Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Markovian lifting and optimal control for integral stochastic Volterra equations with completely monotone kernels

Stefano Bonaccorsi, Fulvia Confortola

Abstract

In this paper, we focus on solving the optimal control problem for integral stochastic Volterra equations in a finite dimensional setting. In our setting, the noise term is driven by a pure jump Lévy noise and the control acts on the intensity of the jumps. We use recent techniques proposed by Hamaguchi, where a crucial requirement is that the convolution kernel should be a completely monotone function. This allows us to use Bernstein's representation and the machinery of Laplace transform to obtain a Markovian lift. It is natural that the Markovian lift, in whatever form constructed, transforms the state equation into a stochastic differential equation in an infinite-dimensional space. This space should be large enough to contain all the information about the history of the process. Hence, although the original equation is taken in a finite dimensional space, the resulting lift is always infinite dimensional. We solve the problem by using the forward-backward approach in the infinite-dimensional setting and prove the existence of the optimal control for the original problem. Under additional assumptions on the coefficients, we see that a control in closed-loop form can be achieved.

Markovian lifting and optimal control for integral stochastic Volterra equations with completely monotone kernels

Abstract

In this paper, we focus on solving the optimal control problem for integral stochastic Volterra equations in a finite dimensional setting. In our setting, the noise term is driven by a pure jump Lévy noise and the control acts on the intensity of the jumps. We use recent techniques proposed by Hamaguchi, where a crucial requirement is that the convolution kernel should be a completely monotone function. This allows us to use Bernstein's representation and the machinery of Laplace transform to obtain a Markovian lift. It is natural that the Markovian lift, in whatever form constructed, transforms the state equation into a stochastic differential equation in an infinite-dimensional space. This space should be large enough to contain all the information about the history of the process. Hence, although the original equation is taken in a finite dimensional space, the resulting lift is always infinite dimensional. We solve the problem by using the forward-backward approach in the infinite-dimensional setting and prove the existence of the optimal control for the original problem. Under additional assumptions on the coefficients, we see that a control in closed-loop form can be achieved.
Paper Structure (8 sections, 13 theorems, 105 equations)

This paper contains 8 sections, 13 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Scalar kernels
  3. Stochastic Volterra equations in finite dimensions
  4. Noise term
  5. An abstract equation for the lift
  6. Existence of the solution
  7. Existence of the solution of the Volterra equation
  8. The control problem

Key Result

Lemma 3

The operator $B : D(B) \subset \mathcal{H} \to \mathcal{H}$ given by $B y(x) = - x y(x)$ with domain generates a linear contraction semigroup $S(t) = e^{t B}$ on $\mathcal{H}$ given by Moreover, $B$ can be extended to an operator $B : \cV \to \cV^\star$ and for any $y \in \cV$.

Theorems & Definitions (30)

  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 20 more