999 research outputs found

    A maximum principle for progressive optimal control of mean-filed forward-backward stochastic system involving random jumps and impulse controls

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    In this paper, we study an optimal control problem of a mean-field forward-backward stochastic system with random jumps in progressive structure, where both regular and singular controls are considered in our formula. In virtue of the variational technology, the related stochastic maximum principle (SMP) has been obtained, and it is essentially different from that in the classical predictable structure. Specifically, there are three parts in our SMP, i.e. continuous part, jump part and impulse part, and they are respectively used to characterize the characteristics of the optimal controls at continuous time, jump time and impulse time. This shows that the progressive structure can more accurately describe the characteristics of the optimal control at the jump time. We also give two linear-quadratic (LQ) examples to show the significance of our results

    Sur un problem de contrˆole optimal stochastique pour certain aspect des ´equations differentielles stochastiques de type mean-field et applications

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    Cette thèse de doctorat s’inscrit dans le cadre de l’analyse stochastique dont le thème central est: les conditions necessaires et suffisantes sous forme du maximum stochastique de type champ moyen d’optimalite et de presque optimalite et ces applications. L’objectif de ce travail est d’etudier des problemes d’optimisation stochastique. Il s’agira ensuite de faire le point sur les conditions necessaires et suffisantes d’optimalite et de presque optimalite pour un system gouverne par des equations differentielles stochastiques de type champ moyen. Cette these s’articule autour de qua¬tre chapitres: Le chapitre 1 est essentiellement un rappel. La candidate présente quelques concepts et résultats qui lui permettent d’aborder son travail; tels que les processus stochastiques, l’esperance condition-nelle, les martingales, les formules d’Ito, les classes de contrôle stochastique,... etc. Dans le deuxieme chapitre, on a etablie et on a prouve les conditions necessaires et suffisantes de presque optimalite d’order 3b5{ 3bb} verifiees par un contrôle optimal stochastique, pour un system différentiel gouverne par des equations differentielles stochastiques EDSs. Le domaine de contrôle stochastique est suppose convexe. La methode utilisee est basee sur le lemme d’Ekeland. Les résultats obtenus dans le chapitre 2, sont tous nouveaux et font l’objet d’un premier article intitule : Boukaf Samira & Mokhtar Hafayed, & Ghebouli Messaoud: A study on optimal control problem with ex-error bound for stochastic systems with application to linear quadratic problem, International Journal of Dynamics and Control, Springer DOI: 10.1007/s40435-015-0178-x (2015). Dans le troisieme chapitre, on a demontré le principe du maximum stochastique de presque optimalite, oh le system est gouverne par des equations differentielles stochastiques progressive rétrogrades avec saut (FBSDEs). Ces resultats ont ete appliques pour résoudre un probleme d’optimisation en finance. Ces resultats generalisent le principe du maximum de Zhou (SIAM. Control. Optim. (36)-3, 929-947 (1998)). Les resultats obtenus dans le chapitre 3 sont tous nouveaux et font l’objet d’un deuxieme article intitule: Mokhtar Hafayed, & Abdelmadjid Abba & Samira Boukaf: On Zhou’s maximumprinciple for near- optimal control of mean-field forward-backward stochastic systems with jumps and its applications International Journal of Modelling, Identification and Control. 25 (1), 1-16, (2016). De plus, et dans le chapitre 4, on a prouve un principe du maximum stochastique de type de Pontryagin pour des systems gouvernes par FBSDEs avec saut. Ces resultats ont ete etabli avec M. Hafayed, et M. Tabet, sous le titre : Mokhtar Hafayed, & Moufida Tabet & Samira Boukaf: Mean-field maximum principle for optimal control of forward-backward stochastic systems with jumps and its application to mean-variance portfolio problem, Communication in Mathematics and Statistics, Springer, Doi: 10.1007/s40304- 015-0054-1, Volume 3, Issue 2, pp 163-186 (2015). Dans le chapitre 5, on a aborde un problème de contrôle singulier, où le problème est d’établir des conditions necessaires et suffisantes d’optimalite pour un control singulier ou le system est gouverne par des equations differentielles stochastiques progressive retrograde de type McKean-Vlasov. Dans ces cas, le domaine de contrôle admissible est suppose convexe. Les résultats obtenus dans le chapitre 5 sont tous nouveaux et font l’objet d’un article intitule : Mokhtar Hafayed, & Samira Boukaf & Yan Shi, & Shahlar Meherrem.: A McKean-Vlasov optimal mixed regular-singular control problem, for nonlinear stochastic systems with Poisson jump pro-cesses, Neurocomputing. Doi 10.1016/j.neucom.2015.11.082, Volume 182,19, pages 133-144 (2016

    Maximum principle for a stochastic delayed system involving terminal state constraints

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    We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtue of Ekeland's variational principle. Finally, applications to a state constrained stochastic delayed linear-quadratic control model and a production-consumption choice problem are studied to illustrate the main obtained result.Comment: 16 page

