) dr + \text{martingale}.$$

When $u_t + L_t u = 0$, the process is a martingale. Stochastic Perron's method generalizes this principle: even if the constructed viscosity solution is not differentiable, it is a martingale along the SDE. This is a consequence of the martingale characterization present in the definitions of $\mathcal{U}^+$ and $\mathcal{U}^-$ and does not require prior identification of regularity properties of the value function.

Thus, uniqueness via the comparison principle, combined with the squeezing by stochastic semi-solutions, provides a path to verification (the identification of the value function with the viscosity solution) in complete absence of smoothness.

5. Trade-offs, Implementation, and Robustness

Stochastic Perron's method is robust to weak assumptions. Notably, it:

• Does not require the SDE to be Markovian, nor uniqueness in law.
• Requires no direct proof that the value function is a viscosity solution—this property emerges from the construction.
• Relies on the existence of sufficiently “rich” families of stochastic sub- and super-solutions; these are readily available via the probabilistic (martingale/submartingale) property.
• Is applicable when the comparison principle holds, which is generally true under structural PDE conditions (monotonicity, proper bounds on coefficients).

The primary technical requirement is verifying comparison, which is standard in much of viscosity solution theory but may fail, for example, in degenerate or non-proper settings.

6. Applications, Extensions, and Future Perspectives

The stochastic Perron method provides a foundational tool for the broader program of verification in stochastic analysis, stochastic control, and game theory. Its transparent martingale-based definitions have facilitated extensions to nonlinear settings (e.g., obstacle problems, Dynkin games (Bayraktar et al., 2011), HJB equations in stochastic control (Bayraktar et al., 2012), Isaacs equations in zero-sum differential games (Sîrbu, 2013), and problems with transaction costs (Bayraktar et al., 2014)) and have influenced subsequent developments in the theory of viscosity solutions for non-classical or non-Markovian problems.

Open avenues include:

• Extension to fully nonlinear second-order PDEs under minimal regularity.
• Broadening to exit time, infinite horizon, and state constraint problems.
• Further development of localized stopping time/switching arguments for generality and extension to non-right-continuous paths.

7. Summary Table: Core Constructs

Object/Definition	Role in Method	Viscosity Property/Result
$\mathcal{U}^+$ (super-solutions)	Bounded USC, $u(T,x)\geq g(x)$, supermartingale on SDE	Viscosity subsolution of PDE
$\mathcal{U}^-$ (sub-solutions)	Bounded LSC, $u(T,x)\leq g(x)$, submartingale on SDE	Viscosity supersolution of PDE
$v^-$	Supremum over $\mathcal{U}^-$	Viscosity supersolution
$v^+$	Infimum over $\mathcal{U}^+$	Viscosity subsolution
Comparison Principle $CP(T,g)$	If satisfied, ensures $v^-\leq v^+$ everyehere	Uniqueness, $v^-=v^+=v$
Martingale property for $v$	$v(t, X_t)$ is martingale along any SDE solution	Non-smooth extension of Itô’s lemma

This methodology bridges analytic viscosity theory and probabilistic representations, forming a foundation for confirmation and construction of value functions in a range of stochastic optimization and control frameworks, particularly in the absence of regularity and without recourse to the dynamic programming principle.