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Optimal control of Volterra integral diffusions and application to contract theory

Dylan Possamaï, Mehdi Talbi

TL;DR

The paper develops a Sobolev-space lifting of stochastic Volterra control problems, enabling a DP approach and viscosity-solution characterization of the value function on a Hilbert space. It defines an infinite-dimensional flow with B and Σ operators, proves existence and uniqueness under a Sobolev-structure assumption, and shows how the original Volterra dynamics are recovered from the lifted system. When volatility is uncontrolled, a BSDE representation links the DP equation to backward dynamics, and the framework is applied to time-inconsistent contract theory with explicit reformulations as stochastic-target problems on Hilbert spaces. The authors also provide a Markovian representation and convergence results for finite-dimensional approximations, compare with existing lifting and PDE methodologies, and discuss extensions to singular kernels. The approach yields both theoretical insights and practical approximation schemes for Volterra-type control problems with broad potential applications in economics and finance.

Abstract

This paper focuses on the optimal control of a class of stochastic Volterra integral equations. Here the coefficients are regular and not assumed to be of convolution type. We show that, under mild regularity assumptions, these equations can be lifted in a Sobolev space, whose Hilbertian structure allows us to attack the problem through a dynamic programming approach. We are then able to use the theory of viscosity solutions on Hilbert spaces to characterise the value function of the control problem as the unique solution of a parabolic equation on Sobolev space. We provide applications and examples to illustrate the usefulness of our theory, in particular for a certain class of time inconsistent principal agent problems. As a byproduct of our analysis, we introduce a new Markovian approximation for Volterra type dynamics.

Optimal control of Volterra integral diffusions and application to contract theory

TL;DR

The paper develops a Sobolev-space lifting of stochastic Volterra control problems, enabling a DP approach and viscosity-solution characterization of the value function on a Hilbert space. It defines an infinite-dimensional flow with B and Σ operators, proves existence and uniqueness under a Sobolev-structure assumption, and shows how the original Volterra dynamics are recovered from the lifted system. When volatility is uncontrolled, a BSDE representation links the DP equation to backward dynamics, and the framework is applied to time-inconsistent contract theory with explicit reformulations as stochastic-target problems on Hilbert spaces. The authors also provide a Markovian representation and convergence results for finite-dimensional approximations, compare with existing lifting and PDE methodologies, and discuss extensions to singular kernels. The approach yields both theoretical insights and practical approximation schemes for Volterra-type control problems with broad potential applications in economics and finance.

Abstract

This paper focuses on the optimal control of a class of stochastic Volterra integral equations. Here the coefficients are regular and not assumed to be of convolution type. We show that, under mild regularity assumptions, these equations can be lifted in a Sobolev space, whose Hilbertian structure allows us to attack the problem through a dynamic programming approach. We are then able to use the theory of viscosity solutions on Hilbert spaces to characterise the value function of the control problem as the unique solution of a parabolic equation on Sobolev space. We provide applications and examples to illustrate the usefulness of our theory, in particular for a certain class of time inconsistent principal agent problems. As a byproduct of our analysis, we introduce a new Markovian approximation for Volterra type dynamics.

Paper Structure

This paper contains 23 sections, 11 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. The infinite dimensional problem
  3. Choice of the state space
  4. The infinite dimensional dynamics
  5. The infinite dimensional control problem
  6. Dynamic programming equation
  7. The value function
  8. Viscosity solutions
  9. The case of uncontrolled volatility
  10. Examples
  11. A---very---simple starter
  12. Linear--quadratic control problem with kernel
  13. Time-inconsistent contract theory
  14. The agent's and principal's problems
  15. Reformulation as a control problem with stochastic target constraints
  16. ...and 8 more sections

Key Result

Lemma 1

We have $H \subset {\cal C}$. Moreover, the injection is compact, that is to say that there exists a constant $C \ge 0$ such that where $\tilde{{\mathbf{x}}}$ denotes the continuous representative of ${\mathbf{x}}$.

Theorems & Definitions (34)

  • Lemma 1
  • proof
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 2
  • Remark 3
  • Proposition 2
  • proof
  • ...and 24 more