# Monotonic Limit Properties for Solutions of BSDEs with Continuous Coefficients

Authors: ShengJun Fan; Xing Song; Ming Ma
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

This paper investigates the monotonic limit properties for the minimal and maximal solutions of certain one-dimensional backward stochastic differential equations with continuous coefficients.


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## Body

## 1. Introduction

Let(Ω,ℱ,P) be a probability space carrying a standard d-dimensional Brownian motion (Bt)t≥0. Fix a terminal time T>0, let (ℱt)t≥0 be the natural σ-algebra generated by (Bt)t≥0, and assume ℱT=ℱ. For every positive integer n, we use |·| to denote norm of Euclidean space Rn. For t∈[0,T], let L2(Ω,ℱt,P) denote the set of all ℱt-measurable random variables ξ such that E|ξ|2<+∞. Let Lℱ2(0,T;Rn) denote the set of ℱt-progressively measurable Rn-valued processes {Xt,t∈[0,T]} such that(1.1)∥X∥2=̂(E∫0T|Xt|2dt)1/2<+∞.This paper is concerned with the following one-dimensional BSDE:(1.2)yt=ξ+∫tTg(s,ys,zs)ds-∫tTzs·dBs,t∈[0,T],
where the random function g(ω,t,y,z):Ω×[0,T]×R×Rd→R is progressively measurable for each (y,z) in R×Rd, termed the generator of the BSDE(1), and ξ is an ℱT-measurable random variables termed the terminal condition. The triple (ξ,T,g) is called the parameters of the BSDE(1). In this paper, for each (ξ,T,g), by solution to the BSDE(1) we mean a pair of processes (y·,z·) in Lℱ2(0,T;R1+d) which satisfies the BSDE(1) and y is a continuous process. Such equations, in nonlinear case, have been introduced by Pardoux and Peng [1]; they established an existence and uniqueness result of solutions of the BSDE(1) under the Lipschitz assumption of the generator g. Since then, these equations and their generalizations have been the subject of a great number of investigations, such as [2, 3]. Particularly, Lepeltier and San Martin [4] obtained the following result when the generator g is only continuous with a linear growth.Proposition 1.1 (see [4,Theorem 1]).
Assume that the generatorg satisfies(H1)
linear growth: there existsK<+∞,for allω,t,y,z,|g(ω,t,y,z)|≤K(1+|y|+|z|);(H2)
for fixedω,t, g(ω,t,·,·) is continuous.
Then, ifξ∈L2(Ω,ℱT,P), the BSDE(1) has a unique minimal solution (Y̲·,Z̲·) and a unique maximal solution (Y¯·,Z¯·), which means that both (Y̲·,Z̲·) and (Y¯·,Z¯·) are the solution of (1.2), and for any other solution (Y·,Z·) of (1.2) one has Y̲·≤Y·≤Y¯· For convenience, for each t∈[0,T], one denoted Y̲t by ℰt,Tg̲[ξ], and Y¯t by ℰt,Tg¯[ξ].This paper will work on the assumptions (H1) and (H2) and investigate the monotonic limit properties on the operatorsℰt,Tg̲[·] and ℰt,Tg¯[·].


