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1.
Rainer Buckdahn Juan Li 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(3):381-420
In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly
controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which
can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games
are interpreted as games of the type “admissible strategy” against “admissible control”, and the associated lower and upper
value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique
viscosity solutions of the associated upper and the lower Bellman–Isaacs equations with two barriers, respectively. For the
proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end
we also prove new estimates for RBSDEs with two barriers, which are sharper than those in Hamadène, Hassani (Probab Theory
Relat Fields 132:237–264, 2005). Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting
barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also
directly by the viscosity solutions of penalized Isaacs equations without barrier.
Partially supported by the NSF of P.R.China (No. 10701050; 10671112), Shandong Province (No. Q2007A04), and National Basic
Research Program of China (973 Program) (No. 2007CB814904). 相似文献
2.
In this paper we study the integral–partial differential equations of Isaacs’ type by zero-sum two-player stochastic differential games (SDGs) with jump-diffusion. The results of Fleming and Souganidis (1989) [9] and those of Biswas (2009) [3] are extended, we investigate a controlled stochastic system with a Brownian motion and a Poisson random measure, and with nonlinear cost functionals defined by controlled backward stochastic differential equations (BSDEs). Furthermore, unlike the two papers cited above the admissible control processes of the two players are allowed to rely on all events from the past. This quite natural generalization permits the players to consider those earlier information, and it makes more convenient to get the dynamic programming principle (DPP). However, the cost functionals are not deterministic anymore and hence also the upper and the lower value functions become a priori random fields. We use a new method to prove that, indeed, the upper and the lower value functions are deterministic. On the other hand, thanks to BSDE methods (Peng, 1997) [18] we can directly prove a DPP for the upper and the lower value functions, and also that both these functions are the unique viscosity solutions of the upper and the lower integral–partial differential equations of Hamilton–Jacobi–Bellman–Isaacs’ type, respectively. Moreover, the existence of the value of the game is got in this more general setting under Isaacs’ condition. 相似文献
3.
Nonlinear BSDEs were first introduced by Pardoux and Peng, 1990, Adapted solutions of backward stochastic differential equations, Systems and Control Letters, 14, 51–61, who proved the existence and uniqueness of a solution under suitable assumptions on the coefficient. Fully coupled forward–backward stochastic differential equations and their connection with PDE have been studied intensively by Pardoux and Tang, 1999, Forward–backward stochastic differential equations and quasilinear parabolic PDE's, Probability Theory and Related Fields, 114, 123–150; Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569; Hamadème, 1998, Backward–forward SDE's and stochastic differential games, Stochastic Processes and their Applications, 77, 1–15; Delarue, 2002, On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case, Stochastic Processes and Their Applications, 99, 209–286, amongst others. Unfortunately, most existence or uniqueness results on solutions of forward–backward stochastic differential equations need regularity assumptions. The coefficients are required to be at least continuous which is somehow too strong in some applications. To the best of our knowledge, our work is the first to prove existence of a solution of a forward–backward stochastic differential equation with discontinuous coefficients and degenerate diffusion coefficient where, moreover, the terminal condition is not necessary bounded. The aim of this work is to find a solution of a certain class of forward–backward stochastic differential equations on an arbitrary finite time interval. To do so, we assume some appropriate monotonicity condition on the generator and drift coefficients of the equation. The present paper is motivated by the attempt to remove the classical condition on continuity of coefficients, without any assumption as to the non-degeneracy of the diffusion coefficient in the forward equation. The main idea behind this work is the approximating lemma for increasing coefficients and the comparison theorem. Our approach is inspired by recent work of Boufoussi and Ouknine, 2003, On a SDE driven by a fractional brownian motion and with monotone drift, Electronic Communications in Probability, 8, 122–134; combined with that of Antonelli and Hamadène, 2006, Existence of the solutions of backward–forward SDE's with continuous monotone coefficients, Statistics and Probability Letters, 76, 1559–1569. Pursuing this idea, we adopt a one-dimensional framework for the forward and backward equations and we assume a monotonicity property both for the drift and for the generator coefficient. At the end of the paper we give some extensions of our result. 相似文献
4.
We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale. 相似文献
5.
N. V. Krylov 《Probability Theory and Related Fields》2014,158(3-4):751-783
We prove the dynamic programming principle for uniformly nondegenerate stochastic differential games in the framework of time-homogeneous diffusion processes considered up to the first exit time from a domain. In contrast with previous results established for constant stopping times we allow arbitrary stopping times and randomized ones as well. There is no assumption about solvability of the the Isaacs equation in any sense (classical or viscosity). The zeroth-order “coefficient” and the “free” term are only assumed to be measurable in the space variable. We also prove that value functions are uniquely determined by the functions defining the corresponding Isaacs equations and thus stochastic games with the same Isaacs equation have the same value functions. 相似文献
6.
