Abstract
We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDEs). This class is characterized by constraints on some uniform a priori estimates on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.
Highlights
Backward Stochastic Differential Equations Backward stochastic differential equations (BSDEs) have been first introduced in a linear version by Bismut [Bis73], but since the early nineties and the seminal work of Pardoux and Peng [PP90], there has been an increasing interest for these equations due to their wide range of applications in stochastic control, in finance or in the theory of partial differential equations
Let us recall that, solving a BSDE consists in finding an adapted pair of processes (Y, Z), where Y is a Rd-valued continuous process and Z is a Rd×k-valued progressively measurable process, satisfying the equation
Concerning the scalar case,i.e. d = 1, existence and uniqueness of solutions for quadratic BSDEs has been first proved by Kobylanski in [Kob00]
Summary
Backward Stochastic Differential Equations Backward stochastic differential equations (BSDEs) have been first introduced in a linear version by Bismut [Bis73], but since the early nineties and the seminal work of Pardoux and Peng [PP90], there has been an increasing interest for these equations due to their wide range of applications in stochastic control, in finance or in the theory of partial differential equations. Framework and first assumptions In this paper we consider the following quadratic BSDE on Rd: Yt = ξ + f (s,Ys, Zs)ds − ZsdWs, 0 t T, a.s. where f is a random function Ω × [0, T ] × Rd × Rd×k → Rd called the generator of the BSDE such that for all (y, z) ∈ Rd × Rd×k and t ∈ [0, T ], (f (t, y, z))0tT is progressively measurable, (Y, Z) is a process with values in Rd × Rd×k and ξ ∈ L2 FT , Rd. Definition – 1.1. We consider an approximation of the BSDE (1.5) To this purpose let us introduce a localisation of f defined by f M(t, y, z) = f (t, y, ρM(z)) where ρM : Rd×k → Rd×k satisfies the following properties :.
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