    Strategically-Timed Actions in Stochastic Differential Games

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    Financial systems are rich in interactions amenable to description by stochastic control theory. Optimal stochastic control theory is an elegant mathematical framework in which a controller, profitably alters the dynamics of a stochastic system by exercising costly control inputs. If the system includes more than one agent, the appropriate modelling framework is stochastic differential game theory — a multiplayer generalisation of stochastic control theory. There are numerous environments in which financial agents incur fixed minimal costs when adjusting their investment positions; trading environments with transaction costs and real options pricing are important examples. The presence of fixed minimal adjustment costs produces adjustment stickiness as agents now enact their investment adjustments over a sequence of discrete points. Despite the fundamental relevance of adjustment stickiness within economic theory, in stochastic differential game theory, the set of players’ modifications to the system dynamics is mainly restricted to a continuous class of controls. Under this assumption, players modify their positions through infinitesimally fine adjustments over the problem horizon. This renders such models unsuitable for modelling systems with fixed minimal adjustment costs. To this end, we present a detailed study of strategic interactions with fixed minimal adjustment costs. We perform a comprehensive study of a new stochastic differential game of impulse control and stopping on a jump-diffusion process and, conduct a detailed investigation of two-player impulse control stochastic differential games. We establish the existence of a value of the games and show that the value is a unique (viscosity) solution to a double obstacle problem which is characterised in terms of a solution to a non-linear partial differential equation (PDE). The study is contextualised within two new models of investment that tackle a dynamic duopoly investment problem and an optimal liquidity control and lifetime ruin problem. It is then shown that each optimal investment strategy can be recovered from the equilibrium strategies of the corresponding stochastic differential game. Lastly, we introduce a dynamic principal-agent model with a self-interested agent that faces minimally bounded adjustment costs. For this setting, we show for the first time that the principal can sufficiently distort that agent’s preferences so that the agent finds it optimal to execute policies that maximise the principal’s payoff in the presence of fixed minimal costs

    Impulse Control in Finance: Numerical Methods and Viscosity Solutions

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    The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. The dynamic programming equations associated with impulse control problems are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities

    Optimal market making under partial information and numerical methods for impulse control games with applications

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    The topics treated in this thesis are inherently two-fold. The first part considers the problem of a market maker who wants to optimally set bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. This stochastic control problem under partial information is solved by means of stochastic filtering, control and piecewise-deterministic Markov processes theory. The value function is characterized as the unique continuous viscosity solution of its dynamic programming equation. Afterwards, the analogous full information problem is solved and results are compared numerically through a concrete example. The optimal full information spreads are shown to be biased when the exact market regime is unknown, as the market maker needs to adjust for additional regime uncertainty in terms of P&L sensitivity and observable order ow volatility. The second part deals with numerically solving nonzero-sum stochastic differential games with impulse controls. These offer a realistic and far-reaching modelling framework for applications within finance, energy markets and other areas, but the diffculty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining very particular cases. To the author's best knowledge, there are no numerical methods available in the literature. A policy-iteration-type solver is proposed to solve an underlying system of quasi-variational inequalities, and it is validated numerically with reassuring results. In particular, it is observed that the algorithm does not enjoy global convergence and a heuristic methodology is proposed to construct initial guesses. Eventually, the focus is put on games with a symmetric structure and a substantially improved version of the former algorithm is put forward. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A provably convergent single-player impulse control solver, often outperforming classical policy iteration, is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise too challenging problems, and even some for which results go beyond the scope of all the currently available theory

    Impulse Control in Finance: Numerical Methods and Viscosity Solutions

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    The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. The dynamic programming equations associated with impulse control problems are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities

    Applications of Stochastic Control in Energy Real Options and Market Illiquidity

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    We present three interesting applications of stochastic control in finance. The first is a real option model that considers the optimal entry into and subsequent operation of a biofuel production facility. We derive the associated Hamilton Jacobi Bellman (HJB) equation for the entry and operating decisions along with the econometric analysis of the stochastic price inputs. We follow with a Monte Carlo analysis of the risk profile for the facility. The second application expands on the analysis of the biofuel facility to account for the associated regulatory and taxation uncertainty experienced by players in the renewables and energy industries. A federal biofuel production subsidy per gallon has been available to producers for many years but the subsidy price level has changed repeatedly. We model this uncertain price as a jump process. We present and solve the HJB equations for the associated multidimensional jump diffusion problem which also addresses the model uncertainty pervasive in real option problems such as these. The novel real option framework we present has many applications for industry practitioners and policy makers dealing with country risk or regulatory uncertainty---which is a very real problem in our current global environment. Our final application (which, although apparently different from the first two applications, uses the same tools) addresses the problem of producing reliable bid-ask spreads for derivatives in illiquid markets. We focus on the hedging of over the counter (OTC) equity derivatives where the underlying assets realistically have transaction costs and possible illiquidity which standard finance models such as Black-Scholes neglect. We present a model for hedging under market impact (such as bid-ask spreads, order book depth, liquidity) using temporary and permanent equity price impact functions and derive the associated HJB equations for the problem. This model transitions from continuous to impulse trading (control) with the introduction of fixed trading costs. We then price and hedge via the economically sound framework of utility indifference pricing. The problem of hedging under liquidity impact is an on-going concern of market makers following the Global Financial Crisis
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