Optimal Stopping in General Predictable Framework

In this paper, we study the optimal stopping problem in the case where the reward is given by a family (\phi(\tau ),\;\;\tau \in \stopo) of non negative random variables indexed by predictable stopping times. We treat the problem by means of Snell's envelope techniques. We prove some properties of the value function family associated to this setting

IDT processes and associated L\'evy processes

This article deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. Given an IDT process $(X_{t},\,t\geq0)$, there exists a unique (in law) L\'evy process $(L_{t}; t\geq0)$ which has the same one-dimensional marginals distributions, i.e for any $t\geq0$ fixed, we have $$X_{t}\stackrel{(law)}{=}L_{t}.$$ Such processes are said to be associated. The main objective of this work is to exhibit numerous examples of IDT processes and their associated L\'evy processes. To this end, we take up ideas of the monograph \textit{Peacocks and associated martingales} from F. Hirsch, C. Profeta, B. Roynette and M. Yor (L\'evy, Sato and Gaussian sheet methods) and apply them in the framework of IDT processes. This gives a new interesting outlook to the study of processes whose only one-dimensional marginals are known. Also, we give an integrated weak It\^o type formula for IDT processes (in the same spirit as the one for Gaussian processes) and some links between IDT processes and selfdecomposability. The last sections are devoted to the study of some extensions of the notion of IDT processes in the weak sense as well as in the multiparameter sense. In particular, a new approach for multiparameter IDT processes is introduced and studied. Main examples of this kind of processes are the $\mathbb{R}_{+}^{N}$-parameter L\'{e}vy process and the L\'{e}vy's $\mathbb{R}^{M}$-parameter Brownian motion. These results give some better understanding of IDT processes, and may be seen as some continuation of the works of K. Es-Sebaiy and Y. Ouknine [\textit{How rich is the class of processes which are infinitely divisible with respect to time ?}] and R. Mansuy [\textit{On processes which are infinitely divisible with respect to time}]

Strong envelope and strong supermartingale: application to reflected bsdes

We provide several characterizations to identify Strong envelop (for bounded measurable process) and Strong super-martingale (for non-negative right upper semi-continuous process of the class \Dc). As examples of application, we prove existence and uniqueness of reflected backward stochastic differential equation with lower barrier (RBSDB in short) in two cases: $i)$. the obstacle is a measurable bounded process; $ii)$. the obstacle is a right upper semicontinuous optional process of class \Dc

Non linear optimal stopping problem and Reflected BSDEs in the predictable setting

In the first part of this paper, we study RBSDEs in the case where the filtration is not quasi-left continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from the general theory of processes as the Merten's decomposition of predictable strong supermartingale. In the second part we introduce an optimal stopping problem indexed by predictable stopping times with the non linear predictable $g$ expectation induced by an appropriate BSDE. We establish some useful properties of ${\cal{E}}^{p,g}$-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part

Reflected backward stochastic differential equations with jumps in time-dependent random convex domains

In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domain ${\cal{D}}=\{D_t, t\in[0,T]\}$. We prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of $\cal{D}$, via a penalization method.Comment: 43 pages. arXiv admin note: text overlap with arXiv:1307.2124 by other author

A linear stochastic differential equation driven by a fractional Brownian motion with Hurst parameter >1/2

Given a fractional Brownian motion \,\,$(B_{t}^{H})_{t\geq 0}$,\, with Hurst parameter \,$> 1/2$\,\,we study the properties of all solutions of \,\,: {equation} X_{t}=B_{t}^{H}+\int_0^t X_{u}d\mu(u), \;\; 0\leq t\leq 1{equation} A different stochastic calculus is required for the process because it is not a semimartingale

Quadratic BSDEs with L2\mathbb{L}^2--terminal data Existence results, Krylov's estimate and It\^o--Krylov's formula

In a first step, we establish the existence (and sometimes the uniqueness) of solutions for a large class of quadratic backward stochastic differential equations (QBSDEs) with continuous generator and a merely square integrable terminal condition. Our approach is different from those existing in the literature. Although we are focused on QBSDEs, our existence result also covers the BSDEs with linear growth, keeping $\xi$ square integrable in both cases. As byproduct, the existence of viscosity solutions is established for a class of quadratic partial differential equations (QPDEs) with a square integrable terminal datum. In a second step, we consider QBSDEs with measurable generator for which we establish a Krylov's type a priori estimate for the solutions. We then deduce an It\^o--Krylov's change of variable formula. This allows us to establish various existence and uniqueness results for classes of QBSDEs with square integrable terminal condition and sometimes a merely measurable generator. Our results show, in particular, that neither the existence of exponential moments of the terminal datum nor the continuity of the generator are necessary to the existence and/or uniqueness of solutions for quadratic BSDEs. Some comparison theorems are also established for solutions of a class of QBSDEs.Comment: 23 pages: Most of the results have been announced in the CRAS note: C.R. Acad. Sci. Paris, Ser. I. 351, (2013) 229-233. The results were presented by Khaled Bahlali at the "7th International Symposium on BSDEs (22-27 June 2014)" in Shandong University, Weihai (China) on June 26, 201

On one-dimensional stochastic differential equations involving the maximum process

We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation \label{eq1} X_{t}=\int_{0}^{t}\sigma (s,X_{s})dW_{s}+\int_{0}^{t}b(s,X_{s})ds+\alpha \max_{0\leq s\leq t}X_{s}. The second type is the equation \label{eq2} {l} X_{t} =\ig{0}{t}\sigma (s,X_{s})dW_{s}+\ig{0}{t}b(s,X_{s})ds+\alpha \max_{0\leq s\leq t}X_{s}\,\,+L_{t}^{0}, X_{t} \geq 0, \forall t\geq 0. The third type is the equation \label{eq3} X_{t}=x+W_{t}+\int_{0}^{t}b(X_{s},\max_{0\leq u\leq s}X_{u})ds. We end the paper by establishing the existence of strong solution and pathwise uniqueness, under Lipschitz condition, for the SDE \label{e2} X_t=\xi+\int_0^t \si(s,X_s)dW_s +\int_0^t b(s,X_s)ds +\al\max_{0\leq s\leq t}X_s +\be \min_{0\leq s \leq t}X_s.Comment: 16 pages, published in at this http://www.worldscinet.com/sd/09/0902/S0219493709002671.html Stochastics and Dynamic

On semimartingale local time inequalities and applications in SDE's

Using the balayage formula, we prove an inequality between the measures associated to local times of semimartingales. Our result extends the "comparison theorem of local times" of Ouknine $(1988)$, which is useful in the study of stochastic differential equations. The inequality presented in this paper covers the discontinuous case. Moreover, we study the pathwise uniqueness of some stochastic differential equations involving local time of unknown process

Estimation of the drift of fractional Brownian motion

We consider the problem of efficient estimation for the drift of fractional Brownian motion $B^H:=(B^H_t)_{t\in[0,T]}$ with hurst parameter $H$ less than 1/2. We also construct superefficient James-Stein type estimators which dominate, under the usual quadratic risk, the natural maximum likelihood estimator