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An Itô-type formula for some measure-valued processes and its application on controlled superprocesses

Shang Li

Abstract

We derive an Itô-type formula for a measure-valued process that has a decomposition analogous to a classical semimartingale. The derivation begins with a time partitioning approach similar to the classical proof of Itô's formula. To address the new challenges arising from the measure-valued setting, we employ symmetric polynomials to approximate the second-order linear derivative of the functional on finite measures, alongside certain localization techniques. A controlled superprocess with a binary branching mechanism can be interpreted as a weak solution to a controlled stochastic partial differential equation (SPDE), which naturally leads to such a decomposition. Consequently, this Itô-type formula makes it possible to derive the Hamilton-Jacobi-Bellman (HJB) equation and the verification theorem for controlled superprocesses with a binary branching mechanism. Additionally, we propose a heuristic definition for the viscosity solution of an equation involving derivatives on finite measures. We prove that a continuous value function is a viscosity solution in this sense and demonstrate the uniqueness of the viscosity solution when the second-order derivative term on the measure vanishes.

An Itô-type formula for some measure-valued processes and its application on controlled superprocesses

Abstract

We derive an Itô-type formula for a measure-valued process that has a decomposition analogous to a classical semimartingale. The derivation begins with a time partitioning approach similar to the classical proof of Itô's formula. To address the new challenges arising from the measure-valued setting, we employ symmetric polynomials to approximate the second-order linear derivative of the functional on finite measures, alongside certain localization techniques. A controlled superprocess with a binary branching mechanism can be interpreted as a weak solution to a controlled stochastic partial differential equation (SPDE), which naturally leads to such a decomposition. Consequently, this Itô-type formula makes it possible to derive the Hamilton-Jacobi-Bellman (HJB) equation and the verification theorem for controlled superprocesses with a binary branching mechanism. Additionally, we propose a heuristic definition for the viscosity solution of an equation involving derivatives on finite measures. We prove that a continuous value function is a viscosity solution in this sense and demonstrate the uniqueness of the viscosity solution when the second-order derivative term on the measure vanishes.

Paper Structure

This paper contains 13 sections, 17 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Space of finite measures
  4. Linear functional derivative
  5. Martingale measures
  6. Controlled superprocesses
  7. Itô's formula for measure-valued processes
  8. Itô's formula for controlled superprocesses
  9. proof of Theorem \ref{['theorem1']}
  10. Some corollaries
  11. HJB equation and verification theorem for controlled superprocesses
  12. Viscosity solutions for some measure-valued control problem
  13. Conclusion

Key Result

Proposition 2.5

For a function $f\in\mathscr{P}_{\mathcal{M}}$, $f\cdot \mathcal{M}$ is a martingale measure with density measure $f(x,t)^2 Q(dx dt).$

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 22 more