$C^{ 0,1}$ -It{ô} chain rules and generalized solutions of parabolic PDEs

TL;DR

The paper develops two Itô-type chain rules to handle non-regular transformations of stochastic processes: (i) for a finite quadratic variation process $X$ with $f \in C^{0,2}_{ac}$; (ii) for a continuous semimartingale $X$ where $f$ is a quasi-strong solution of a parabolic PDE, using a Fukushima-Dirichlet decomposition. It introduces and analyzes quasi-strict, quasi-strong, and mild solutions for backward parabolic equations, and proves an explicit $C^{0,1}$ chain rule with a concrete martingale-orthogonal term, enabling applications when classical $C^{1,2}$ regularity is unavailable. The framework leverages forward integrals, weak Dirichlet processes, and the Fukushima decomposition to yield explicit representations of non-martingale components, linking PDE solution concepts to stochastic calculus. These results pave the way for verification theorems in stochastic control and differential games even when PDE data and value functions lack high regularity, as illustrated for subsequent applications in companion work.

Abstract

In this paper we first establish an Itô formula for a finite quadratic variation process $X$ expanding $f(t,X_t),$ when $f$ is of class $C^2$ in space and is absolutely continuous in time. Second, via a Fukushima-Dirichlet decomposition we obtain an explicit chain rule for $f(t,X_t)$, when $X$ is a continuous semimartingale and $f$ is a ``quasi-strong solution'' (in the sense of approximation of classical solutions) of a parabolic PDE.

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Proposition 2.7
• Remark 2.8
• Remark 2.9
• proof : Proof of Proposition \ref{['ItoGen']}