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Fulvia Confortola

Publications and source records attributed to Fulvia Confortola.

16 recordsLinked to original sources
arXiv2026

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

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.

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arXiv2026

Partially observed controlled Markov chains and optimal control of the Wonham filter

We consider a class of optimal control problems, with finite or infinite horizon, for a continuous-time Markov chain with finite state space. In this case, the control process affects the transition rates. We suppose that the controlled process can not be observed, and at any time the control actions are chosen based on the observation of a related stochastic process perturbed by an exogenous Brownian motion. We describe a construction of the controlled Markov chain, having stochastic transition rates adapted to the observation filtration. By a change of probability measure of Girsanov type, we introduce the so-called separated optimal control problem, where the state is the conditional (unnormalized) distribution of the controlled Markov chain and the observation process becomes a driving Brownian motion, and we prove the equivalence with the original control problem. The controlled equations for the separated problem are an instance of the Wonham filtering equations. Next we present an analysis of the separated problem: we characterize the value function as the unique viscosity solution to the dynamic programming equations (both in the parabolic and the elliptic case) we prove verifications theorems and a version of the stochastic maximum principle in the form of a necessary conditions for optimality.

math.OC↗
arXiv2021

Progressively Enlargement of Filtrations and Control Problems for Step Processes

In the present paper we address stochastic optimal control problems for a step process $(X,\mathbb{F})$ under a progressive enlargement of the filtration. The global information is obtained adding to the reference filtration $\mathbb{F}$ the point process $H=1_{[τ,+\infty)}$. Here $τ$ is a random time that can be regarded as the occurrence time of an external shock event. We study two classes of control problems, over $[0,T]$ and over the random horizon $[0,T \wedge τ]$. We solve these control problems following a dynamical approach based on a class of BSDEs driven by the jump measure $μ^ Z$ of the semimartingale $Z=(X,H)$, which is a step process with respect to the enlarged filtration $\mathbb G$. The BSDEs that we consider can be solved in $\mathbb{G}$ thanks to a martingale representation theorem which we also establish here. To solve the BSDEs and the control problems we need to ensure that $Z$ is quasi-left continuous in the enlarged filtration $\mathbb{G}$. Therefore, in addition to the $\mathbb{F}$-quasi left continuity of $X$, we assume some further conditions on $τ$: the {\it avoidance} of $\mathbb{F}$-stopping times and the {\it immersion} property, or alternatively {\it Jacod's absolutely continuity} hypothesis.

math.PR↗
arXiv2018

BSDE Representation and Randomized Dynamic Programming Principle for Stochastic Control Problems of Infinite-Dimensional Jump-Diffusions

We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.

math.PR↗
arXiv2016

$L^p$ solution of backward stochastic differential equations driven by a marked point process

We obtain existence and uniqueness in L^p, p>1 of the solutions of a backward stochastic differential equations (BSDEs for short) driven by a marked point process, on a bounded interval. We show that the solution of the BSDE can be approximated by a finite system of deterministic differential equations. As application we address an optimal control problems for point processes of general non-Markovian type and show that BSDEs can be used to prove existence of an optimal control and to represent the value function.

math.PR↗
arXiv2016

Linear-quadratic optimal control under non-Markovian switching

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which a backward stochastic differential equation driven by the Bronwian motion and by the random measure associated to the marked point process.

math.OC↗
arXiv2016

Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.

math.PR↗
arXiv2015

Optimal control of semi-Markov processes with a backward stochastic differential equations approach

In the present work we employ, for the first time, backward stochastic differential equations (BSDEs) to study the optimal control of semi-Markov processes on finite horizon, with general state and action spaces. More precisely, we prove that the value function and the optimal control law can be represented by means of the solution of a class of BSDEs driven by a semi-Markov process or, equivalently, by the associated random measure. The peculiarity of the semi-Markov framework, with respect to the pure jump Markov case, consists in the proof of the relation between BSDE and optimal control problem. This is done, as usual, via the Hamilton-Jacobi-Bellman (HJB) equation, which however in the semi-Markov case is characterized by an additional differential term $\partial_a$. Taking into account the particular structure of semi-Markov processes we rewrite the HJB equation in a suitable integral form which involves a directional derivative operator $D$ related to $\partial_a$. Then, using a formula of Ito type tailor-made for semi-Markov processes and the operator $D$, we are able to prove that the BSDE provides the unique classical solution to the HJB equation, which is shown to be the value function of our control problem.

math.OC↗
arXiv2013

Backward stochastic differential equations associated to jump Markov processes and applications

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space K. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K, that generalize the Kolmogorov equation of X. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton-Jacobi-Bellman equation and to identify it with the value function.

math.PR↗
arXiv2012

Backward stochastic differential equations and optimal control of marked point processes

We study a class of backward stochastic differential equations (BSDEs) driven by a random measure or, equivalently, by a marked point process. Under appropriate assumptions we prove well-posedness and continuous dependence of the solution on the data. We next address optimal control problems for point processes of general non-markovian type and show that BSDEs can be used to prove existence of an optimal control and to represent the value function. Finally we introduce a Hamilton-Jacobi-Bellman equation, also stochastic and of backward type, for this class of control problems: when the state space is finite or countable we show that it admits a unique solution which identifies the (random) value function and can be represented by means of the BSDEs introduced above.

math.PR↗
arXiv2011

Feedback Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

In this paper we are concerned with a class of stochastic Volterra integro-differential problems with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We start by reformulating the state equation into a semilinear evolution equation which can be treated by semigroup methods. The application to optimal control provide other interesting result and require a precise descriprion of the properties of the generated semigroup. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The main technical point consists in the differentiability of the BSDE associated with the reformulated equation with respect to its initial datum x.

math.PR↗
arXiv2010

Filtering of continuous-time Markov chains with noise-free observation and applications

Let X be a continuous-time Markov chain in a finite set I, let h be a mapping of I onto another set, and let Y be defined by Y_t=h(X_t), (for t nonnegative). We address the filtering problem for X in terms of the observation Y, which is not directly affected by noise. We write down explicit equations for the filtering process and show that this is a Markov process with the Feller property. We also prove that it is a piecewise-deterministic Markov process in the sense of Davis, and we identify its characteristics explicitly. We finally solve an optimal stopping problem for X with partial observation, i.e. where the moment of stopping is required to be a stopping time with respect to the natural filtration of Y.

math.PR↗
arXiv2007

Quadratic BSDEs with random terminal time and elliptic PDEs in infinite dimension

In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F(t,Y,Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces.

math.PR↗
arXiv2007

Dissipative backward stochastic differential equations with locally Lipschitz nonlinearity

In this paper we study a class of backward stochastic differential equations (BSDEs) of the form dY(t)= -AY(t)dt -f_0(t,Y(t))dt -f_1(t,Y(t),Z(t))dt + Z(t)dW(t) on the interval [0,T], with given final condition at time T, in an infinite dimensional Hilbert space H. The unbounded operator A is sectorial and dissipative and the nonlinearity f_0(t,y) is dissipative and defined for y only taking values in a subspace of H. A typical example is provided by the so-called polynomial nonlinearities. Applications are given to stochastic partial differential equations and spin systems.

math.PR↗
arXiv2007

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations.

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