We study a forward-backward system
of stochastic differential equations in an
infinite-dimensional framework and its relationships
with a semilinear parabolic differential equation on a Hilbert space,
in the spirit of the approach of Pardoux-Peng.
We prove that the stochastic system
allows us to construct a unique
solution of the parabolic equation in
a suitable class of locally Lipschitz real
functions. The parabolic equation is understood in
a mild sense which requires the notion
of a generalized directional gradient, that
we introduce by a probabilistic approach
and prove to exist for locally Lipschitz
functions.
The use of the generalized directional gradient
allows us to cover various applications to option
pricing problems and to optimal stochastic control problems
(including control of delay equations and
reaction--diffusion equations),
where the lack of differentiability of the coefficients
precludes differentiability of solutions to the associated
parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman
type. 相似文献
7.
This paper deals with a class of anticipated backward stochastic differential equations. We extend results of Peng and Yang (2009) to the case in which the generator satisfies non-Lipschitz condition. The existence and uniqueness of solutions for anticipated backward stochastic differential equations as well as a comparison theorem are obtained. The existence and uniqueness of Lp(p>2) solutions for anticipated backward stochastic differential equations are also studied. 相似文献
8.
9.
10.
Qian Lin 《Stochastic Processes and their Applications》2012,122(1):357-385
In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach. 相似文献
11.
《数学物理学报(B辑英文版)》2017,(5)
We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs' condition. 相似文献
12.
Shaokuan Chen 《随机分析与应用》2013,31(5):820-841
In this article, we study one-dimensional backward stochastic differential equations with continuous coefficients. We show that if the generator f is uniformly continuous in (y, z), uniformly with respect to (t, ω), and if the terminal value ξ ∈L p (Ω, ? T , P) with 1 < p ≤ 2, the backward stochastic differential equation has a unique L p solution. 相似文献
13.
14.
Abstract In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 p -consistency, numerical 𝕃 p -stability and 𝕃 p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency. 相似文献
15.
The authors discuss one type of general forward-backward stochastic differential
equations (FBSDEs) with It?o’s stochastic delayed equations as the forward equations and
anticipated backward stochastic differential equations as the backward equations. The
existence and uniqueness results of the general FBSDEs are obtained. In the framework
of the general FBSDEs in this paper, the explicit form of the optimal control for linearquadratic
stochastic optimal control problem with delay and the Nash equilibrium point
for nonzero sum differential games problem with delay are obtained. 相似文献
16.
In this paper, we establish a local representation theorem for generators of reflected backward stochastic differential equations (RBSDEs), whose generators are continuous with linear growth. It generalizes some known representation theorems for generators of backward stochastic differential equations (BSDEs). As some applications, a general converse comparison theorem for RBSDEs is obtained and some properties of RBSDEs are discussed. 相似文献
17.
Juliang Yin 《Bulletin des Sciences Mathématiques》2010,134(8):799-815
This paper investigates a class of multi-dimensional stochastic differential equations with one reflecting lower barrier (RBSDEs in short), where the random obstacle is described as an Itô diffusion type of stochastic differential equation. The existence and uniqueness results for adapted solutions to such RBSDEs are established based on a penalization scheme and some higher moment estimates for solutions to penalized BSDEs under the Lipschitz condition and a higher moment condition on the coefficients. Finally, two examples are given to illustrate our theory and their applications. 相似文献
18.
Jonathan Bennett 《随机分析与应用》2013,31(3):471-494
Abstract In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton–Jacobi–Bellman equation. 相似文献
19.
We prove a convergence theorem for a family of value functions associated with
stochastic control problems whose cost functions are defined by backward stochastic
differential equations. The limit function is characterized as a viscosity solution
to a fully nonlinear partial differential equation of second order. The key
assumption we use in our approach is shown to be a necessary and sufficient assumption
for the homogenizability of the control problem. The results generalize partially
homogenization problems for Hamilton–Jacobi–Bellman equations treated recently by
Alvarez and Bardi by viscosity solution methods. In contrast to their approach, we
use mainly probabilistic arguments, and discuss a stochastic control interpretation
for the limit equation. 相似文献
20.
We construct, for various classes of p-adic-valued functions, stochastic integrals with respect to the Poisson random measure. This leads to the construction of Markov processes over the field of p-adic numbers by means of stochastic differential equations. 相似文献