
[+] N. Baradel. Continuoustime modeling and bootstrap for chain ladder reserving, 2024.
Abstract : We revisit the famous Mack's model which gives an estimate for the mean square error of prediction of the chain ladder claims reserves. We introduce a stochastic differential equation driven by a Brownian motion to model accumulated total claims amount for the chain ladder method. Within this continuoustime framework, we propose a bootstrap technique for estimating the distribution of claims reserves. It turns out that our approach leads to inherently capturing asymmetry and nonnegativity, eliminating the necessity for additional assumptions. We conclude with a case study and comparative analysis against alternative methodologies based on Mack's model.

[+] N. Baradel. Modeling frequency distribution above a priority in presence of IBNR, 2024.
Abstract : In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach.

[+] N. Baradel. Optimal control under uncertainty: Application to the issue of CAT bonds, Insurance: Mathematics and Economics, vol. 117, p. 1644, 2024.
Abstract : We propose a general framework for studying optimal issue of CAT bonds in the presence of uncertainty of the parameters. In particular, the intensity of arrival of natural disasters is inhomogeneous and may depend on unknown parameters. Given a prior on the distribution of the unknown parameters, we explain how it should evolve according to the classical Bayes rule. Taking these progressive prioradjustments into account, we characterize the optimal policy through a quasivariational parabolic equation, which can be solved numerically. We provide examples of application in the context of hurricanes in Florida.

[+] N. Baradel, B. Bouchard, D. Evangelista, O. Mounjid. Optimal inventory management and order book modeling, ESAIM: Proceedings and Surveys, vol. 65, p. 145181, 2019.
Abstract : We model the behavior of three agent classes acting dynamically in a limit order book of a financial asset. Namely, we consider market makers (MM), highfrequency trading (HFT) firms, and institutional brokers (IB). Given a prior dynamic of the order book, similar to the one considered in the QueueReactive models [14, 20, 21], the MM and the HFT define their trading strategy by optimizing the expected utility of terminal wealth, while the IB has a prescheduled task to sell or buy many shares of the considered asset. We derive the variational partial differential equations that characterize the value functions of the MM and HFT and explain how almost optimal control can be deduced from them. We then provide a first illustration of the interactions that can take place between these different market participants by simulating the dynamic of an order book in which each of them plays his own (optimal) strategy.

[+] N. Baradel, B. Bouchard, N. M. Dang. Optimal trading with online parameters revisions, Market Microstructure and Liquidity, 2(03n04), 2016.
Abstract : The aim of this paper is to explain how parameters adjustments can be integrated in the design or the control of automates of trading. Typically, we are interested by the online estimation of the market impacts generated by robots or single orders, and how they/the controller should react in an optimal way to the informations generated by the observation of the realized impacts. This can be formulated as an optimal impulse control problem with unknown parameters, on which a prior is given. We explain how a mix of the classical Bayesian updating rule and of optimal control techniques allows one to derive the dynamic programming equation satisfied by the corresponding value function, from which the optimal policy can be inferred. We provide an example of convergent finite difference scheme and consider typical examples of applications.

[+] N. Baradel, B. Bouchard, N. M. Dang. Optimal control under uncertainty and Bayesian parameters adjustments, SIAM Journal on Control and Optimization, 56(2) :10381057, 2018.
Abstract : We propose a general framework for studying optimal impulse control problem in the presence of uncertainty on the parameters. Given a prior on the distribution of the unknown parameters, we explain how it should evolve according to the classical Bayesian rule after each impulse. Taking these progressive prioradjustments into account, we characterize the optimal policy through a quasivariational parabolic equation, which can be solved numerically. The derivation of the dynamic programming equation seems to be new in this context. The main difficulty lies in the nature of the set of controls which depends in a non trivial way on the initial data through the filtration itself.