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Jean-François Renaud

Publications and source records attributed to Jean-François Renaud.

At least 19 recordsLinked to original sources
arXiv2026

A new perspective on the Løkka-Zervos dichotomy for absolutely continuous dividend strategies

We revisit the optimization problem (and its dichotomous solution) analyzed in Renaud et al. (2026). By choosing affine functions (of the surplus level) to bound the dividend rates, we are able to provide a more explicit derivation of the solution, avoiding the use of viscosity solutions. Moreover, with explicitly parameterized expressions, we are able to study analytical properties of the optimal thresholds and to refine the statement of the dichotomy. To reach these objectives, we add a penalty-at-ruin parameter in the performance function, allowing us to unify the expressions of the value functions of the two subproblems appearing in the dichotomous solution; as a by-product, the dichotomy can also be expressed in terms of this newly added parameter. Finally, we perform sensitivity analyses to illustrate that the range of values generated in an affine-bound framework is flexible and can span the range from a constant bound to the singular version of this problem.

math.OC↗
arXiv2026

Optimization of capital injections and absolutely continuous dividend payments in a diffusion model

We investigate a joint optimization problem of dividend payments and capital injections for a surplus process driven by a general diffusion. Dividend payments are assumed to be absolutely continuous in time, with the dividend rate bounded by a nonnegative concave function of the current surplus; while capital injections are modelled by a general nondecreasing process. We first analyze an auxiliary bail-out problem in which capital injections are required to keep the surplus nonnegative at all times. Under a concavity assumption on the drift, we prove that the associated value function is concave and is a classical solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We further characterize an optimal policy as a refraction-reflection strategy: the surplus is reflected at zero by capital injections, while dividends are paid at the maximal admissible rate whenever the surplus exceeds an optimal threshold. Our main contribution establishes that the general optimization problem exhibits a Lokka-Zervos type dichotomy. More precisely, an optimal policy is either a dividend refraction strategy without injections, in which ruin occurs, or a refraction-reflection dividend-injection strategy. This optimal injection decision is characterized through a simple comparison of the auxiliary value functions at the origin.

q-fin.MF↗
arXiv2026

Optimality of a barrier strategy in a spectrally negative Lévy model with a level-dependent intensity of bankruptcy

We consider de Finetti's stochastic control problem for a spectrally negative Lévy process in an Omega model. In such a model, the (controlled) process is allowed to spend time under the critical level but is then subject to a level-dependent intensity of bankruptcy. First, before considering the control problem, we derive some analytical properties of the corresponding Omega scale functions. Second, we prove that exists a barrier strategy that is optimal for this control problem under a mild assumption on the Lévy measure. Finally, we analyse numerically the impact of the bankruptcy rate function on the optimal strategy.

math.PR↗
arXiv2026

Insurance risk models in a heterogeneous time-dependent population: scaling limits and ruin probabilities

Epidemic dynamics introduce time-varying heterogeneity into insured populations, as individuals' risk profiles depend on their evolving health status, thereby challenging classical insurance models based on homogeneity. Motivated by this phenomenon, we develop in this paper actuarial risk models in which the claim process is directly linked to an underlying general stochastic population dynamics, resulting in interacting subpopulations with different claim frequencies and severities. We propose both collective and individual modeling frameworks that track the changing composition of the insured population and its impact on aggregate claims. For these models, we derive scaling limits and provide bounds and approximations for ruin probabilities, offering tractable tools for risk assessment. Applications in an SIS context are detailed for different classes of risk processes, thereby contributing to a better understanding of how contagion-driven changes in population structure affect insurer solvency and supporting more realistic modeling of insurance portfolios in the presence of epidemic risk.

math.PR↗
arXiv2024

Optimality of a refraction strategy in the optimal dividends problem with absolutely continuous controls subject to Parisian ruin

We consider de Finetti's optimal dividends problem with absolutely continuous strategies in a spectrally negative Lévy model with Parisian ruin as the termination time. The problem considered is essentially a generalization of both the control problems considered by Kyprianou, Loeffen & Pérez (2012) and by Renaud (2019). Using the language of scale functions for Parisian fluctuation theory, and under the assumption that the density of the Lévy measure is completely monotone, we prove that a refraction dividend strategy is optimal and we characterize the optimal threshold. In particular, we study the effect of the rate of Parisian implementation delays on this optimal threshold.

math.PR↗
arXiv2024

Optimal withdrawals in a general diffusion model with control rates subject to a state-dependent upper bound

We consider a classical stochastic control problem in which a diffusion process is controlled by a withdrawal process up to a termination time. The objective is to maximize the expected discounted value of the withdrawals until the first-passage time below level zero. In this work, we are considering absolutely continuous control strategies in a general diffusion model. Our main contribution is a solution to the control problem under study, which is achieved by using a probabilistic guess-and-verify approach. We prove that the optimal strategy belongs to the family of bang-bang strategies, i.e. strategies in which, above an optimal barrier level, we withdraw at the highest-allowed rate, while no withdrawals are made below this barrier. Some nontrivial examples are studied numerically.

