Tahir Choulli

Classical and Impulse Stochastic Control for the Optimization of the Dividend and Risk Policies of an Insurance Firm

This paper deals with the dividend optimization problem for a financial or an insurance entity which can control its business activities, simultaneously reducing the risk and potential profits. It also controls the timing and the amount of dividends paid out to the shareholders. The objective of the corporation is to maximize the expected total discounted dividends paid out until the time of bankruptcy. Due to the presence of a fixed transaction cost, the resulting mathematical problem becomes a mixed classical-impulse stochastic control problem. The analytical part of the solution to this problem is reduced to quasivariational inequalities for a second-order nonlinear differential equation. We solve this problem explicitly and construct the value function together with the optimal policy. We also compute the expected time between dividend payments under the optimal policy.

A Diffusion Model for Optimal Dividend Distribution for a Company with Constraints on Risk Control

This paper investigates a model of a corporation which faces constant liability payments and which can choose a production/business policy from an available set of control policies with different expected profits and risks. The objective is to find a business policy and a dividend distribution scheme so as to maximize the expected present value of the total dividend distributions. The main feature of this paper is that there are constraints on business activities such as inability to completely eliminate risk (even at the expense of reducing the potential profit to zero) or when such a risk cannot exceed a certain level. The case in which there is no restriction on the dividend pay-out rates is dealt with. This gives rise to a mixed regular-singular stochastic control problem. First the value function is analyzed in great detail and in particular is shown to be a viscosity solution of the corresponding Hamilton--Jacobi--Bellman (HJB) equation. Based on this it is further proved that the value function must be twice continuously differentiable. Then a delicate analysis is carried out on the HJB equation, leading to an explicit expression of the value function as well as the optimal policies.

Minimal Hellinger martingale measures of order q

Abstract This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q≠1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Föllmer and Schweizer (here q=2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in ℝd. Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market’s exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.

Excess-of-loss reinsurance for a company with debt liability and constraints on risk reduction

We consider a problem of risk control and dividend optimization for a financial corporation facing a constant liability payment. More specifically we investigate the case of excess-of-loss reinsurance for an insurance company. In this scheme the insurance company diverts a part of its premium stream to another company, the reinsurer, in exchange for an obligation to pick up that amount of each claim which exceeds a certain level a. The objective of the insurer is to maximize the expected present value of total future dividend pay-outs. We consider cases when there is restriction on the rate of dividend pay-outs and when there is no restriction. In both cases we describe explicitly the optimal return function as well as the optimal policy.

How non-arbitrage, viability and numéraire portfolio are related

This paper proposes two approaches that quantify the exact relationship among viability, absence of arbitrage, and/or existence of the numéraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence between the no-unbounded-profit-with-bounded-risk condition (NUPBR hereafter), the existence of the numéraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure for any “nice” utility and positive initial capital. Herein, a “nice” utility is any smooth von Neumann–Morgenstern utility satisfying Inada’s conditions and the elasticity assumptions of Kramkov and Schachermayer. Furthermore, the equivalent probability measure—under which the utility maximization problems have solutions—can be chosen as close to the real-world probability measure as we want (but might not be equal). Without changing the underlying probability measure and under mild assumptions, our second contribution proves that NUPBR is equivalent to the “local” existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities.

The Role of Hellinger Processes in Mathematical Finance

This paper illustrates the natural role that Hellinger processes can play in solving problems from ¯nance. We propose an extension of the concept of Hellinger process applicable to entropy distance and f-divergence distances, where f is a convex logarithmic function or a convex power function with general order q, 0 6= q < 1. These concepts lead to a new approach to Mertons optimal portfolio problem and its dual in general L¶evy markets.

On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration

Given a reference filtration \(\mathbb{F}\), we consider the cases where an enlarged filtration \(\mathbb{G}\) is constructed from \(\mathbb{F}\) in two different ways: progressively with a random time or initially with a random variable. In both situations, under suitable conditions, we present a \(\mathbb{G}\)-optional semimartingale decomposition for \(\mathbb{F}\)-local martingales. Our study is then applied to the question of how an arbitrage-free semimartingale model is affected when stopped at the random time in the case of progressive enlargement or when the random variable used for initial enlargement satisfies Jacod’s hypothesis. More precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition, also called non arbitrages of the first kind in the literature. We provide alternative proofs of some results from Aksamit et al. (Non-arbitrage up to random horizon for semimartingale models, short version, preprint, 2014 [arXiv:1310.1142]), incorporating a different methodology based on our optional semimartingale decomposition.

No-arbitrage under a class of honest times

This paper quantifies the interplay between the no-arbitrage notion of no unbounded profit with bounded risk (NUPBR) and additional progressive information generated by a random time. This study complements the one of Aksamit et al. (Finance Stoch. 21:1103–1139, 2017) in which the authors have studied similar topics for the model stopped at the random time, while here we deal with the question of what happens after the random time. Given that the existing literature proves that NUPBR is always violated after honest times that avoid stopping times in a continuous filtration, we propose here a new class of honest times for which NUPBR can be preserved for some models. For these honest times, we obtain two principal results. The first result characterizes the pairs of initial market and honest time for which the resulting model preserves NUPBR, while the second result characterizes honest times that do not affect NUPBR of any quasi-left-continuous model (i.e., in which the asset price process has no predictable jump times). Furthermore, we construct explicitly local martingale deflators for a large class of models.

Non-arbitrage for Informational Discrete Time Market Models

Abstract This paper focuses on the stability of no-arbitrage, for discrete time market models, under additional uncertainty generated by a random time . At the practical level, this random time represents the death time, the default time of a firm, or any occurrence time of an event that might affect the market somehow. We address the no-arbitrage issue for the resulting new flow of information (filtration) which makes the random time either a nontrivial stopping time (progressive enlargement) or a known time from the beginning (initial enlargement). Our main conclusions are twofold. On the one hand, for a fixed initial market S, we completely and precisely characterize the interplay between S and such that the no-arbitrage is preserved for the new market model. On the other hand, we give the necessary and sufficient conditions on to ensure the preservation of the no-arbitrage under the additional uncertainty of for any market. Two concrete examples are presented to illustrate the results.

Three Essays on Exponential Hedging with Variable Exit Times

This paper addresses three main problems that are intimately related to exponential hedging with variable exit times. The first problem consists of explicitly parameterizing the exponential forward performances and describing the optimal solution for the corresponding utility maximization problem. The second problem deals with the horizon-unbiased exponential hedging. Precisely, we are interested in describing the dynamic payoffs for which there exists an admissible strategy that minimizes the risk—in the exponential utility framework—whenever the investor exits the market at stopping times. Furthermore, we explicitly describe this optimal strategy when it exists. Our last contribution is concerned with the optimal sale problem, where the investor is looking simultaneously for the optimal portfolio and the optimal time to liquidate her assets.