Singular recursive utility

We introduce the concept of singular recursive utility. This leads to a kind of singular BSDE which, to the best of our knowledge, has not been studied before. We show conditions for existence and uniqueness of a solution for this kind of singular BSDE. Furthermore, we analyze the problem of maximizing the singular recursive utility. We derive sufficient and necessary maximum principles for this problem, and connect it to the Skorohod reflection problem. Finally, we apply our results to a specific cash flow. In this case, we find that the optimal consumption rate is given by the solution to the corresponding Skorohod reflection problem

Model Setting for Optimal Stopping and Singular Stochastic Control for Optimal Investment Strategy in Oil Field Project

Investing in projects involving huge financial risks demands great care. Dealing with market uncertainty and taking effective investment decision in oil field project, therefore, requires a reliable guide. The strategy emerged from addressing a problem involving an optimal stopping time with singular stochastic control for jump diffusions. The strategy identified two unique thresholds, one indicating when to apply the control and the other showing when to quit. Optimal strategy for investment in oil field project were obtained. Two particular cases of Brownian motion and Geometric Brownian motion are presented. The model is set to include jumps in the analysis: to obtain better investment strategies in oil field project

Pointwise Second Order Necessary Conditions for Stochastic Optimal Control with Jump Diffusion

Stochastic maximum principle is one of the important major approaches to discuss stochastic control problems. A lot of work has been done on this kind of problem, see, for example, Bensoussan [3], Cadenillas and Karatzas [10], Kushner [31], Peng [41]. Recently, another kind of stochastic maximum principle, pointwise second order necessary conditions for stochastic optimal controls has been established and studied for its applications in the financial market by Zhang and Zhang [58] when the control region is assumed to be convex. In Zhang and Zhang [59], the authors extended the pointwise second order necessary conditions for stochastic optimal controls in the general cases when the control region is allowed to be non convex. Second order necessary conditions for optimal control with recursive utilities was proved by Dong and Meng [13]. In this thesis, we generalizes the work of Zhang and Zhang [58] for jump diffusions, we establish a second order necessary conditions where the controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes. Our stochastic control problem provides also an interesting models in many applications such as economics and mathematical finance

Mean-Field Optimal Control of Di¤usion with Regime Switching

The objective of this thesis is to study a problem of optimal control with regime switching jump-di¤usion model of mean-…eld type. In the …rst part we recall a result in the stochastic maximum principle whose horizon is …nite. In the second part, we devote ourselves to presenting the two main results of this thesis, in the …rst result we give the necessary and su¢ cient conditions of optimality whose control system is governed by a stochastic di¤erential equation with regime switching of in…nite horizon and by way of illustration, we have given two examples where in both cases the equation of state is linear and the objective function is of utility form. The second contribution on the maximum principle for a control problem of conditional mean …eld type of …nite horizon, we illustrate our result by a model which gives an explicit solutio

Contributions to Optimal Stopping and Long-Term Average Impulse Control

In this thesis we consider undiscounted, infinite time horizon optimal stopping problems with generalized linear costs and long-term average impulse control problems. The main goal is to find (semi-)explicit solutions in case the underlying process contains jumps. In order to solve the stopping problems, we utilize embedded monotone problems to find sufficient conditions, that are easy to handle, for a threshold time to be optimal. Further, we characterize the threshold for one-dimensional Markov processes in both discrete and continuous time. While in the discrete time case the concept of ladder times can be used to exploit inherent monotone structures, in continuous time we develop an integral type maximum representation to enable a comparable line of argument. The findings on long-term average impulse control problems are structured in two main areas. First, for a general one-dimensional Markov process we characterize the problem’s value and possible optimal strategies by an associated stopping problem. Then, we develop a step-by-step solution technique in case the process is a Lévy process and demonstrate its usefulness by applying it to relevant examples, among others problems from inventory control and optimal harvesting. Apart from these direct applications we use our theoretical findings to investigate the influence of varying fixed costs on the impulse control problem, study a control problem with a restriction to the impulse frequency and treat mean field games and problems of impulse control

On the variational principle for a class of stochastic control for systems governed by stochastic differential equations of mean-field type with applications

