# Probabilistic interpretation of HJB equations by the representation theorem for generators of BSDEs

## TL;DR

This paper introduces a new probabilistic interpretation of HJB equations in stochastic control by leveraging the representation theorem for BSDE generators, offering broader applicability than existing methods.

## Contribution

It presents a novel approach for interpreting HJB equations using the representation theorem, expanding the applicability in stochastic recursive control problems.

## Key findings

- The approach effectively transmits signs between solutions and generators.
- It demonstrates broader applicability compared to previous methods.
- Provides a new application of the representation theorem for BSDEs.

## Abstract

The purpose of this note is to propose a new approach for the probabilistic interpretation of Hamilton-Jacobi-Bellman equations associated with stochastic recursive optimal control problems, utilizing the representation theorem for generators of backward stochastic differential equations. The key idea of our approach for proving this interpretation consists of transmitting the signs between the solution and generator via the identity given by representation theorem. Compared with existing methods, our approach seems to be more applicable for general settings. This can also be regarded as a new application of such representation theorem.