Mean-field Backward Stochastic Differential Equations With Quadratic Growth

Since Buckdahn, Djehiche, Li, and Peng [1] firstly introduced mean-field backward stochastic differential equations, lots of attention have been attract-ed. Du, Li and Wei [2] studied one dimensional mean-field backward stochastic differential equations with continuous coefficients. Scholars found the connec-tions between such equations and problems in various field, such as partial differential equations, stochastic control and stochastic differential games, etc. (for instance, [1], [3], [4], [5]). However, assumptions on coefficients are mostly too strict in the former works.In this paper, we firstly prove the existence of the solutions for mean-field backward stochastic differential equations with quadratic growth under different conditions when terminal values are bounded. On the other hand, classical mean-field backward stochastic differential equations are all based on natural filtrations (generated by brownian motion) and we consider a class of mean-field backward stochastic differential equations with respect to general fitrations. Therefore, this paper is divided into two parts according to the contents.Part I:One dimensional mean-field backward stochastic differential equa-tions with quadratic growth in z under different conditions are studied, Yy=(?)+(?)TtE’[f(s,Y’s,Ys,Zs)]ds-(?)ZsdWs,0≤t≤T, (1) where f(t, y, y’, z) is quadratic in z. Since the case that the terminal value (?) unbounded is more complex and difficult to deal with, our disscussions on (1) are based on the framework of bounded terminal value.First, we prove the existence of the solutions for mean-field backward stochastic differential equation (1) whose coefficient f is continuous, nonde-creasing in y1 super-linear in y, quadratic in z, i.e., |f(t,y1,y,z)|≤<l(y)+C|z|2, where l is a strictly positive continuous function which belongs to a functional class. The structure of quadratic mean-field backward stochastic differential equation is more special. We study the solutions of (1) by considering its e-quivalent equation’s solutions in method of exponential transform. We choose a sequence of continuous functions to approximate the coefficient of equivalent equation, which is different from the classical method of using Lipchitz func-tional sequence. Meanwhile, comparison theorem is obtained for (1) under the same assumptions, which is crucial in the following discussions.Second, we get that (1) has a maximal bounded solution when f satisfies monotonic condition in y, (f (t,y1,y,z)—f (t,y1,0,z))y≤μy2 for some constant μ≥0, and |f(t,y1,y,z)|≤)(?)(|y|)＋A|z|2 for some non-decreasing function (?):R+â†'> R+and constant A≥ 0.Third, we have a general result that the maximal solution also exists when the coefficient f(t, y1,y, z) is non-decreasing in y’, linear in (y1, y) and quadratic m z, |f(t,y’,y,z)|<C(1+|y’|+|y|+|z|2).Part â…¡:We extend a class of mean-field backward stochastic differential equations with respect to general filtrations with the form Yt=(?)+(E’[f(s, Y’s, Ys)]ds-(MT-Mt), (2) where M is an RCLL martingale adapted to such filtrations. We consider the coefficient f satisfies the following assumptions:(M1)E[(?)0T|f(t,0,0)|2]<∞;(M2)there exists C>0 such that |f(t,y1,z1)—f(t,y2,z2)|≤C(|y1-y2|+|z1+z2|),P-a.s., for all(y1,y2,z1,z1)∈R4;(M3)there exist two positive non-random functions u(t)and v(t)such that |f(t,y1,z1)—f(t,y2,z2)|≤u(t)|y1-y2|+v(t)|z1-z2|,P-a.s., for all(y1,y,z1,z2)∈R4,where (?)0T[u(t)+v(t)]dt<+∞;(M4)there exists C>0 such that (y1-y2)(f(t,y1,y1)—f(t,y2,y2))≤C|y1-y2|2,P-a.s.;(M5)f(t,y’,y)is continuous in y’,y and there exists a positive non-random function A(t)such that,|f(t,y’,y)|≤A(t),P-a.s.,P-a.s.,for all y’,y∈R, where (?)0TA(s)ds<∞.We prove the existence and uniqueness of the solution for(2)in s2×M2 under different conditions,i.e.,f satisfies(M1)+(M2),(M1)+(M3),(M1) +(M4)+(M5),respectivel y.Finally, we discuss a class of reflected mean-filed backward stochastic dif-ferential equations with the form and the corresponding existence and uniqueness theorem for(3)is also estab-lished when f satisfies(M1)and(M2).