Cette thèse de doctorat s’inscrit dans le cadre de la théorie de contrôle optimal stochastique. Le thème central est l’optimisation stochastique a…n d’établir des conditions nécessaires d’un contrôle optimal sous forme du principe du maximum stochastique de type de Pontryagin. D’une part, et plus précisement, nous étudions des problèmes de contrôle stochastique optimal singulier partiellement observés de type mean-…eld (McKean-Vlasov) général avec des corrélations entre le système et l’observation Y (�) : Dans ce travail, la variable de contrôle (u (�) ; � (�)) a deux composantes, la première u (�) est absolument continue et la seconde � (�) est une variation bornée, non décroissante continue à droite avec limit à gauche (càdlàg). Le système stochastique étudié est gouverné par une équation di¤érentielle stochastique contrôlée de type Itô où les coe¢ cients de la dynamique dépendent du processus d’état ainsi que de sa loi de probabilité Pxu;�(t) et de la variable de contrôle continue u (�) ; dé…nit par : 8>>>>>>>>>: dxu;� (t) = f(t; xu;� (t) ; Pxu;�(t); u (t))dt + �(t; xu;� (t) ; Pxu;�(t); u (t))dW (t) +g(t; xu;� (t) ; Pxv;�(t); v (t))dWf (t) + G(t)d�(t); xu;� (0) = x0; t 2 [0; T] : Nous supposons que le processus d’état xu;� (t) ne peut pas être observé directement, mais les contrôleurs peuvent observer un processus de bruit associé Y (�), régit par l’équation suivante : 8>: dY (t) = h(t; xu;� (t) ; u (t))dt + dWf (t) Y (0) = 0; où Wf (t) est un processus stochastique dépendant du contrôle u(�), et Y (�) le processus d’observation. On de…nit FY t martingale �u(t) qui est une solution de l’equation suivante : 8>: d�u(t) = �u(t)h (t; xu(t); u(t)) dY (t); �u(0) = 1: D’aprés le théorème de dérivation de Radon-Nikodym, cette martingale a permis de dé…nir une nouvelle probabilité notée Pu, qui dépend de u (�) et donnée par : dPu dP FY t = �u(t). La fonctionnelle de coût J(u(�); �(�)) peut s’écrire sous forme J(u(�); �(�)) = E �Z0T �u(t)l(t; xu;�(t); Pxu;�(t); u(t))dt + �u(T) (xu;�(T); Pxu;�(T )) + Z[0;T ] �u(t)M(t)d�(t)� : Par l’utilisation des techniques variationnelles convexes classiques, nous établissons un ensemble de conditions nécessaires de contrôle singulier optimal sous la forme du principe du maximum. Notre résultat principal est prouvé en appliquant le théorème de Girsanov et les dérivées par rapport à une mesure (ou la loi de probabilité) au sense de P. Lions. D’autre part, nous établissons des conditions nécessaires du second-ordre pour un contrôle stochastique mixed continu-singulier (u (�) ; � (�)), où le système est gouverné par des systèmes di¤érentiels stochastiques contrôlés non linéaires. Le principe du maximum ponctuel du second-ordre en termes de martingale par rapport à la variable de temps est prouvé. Le domaine de contrôle est supposé convexe. Notons que dans ce travail que les termes de dérivée et les termes de di¤usion des systèmes dépendent de la variable de contrôle continue u (�). Notre résultat est prouvé en utilisant des techniques variationnelles sous certaines conditions de convexité

Stochastics of Environmental and Financial Economics

Systems Theory, Contro

Coupling polynomial Stratonovich integrals : the two-dimensional Brownian case

We show how to build an immersion coupling of a two-dimensional Brownian motion (W1,W2) along with (n2)+n=12n(n+1). integrals of the form ∫Wi1Wj2∘dW2, where j=1,…,n and i=0,…,n−j for some fixed n. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)) and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also multiple selected integral functionals of the diffusions

Coupling polynomial Stratonovich integrals : the two-dimensional Brownian case

We show how to build an immersion coupling of a two-dimensional Brownian motion (W1,W2) along with (n2)+n=12n(n+1). integrals of the form ∫Wi1Wj2∘dW2, where j=1,…,n and i=0,…,n−j for some fixed n. The resulting construction is applied to the study of couplings of certain hypoelliptic diffusions (driven by two-dimensional Brownian motion using polynomial vector fields). This work follows up previous studies concerning coupling of Brownian stochastic areas and time integrals (Ben Arous, Cranston and Kendall (1995), Kendall and Price (2004), Kendall (2007), Kendall (2009), Kendall (2013), Banerjee and Kendall (2015), Banerjee, Gordina and Mariano (2016)) and is part of an ongoing research programme aimed at gaining a better understanding of when it is possible to couple not only diffusions but also multiple selected integral functionals of the diffusions