## 2. Main Results

In this section, we always assume that the generatorg satisfies assumptions (H1) and (H2). The following Theorem 2.1 and Remark 2.2 are the main results of this paper.Theorem 2.1.
Assume that the generatorg satisfies assumptions (H1) and (H2). Let t∈[0,T], ξn∈L2(Ω,ℱt,P), n∈N, and E|ξ|2<+∞.
Ifξn↑ξ,P-a.s., then for all s∈[0,t],
(2.1)limn→∞↑ℰs,tg̲[ξn]=ℰs,tg̲[limn→∞ξn]=ℰs,tg̲[ξ],P-a.s.
Ifξn↓ξ,P-a.s., then for all s∈[0,t],
(2.2)limn→∞↓ℰs,tg¯[ξn]=ℰs,tg¯[limn→∞ξn]=ℰs,tg¯[ξ],P-a.s.Remark 2.2.
If the condition “ξn↑ξ” in Theorem 2.1 is replaced by “ξn↓ξ”, the conclusion of the first part of Theorem 2.1 does not hold in general. Similarly, the condition “ξn↓ξ” in Theorem 2.1 cannot be replaced by “ξn↑ξ” in general. For example, we consider the BSDE with
(2.3)g(u,y,z)=7y6/7,ξ=0.
It is easy to see that both
(2.4)(yr,zr)r∈[0,t]=(0,0)r∈[0,t],(Yr,Zr)r∈[0,t]=((t-r)7,0)r∈[0,t]
are solutions of BSDE
(2.5)yr=∫rt7yu6/7du-∫rtzudBu,r∈[0,t].
For each n∈N, set ξn=1/n, then ξn∈L2(Ω,ℱt,P), E|ξ|2<+∞ and ξn↓ξ,P-a.s. However, one can verify that for each s∈[0,t],
(2.6)ℰs,tg̲[ξn]=(t-s+1n7)7.
Consequently,
(2.7)limn→∞ℰs,tg̲[ξn]=Ys≠ℰs,tg̲[ξ]≤0.
So, the conclusion of the first part of Theorem 2.1 does not hold.In order to prove Theorem2.1, we need the following lemmas. Lemma 2.3 is actually a direct corollary of Theorem 1.1 in [5].Lemma 2.3.
Assume that the generatorg satisfies assumptions (H1) and (H2). Let t∈[0,T] and ξ,ξ′∈L2(Ω,ℱt,P). If ξ≤ξ′,P-a.s., then
(2.8)∀s∈[0,t],ℰs,tg̲[ξ]≤ℰs,tg̲[ξ′],P-a.s.,∀s∈[0,t],ℰs,tg¯[ξ]≤ℰs,tg¯[ξ′],P-a.s.From the procedure of the proof of Theorem2.1 in [4], we can obtain the following Lemma 2.4.Lemma 2.4.
If the functiong satisfies (H1) and (H2), and one sets
(2.9)g̲m(t,y,z):=inf(u,v)∈Q1+d{g(t,u,v)+m(|y-u|+|z-v|)},g̅m(t,y,z):=sup(u,v)∈Q1+d{g(t,u,v)-m(|y-u|+|z-v|)},
then for any m>K, g̲m and g̅m are Lipschitz functions with constant m, that is, for any y1,y2∈R,z1,z2∈Rd and t∈[0,T],
(2.10)|g̲m(t,y1,z1)-g̲m(t,y2,z2)|≤m(|y1-y2|+|z1-z2|),|g̅m(t,y1,z1)-g̅m(t,y2,z2)|≤m(|y1-y2|+|z1-z2|).Moreover, lett∈[0,T] and ξ∈L2(Ω,ℱt,P), and let (Y̲rm,Z̲rm)r∈[0,t] and (Y̅rm,Z̅rm)r∈[0,t] be the unique solutions of the BSDEs with parameters (ξ,t,g̲m) and (ξ,t,g̅m), respectively. For convenience, from now on, we denoted Y̲sm by ℰs,tg̲m[ξ], and Y¯sm by ℰs,tg̅m[ξ] for each s∈[0,t], then for each s∈[0,t], we have(2.11)limm→∞↑ℰs,tg̲m[ξ]=ℰs,tg̲[ξ],P-a.s.,limm→∞↓ℰs,tg̅m[ξ]=ℰs,tg¯[ξ],P-a.s.Finally, the following Lemma2.5 can be easily obtained by [6,Lemma 1].Lemma 2.5.
Lett∈[0,T], ξn∈L2(Ω,ℱt,P), n∈N, and E|ξ|2<+∞, and let the generators g,g̲m and g̅m be defined as that in Lemma 2.4.
Ifξn↑ξ,P-a.s., then for all s∈[0,t] and each m>K,
