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.

We investigate 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 di erent expected pro ts and risks. The objective is to maximize the expected present value of the total dividend distributions. The main purpose of this paper is to deal with the impact of constraints on business activities such as inability to completely eliminate risk (even at the expense of reducing the potential pro t to zero) or when such a risk cannot exceed a certain level. We analyze the case in which there is no restriction on the dividend pay-out rates. By delicate analysis on the corresponding Hamilton-Jacobi-Bellman equation we compute explicitly the optimal return function and determine the optimal policy.

This paper addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon. We focus on No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) concept, which is also known in the literature as the first kind of non-arbitrage. For this non-arbitrage notion, we obtain two principal results. The first result lies in describing the pairs of market model and random time for which the resulting stopped model fulfills NUPBR condition. The second main result characterises the random time models that preserve the NUPBR property after stopping for any market model. These results are elaborated in a very general market model, and we also pay attention to some particular and practical models. The analysis that drives these results is based on new stochastic developments in semimartingale theory with progressive enlargement. Furthermore, we construct explicit martingale densities (deflators) for some classes of local martingales when stopped at random time.Remark 2.5. Proposition 2.3 implies that for any process X and any finite stopping time σ, the two concepts of NUPBR(H) (the current concept and the one of the literature) coincide for X σ .Below, we prove that, in the case of infinite horizon, the current NUPBR condition is stable under localization, while this is not the case for the NUPBR condition defined in the literature.Proposition 2.6. Let X be an H adapted process. Then, the following assertions are equivalent. (a) There exists a sequence (T n ) n≥1 of H-stopping times that increases to +∞, such that for each n ≥ 1, there exists a probability Q n on (Ω, H Tn ) such that Q n ∼ P and X Tn satisfies NUPBR(H) under Q n . (b) X satisfies NUPBR(H). (c) There exists an H-predictable process φ, such that 0 < φ ≤ 1 and (φ X) satisfies NUPBR(H).Proof. The proof for (a)⇐⇒(b) follows from the stability of NUPBR condition for a finite horizon under localization which is due to [41] (see also [8] for further discussion about this issue), and the fact that the NUPBR condition is stable under any equivalent probability change. The proof of (b)⇒(c) is trivial and is omitted. To prove the reverse, we assume that (c) holds. Then Proposition 2.3 implies the existence of an H-predictable process ψ such that 0 < ψ ≤ 1 and a positive H-local martingale Z = E(N ) such that Z(ψφ X) is a local martingale. Since ψφ is predictable and 0 < ψφ ≤ 1, we deduce that S satisfies NUPBR(H). This ends the proof of the proposition.We end this section with a simple, but useful result for predictable process with finite variation.Lemma 2.7. Let X be an H-predictable process with finite variation. Then X satisfies NUPBR(H) if and only if X ≡ X 0 (i.e. the process X is constant).Proof. It is obvious that if X ≡ X 0 , then X satisfies NUPBR(H). Suppose that X satisfies NUPBR(H). Consider a positive H-local martingale Y , and an H-predictable process θ such that 0 < θ ≤ 1 and Y (θ X) is a local martingale. Let (T n ) n≥1 be a sequence of H-stopping times that increases to +∞ such that Y Tn and Y Tn (θ...

L'accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

This paper quantifies the interplay between the non-arbitrage notion of No-Unbounded-Profit-with-Bounded-Risk (NUPBR hereafter) and additional information generated by a random time. This study complements the one of Aksamit/Choulli/Deng/Jeanblanc [1] in which the authors studied similar topics for the case of stopping at the random time instead, while herein we are concerned with the part after the occurrence of the random time. Given that all the literature -up to our knowledge-proves that the NUPBR notion is always violated after honest times that avoid stopping times in a continuous filtration, herein we propose a new class of honest times for which the NUPBR notion can be preserved for some models. For this family of honest times, we elaborate two principal results. The first main result characterizes the pairs of initial market and honest time for which the resulting model preserves the NUPBR property, while the second main result characterizes the honest times that preserve the NUPBR property for any quasi-left continuous model. Furthermore, we construct explicitly "the-after-τ " local martingale deflators for a large class of initial models (i.e. models in the small filtration) that are already risk-neutralized.