We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs / M. Fuhrman, M. Morlais. - In: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - ISSN 0304-4149. - 130:5(2020 May), pp. 3120-3153. [10.1016/j.spa.2019.09.008]

Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

M. Fuhrman
Primo
;
2020

Abstract

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton–Jacobi–Bellman equation in our non-Markovian framework.
English
stochastic optimal switching; backward SDEs; randomization of controls
Settore MAT/06 - Probabilita' e Statistica Matematica
Articolo
Esperti anonimi
Pubblicazione scientifica
   Deterministic and stochastic evolution equations
   MINISTERO DELL'ISTRUZIONE E DEL MERITO
   2015233N54_002
mag-2020
Elsevier
130
5
3120
3153
34
Pubblicato
Periodico con rilevanza internazionale
crossref
Aderisco
info:eu-repo/semantics/article
Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs / M. Fuhrman, M. Morlais. - In: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - ISSN 0304-4149. - 130:5(2020 May), pp. 3120-3153. [10.1016/j.spa.2019.09.008]
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Prodotti della ricerca::01 - Articolo su periodico
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Article (author)
Periodico con Impact Factor
M. Fuhrman, M. Morlais
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/728421
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