(2.12)limn→∞↑ℰs,tg̲m[ξn]=ℰs,tg̲m[limn→∞ξn]=ℰs,tg̲m[ξ],P-a.s.
Ifξn↓ξ,P-a.s., then for all s∈[0,t] and each m>K,
(2.13)limn→∞↓ℰs,tg̅m[ξn]=ℰs,tg̅m[limn→∞ξn]=ℰs,tg̅m[ξ],P-a.s.Now, we are in the position to prove Theorem2.1.Proof of Theorem2.1.
We only prove the first part of this theorem, in the same way, one can complete the proof of the second part.
Sinceξn∈L2(Ω,ℱt,P) and limn→∞↑ξn=ξ,P-a.s., one knows that ξ∈ℱt. Thus, by E|ξ|2<+∞, we have ξ∈L2(Ω,ℱt,P), then by Proposition 1.1, for each s∈[0,t], both ℰs,tg̲[ξn] and ℰs,tg̲[ξ] are well defined. Moreover, according to Lemma 2.3, ℰs,tg̲[ξn] is nondecreasing with respect to n and bounded by ℰs,tg̲[ξ] from above, so in the sense of “almost surely,” the limit of the sequence ℰs,tg̲[ξn] must exist. Thus, in order to complete the proof of Theorem 2.1, we need only to prove that this limit is just ℰs,tg̲[ξ].
Let functionsg̲m and g̅m be defined for each m>K as that in Lemma 2.4, then from Lemmas 2.3 and 2.4 one deduce that for each n∈N, m>K and s∈[0,t],
(2.14)0≥ℰs,tg̲[ξn]-ℰs,tg̲[ξ]=ℰs,tg̲[ξn]-ℰs,tg̲m[ξn]+ℰs,tg̲m[ξn]-ℰs,tg̲m[ξ]+ℰs,tg̲m[ξ]-ℰs,tg̲[ξ]≥ℰs,tg̲m[ξn]-ℰs,tg̲m[ξ]+ℰs,tg̲m[ξ]-ℰs,tg̲[ξ],P-a.s.
Letting n→∞ in (2.14), from Lemma 2.5 we get that for each m>K,
(2.15)0≥limn→∞ℰs,tg̲[ξn]-ℰs,tg̲[ξ]≥ℰs,tg̲m[ξ]-ℰs,tg̲[ξ],P-a.s.
Furthermore, letting m→∞ in (2.15), from Lemma 2.4 we can easily deduce that for each s∈[0,t],
(2.16)limn→∞ℰs,tg̲[ξn]=ℰs,tg̲[ξ],P-a.s.
The proof of Theorem 2.1 is completed.According to Theorem2.1, we can obtain the following theorem.Theorem 2.6.
Assume that the generatorg satisfies assumptions (H1) and (H2). Let t∈[0,T] and ξn∈L2(Ω,ℱt,P), n∈N, Let E|η|2<+∞ and E|ζ|2<+∞.
Ifξn≥ζ,P-a.s.(n∈N) with E|limn→∞¯ξn|2<+∞, then for all s∈[0,t],
(2.17)ℰs,tg̲[limn→∞¯ξn]≤limn→∞¯ℰs,tg̲[ξn],P-a.s.
Ifξn≤η,P-a.s.(n∈N) with E|limn→∞¯ξn|2<+∞, then for all s∈[0,t],
(2.18)ℰs,tg¯[limn→∞¯ξn]≥limn→∞¯ℰs,tg¯[ξn],P-a.s.Proof.
We only prove the first part of this theorem, the proof of the second part is similar.
Let us fixs∈[0,t]. Since ξn∈L2(Ω,ℱt,P), then lim̲n→∞ξn∈ℱt, and by the assumption of this theorem one knows that
(2.19)limn→∞¯ξn∈L2(Ω,ℱt,P),
thus by Proposition 1.1, ℰs,tg̲[lim̲n→∞ξn] is well defined.
We setYn=infk≥nξk, then Yn∈ℱt,andζ≤Yn↑lim̲n→∞ξn,P-a.s. So for each n∈N,
(2.20)E|Yn|2≤max{E|ζ|2,E|limn→∞¯ξn|2}<+∞.
Thus,
(2.21)Yn∈L2(Ω,ℱt,P),
then ℰs,tg̲[Yn] is also well defined. Since Yn≤ξn,P-a.s., by Lemma 2.3 we know that ℰs,tg̲[Yn]≤ℰs,tg̲[ξn],P-a.s., then applying Theorem 2.1 to the random variable sequence {Yn}, we get
(2.22)ℰs,tg̲[limn→∞¯ξn]=ℰs,tg̲[limn→∞Yn]=limn→∞ℰs,tg̲[Yn]≤limn→∞¯ℰs,tg̲[ξn],P-a.s.
The proof is completed.



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*Source: 671643-2010-01-20.xml*