Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of optimal controls for stochastic Volterra equations

Andrés Cárdenas, Sergio Pulido, Rafael Serrano

Abstract

We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of stochastic control problems for stochastic Volterra equations (SVEs). Our study can be applied to rough processes that arise when the kernel appearing in the controlled SVE is singular at zero. The existence of relaxed optimal policies relies on the interaction between integrability hypotheses on the kernel and growth conditions on the running cost functional and the coefficients of the controlled SVEs. Under classical convexity assumptions, we can also deduce the existence of optimal strict controls.

Existence of optimal controls for stochastic Volterra equations

Abstract

We provide sufficient conditions that guarantee the existence of relaxed optimal controls in the weak formulation of stochastic control problems for stochastic Volterra equations (SVEs). Our study can be applied to rough processes that arise when the kernel appearing in the controlled SVE is singular at zero. The existence of relaxed optimal policies relies on the interaction between integrability hypotheses on the kernel and growth conditions on the running cost functional and the coefficients of the controlled SVEs. Under classical convexity assumptions, we can also deduce the existence of optimal strict controls.
Paper Structure (12 sections, 16 theorems, 122 equations)

This paper contains 12 sections, 16 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Controlled stochastic Volterra equations (CSVEs)
  3. Relaxed control formulation
  4. Weak formulation of optimal control problem
  5. Main existence result
  6. Existence of strict controls
  7. Proofs of the main theorems
  8. Relaxed controls and Young measures
  9. Proof of Theorem \ref{['thm-main']}
  10. Proof of Theorem \ref{['thm-strict']}
  11. Auxiliary results
  12. Relative compactness and limit theorems for Young measures

Key Result

Theorem 2.1

Suppose that Assumption Assum1 holds and that $K\in L^{r}_{\rm loc}(\mathds{R}_+;\mathds{R}^{d\times d})$ for some $r>2.$ Let $(u_t)_{t\in[0,T]}$ be a $U$-valued adapted control process such that for some $p$ satisfying $\frac{1}{p}+\frac{1}{r}<\frac{1}{2}.$ Let $X$ be a $\mathds{R}^d$-valued solution to the controlled equation (EVC) with initial condition $x_0\in\mathcal{C}(0,T;\mathds{R}^d).$ T

Theorems & Definitions (40)

  • Theorem 2.1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Definition 3.2
  • ...and 30 more