arXiv subjects

Sarah Kaakai

Publications and source records attributed to Sarah Kaakai.

9 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 Pension policy in Pay-As-You-Go systems under forward utility and ageing population

This paper investigates optimal investment and pension policies in a Pay-As-You-Go (PAYG) system supplemented by a buffer fund used as an intergenerational risk-sharing mechanism. The social planner's preference criterion is represented by non-zero volatility forward Constant Relative Risk Aversion (CRRA) utilities, and explicitly accounts for both sustainability and adequacy constraints. The optimal policies are characterized in closed form, and an in-depth analysis of the impact of preference sensitivities on the pension scheme is conducted. A detailed numerical analysis is performed to evaluate the sustainability and benefit adequacy of this hybrid PAYG buffer fund arrangement under a range of demographic, financial, and macroeconomic scenarios.

q-fin.MF↗

arXiv2024

Efficient simulation of individual-based population models: the R Package IBMPopSim

The R Package IBMPopSim aims to simulate the random evolution of heterogeneous populations using stochastic Individual-Based Models (IBMs). The package enables users to simulate population evolution, in which individuals are characterized by their age and some characteristics, and the population is modified by different types of events, including births/arrivals, death/exit events, or changes of characteristics. The frequency at which an event can occur to an individual can depend on their age and characteristics, but also on the characteristics of other individuals (interactions). Such models have a wide range of applications in fields including actuarial science, biology, ecology or epidemiology. IBMPopSim overcomes the limitations of time-consuming IBMs simulations by implementing new efficient algorithms based on thinning methods, which are compiled using the Rcpp package while providing a user-friendly interface.

q-bio.PE↗

arXiv2024

Estimation of Systemic Shortfall Risk Measure using Stochastic Algorithms

Systemic risk measures were introduced to capture the global risk and the corresponding contagion effects that is generated by an interconnected system of financial institutions. To this purpose, two approaches were suggested. In the first one, systemic risk measures can be interpreted as the minimal amount of cash needed to secure a system after aggregating individual risks. In the second approach, systemic risk measures can be interpreted as the minimal amount of cash that secures a system by allocating capital to each single institution before aggregating individual risks. Although the theory behind these risk measures has been well investigated by several authors, the numerical part has been neglected so far. In this paper, we use stochastic algorithms schemes in estimating MSRM and prove that the resulting estimators are consistent and asymptotically normal. We also test numerically the performance of these algorithms on several examples.

math.OC↗

arXiv2024

Time-consistent pension policy with minimum guarantee and sustainability constraint

This paper proposes and investigates an optimal pair investment/pension policy for a pay-as-you-go (PAYG) pension scheme. The social planner can invest in a buffer fund in order to guarantee a minimal pension amount. The model aims at taking into account complex dynamic phenomena such as the demographic risk and its evolution over time, the time and age dependence of agents preferences, and financial risks. The preference criterion of the social planner is modeled by a consistent dynamic utility defined on a stochastic domain, which incorporates the heterogeneity of overlapping generations and its evolution over time. The preference criterion and the optimization problem also incorporate sustainability, adequacy and fairness constraints. The paper designs and solves the social planner's dynamic decision criterion, and computes the optimal investment/pension policy in a general framework. A detailed analysis for the case of dynamic power utilities is provided.

q-fin.MF↗

arXiv2022

Utility Maximization Problem with Uncertainty and a Jump Setting

We study a robust utility maximization problem in the unbounded case with a general penalty term and information including jumps. We focus on time consistent penalties and we prove that there exists an optimal probability measure solution of the robust problem. Then, we characterize the dynamic value process of our stochastic control problem as the unique solution of a Quadratic-Exponential BSDE.

math.OC↗

arXiv2022

Multivariate Optimized Certainty Equivalent Risk Measures and their Numerical Computation

We present a framework for constructing multivariate risk measures that is inspired from univariate Optimized Certainty Equivalent (OCE) risk measures. We show that this new class of risk measures verifies the desirable properties such as convexity, monotonocity and cash invariance. We also address numerical aspects of their computations using stochastic algorithms instead of using Monte Carlo or Fourier methods that do not provide any error of the estimation.

math.OC↗

arXiv2021

Simulating long-term impacts of mortality shocks: learning from the cholera pandemic

The aim of this paper is to study the long-term consequence on longevity of a mortality shock. We adopt an historical and modeling approach to study how the population evolution following a mortality shock such as the COVID-19 pandemic could impact future mortality rates. In the first of part the paper, we study the several cholera epidemics in France and in England starting from the 1830s, and their impact on the major development of public health at the end of the nineteenth century. In the second part, we present the mathematical modeling of stochastic Individual-Based models. Using the R package IBMPopSim, this flexible framework is then applied to simulate the long-term impact of a mortality shock, using a toy model where nonlinear population compositional changes affect future mortality rates.

q-bio.PE↗

arXiv2020

Birth Death Swap population in random environment and aggregation with two timescales

This paper deals with the stochastic modeling of a class of heterogeneous population in a random environment, called birth-death-swap. In addition to demographic events, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the multi-type population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jumps measure of the population, described by a multivariate counting process. This process is defined as a solution of a stochastic differential system with random coefficients, driven by a multivariate Poisson random measure. The solution is obtained under weak assumptions, by the thinning of a strongly dominating point process generated by the same Poisson measure. This key construction relies on a general strong comparison result, of independent interest. The second part is dedicated to averaging results when swap events are significantly more frequent than demographic events. An important ingredient is the stable convergence, which is well-adapted to the general random environment. The pathwise construction by domination yields tightness results straightforwardly. At the limit, the demographic intensity functionals are averaged against random kernels depending on swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a "true" birth-death process in random environment, with non-linear intensity functionals.

math.PR↗