arXiv subjects

Guillaume Broux-Quemerais

Publications and source records attributed to Guillaume Broux-Quemerais.

3 recordsLinked to original sources

arXiv2026

Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

math.NA↗

arXiv2026

Optimal investment and consumption under forward utilities with relative performance concerns

We study a n-player and mean-field portfolio optimization problem under relative performance concerns with non-zero volatility, for wealth and consumption. The consistency assumption defining forward relative performance processes leads to a sufficient characterization of such processes with mean of a Stochastic HJB equations, which highlights the link between wealth and consumption utility, and also characterizes the optimal strategies. In particular, forward relative performance processes with a wealth utility of CRRA type and separable time and space dependence necessarily have a consumption utility of the same form, with the same risk aversion parameter. This characterization gives a better understanding of the drift condition ensuring time consistency. In this setting, we establish closed form of the Nash equilibrium for both the n-player and mean eld problems. We also provide some numerical examples.

math.OC↗

arXiv2024

Deep learning scheme for forward utilities using ergodic BSDEs

In this paper, we present a probabilistic numerical method for a class of forward utilities in a stochastic factor model. For this purpose, we use the representation of dynamic consistent utilities with mean of ergodic Backward Stochastic Differential Equations (eBSDEs) introduced by Liang and Zariphopoulou in [27]. We establish a connection between the solution of the ergodic BSDE and the solution of an associated BSDE with random terminal time $τ$ , defined as the hitting time of the positive recurrent stochastic factor V . The viewpoint based on BSDEs with random horizon yields a new characterization of the ergodic cost $λ$ which is a part of the solution of the eBSDEs. In particular, for a certain class of eBSDEs with quadratic generator, the Cole-Hopf transform leads to a semi-explicit representation of the solution as well as a new expression of the ergodic cost $λ$. The latter can be estimated with Monte Carlo methods. We also propose two new deep learning numerical schemes for eBSDEs, where the ergodic cost $λ$ is optimized according to a loss function at the random horizon $τ$ or taking into account the whole trajectory. Finally, we present numerical results for different examples of eBSDEs and forward utilities along with the associated investment strategies.

math.PR↗