math.PR↗
arXiv2023

An optimization dichotomy for capital injections and absolutely continuous dividend strategies

We consider an optimal stochastic control problem in which a firm's cash/surplus process is controlled by dividend payments and capital injections. Stockholders aim to maximize their dividend stream minus the cost of injecting capital, if needed. We consider absolutely continuous dividend policies subject to a level-dependent upper bound on the dividend rate while we allow for general capital injections behavior. We prove that the optimal strategy can only be of two types: dividends are paid according to a \textit{mean-reverting} strategy with capital injections performed each time the cash process reaches zero; or, dividends are paid according to another \textit{mean-reverting} strategy and no injection of capital is ever made, until ruin is reached. We give a complete solution to this problem and characterize this dichotomy by comparing (the derivatives of) the value functions at zero of two sub-problems. The first sub-problem is concerned solely with the maximization of dividends, while the second sub-problem is the corresponding bail-out optimal dividend problem for which we provide also a complete solution.

math.OC↗
arXiv2022

De Finetti's control problem with a concave bound on the control rate

We consider De Finetti's control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal. In order to solve this problem, we need to deal with a nonlinear Ornstein-Uhlenbeck process. Despite the level of generality of the bound imposed on the rate, an explicit expression for the value function is obtained up to the evaluation of two functions.This optimal control problem has those with control rates bounded by a constant and a linear function, respectively, as special cases.

math.OC↗
arXiv2020

A stochastic control problem with linearly bounded control rates in a Brownian model

Aiming for more realistic optimal dividend policies, we consider a stochastic control problem with linearly bounded control rates using a performance function given by the expected present value of dividend payments made up to ruin. In a Brownian model, we prove the optimality of a member of a new family of control strategies called delayed linear control strategies, for which the controlled process is a refracted diffusion process. For some parameters specifications, we retrieve the strategy initially proposed by Avanzi & Wong (2012) to regularize dividend payments, which is more consistent with actual practice.

math.PR↗
arXiv2019

De Finetti's control problem with Parisian ruin for spectrally negative Lévy processes

We consider de Finetti's stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show that, under mild assumptions on the Lévy measure, an optimal strategy is formed by a barrier strategy and that this optimal barrier level is always less than the optimal barrier level when classical ruin is implemented. Also, we give necessary and sufficient conditions for the barrier strategy at level zero to be optimal.

math.PR↗
arXiv2018

A unified approach to ruin probabilities with delays for spectrally negative Lévy processes

In this paper, we unify two popular approaches for the definition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new definition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the first time an excursion in the red zone lasts longer than an implementation delay with a deterministic and a stochastic component. For this Parisian ruin with mixed delays, we identify the joint distribution of the time of ruin and the deficit at ruin, therefore providing generalizations of many results previously obtained, such as in \cite{baurdoux_et_al_2015} and \cite{loeffenetal2017} for the case of an exponential delay and that of a deterministic delay, respectively.

math.PR↗
arXiv2017

Parisian ruin for a refracted Lévy process

In this paper, we investigate Parisian ruin for a Lévy surplus process with an adaptive premium rate, namely a refracted Lévy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probability of Parisian ruin of a standard Lévy insurance risk process. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.

math.PR↗
arXiv2015

On the distribution of cumulative Parisian ruin

We introduce the concept of cumulative Parisian ruin, which is based on the time spent in the red by the underlying surplus process. Our main result is an explicit representation for the distribution of the occupation time, over a finite-time horizon, for a compound Poisson process with drift and exponential claims. The Brownian ruin model is also studied in details. Finally, we analyze for a general framework the relationships between cumulative Parisian ruin and classical ruin, as well as with Parisian ruin based on exponential implementation delays.

math.PR↗
arXiv2014

Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view

For a spectrally negative Lévy process $X$, we study the following distribution: $$ \mathbb{E}_x \left[ \mathrm{e}^{- q \int_0^t \mathbf{1}_{(a,b)} (X_s) \mathrm{d}s } ; X_t \in \mathrm{d}y \right], $$ where $-\infty \leq a < b < \infty$, and where $q,t>0$ and $x \in \mathbb{R}$. More precisely, we identify the Laplace transform with respect to $t$ of this measure in terms of the scale functions of the underlying process. Our results are then used to price step options and the particular case of an exponential spectrally negative Lévy jump-diffusion model is discussed.

math.PR↗
arXiv2013

On the time spent in the red by a refracted Lévy risk process

In this paper, we introduce an insurance ruin model with adaptive premium rate, thereafter refered to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model, the premium rate is increased as soon as the wealth process falls into the red zone and is brought back to its regular level when the process recovers. The analysis is mainly focused on the time a refracted Lévy risk process spends in the red zone (analogous to the duration of the negative surplus). Building on results from Kyprianou and Loeffen (2010) and Loeffen et al. (2012), we identify the distribution of various functionals related to occupation times of refracted spectrally negative Lévy processes. For example, these results are used to compute the probability of bankruptcy and the probability of Parisian ruin in this model with restructuring.

math.PR↗
arXiv2012

Occupation times of intervals until first passage times for spectrally negative Lévy processes

In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative Lévy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented.

math.PR↗