Monotonic Limit Properties for Solutions of BSDEs with Continuous Coefficients

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.

1. Introduction

Let (Ω,ℱ,𝑃) be a probability space carrying a standard 𝑑-dimensional Brownian motion (𝐵_𝑡)_𝑡≥0. Fix a terminal time 𝑇>0, let (ℱ_𝑡)_𝑡≥0 be the natural 𝜎-algebra generated by (𝐵_𝑡)_𝑡≥0, and assume ℱ_𝑇=ℱ. For every positive integer 𝑛, we use |⋅| to denote norm of Euclidean space 𝐑^𝑛. For 𝑡∈[0,𝑇], let 𝐿²(Ω,ℱ_𝑡,𝑃) denote the set of all ℱ_𝑡-measurable random variables 𝜉 such that 𝐸|𝜉|²<+∞. Let 𝐿²_ℱ(0,𝑇;𝐑^𝑛) denote the set of ℱ_𝑡-progressively measurable 𝐑^𝑛-valued processes {𝑋_𝑡,𝑡∈[0,𝑇]} such that

‖𝑋‖₂=(𝐸^𝑇₀|𝑋_𝑡|²d𝑡)^1/2<+∞. (1.1)

This paper is concerned with the following one-dimensional BSDE:

𝑦_𝑡=𝜉+^𝑇_𝑡𝑔(𝑠,𝑦_𝑠,𝑧_𝑠)d𝑠−^𝑇_𝑡𝑧_𝑠⋅d𝐵_𝑠,𝑡∈[0,𝑇], (1.2) where the random function 𝑔(𝜔,𝑡,𝑦,𝑧)∶Ω×[0,𝑇]×𝐑×𝐑^𝑑→𝐑 is progressively measurable for each (𝑦,𝑧) in 𝐑×𝐑^𝑑, termed the generator of the BSDE(1), and 𝜉 is an ℱ_𝑇-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 𝐿²_ℱ(0,𝑇;𝐑^1+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 generator 𝑔 satisfies(H1) linear growth: there exists𝐾<+∞,for all𝜔,𝑡,𝑦,𝑧,|𝑔(𝜔,𝑡,𝑦,𝑧)|≤𝐾(1+|𝑦|+|𝑧|);(H2) for fixed 𝜔,𝑡, 𝑔(𝜔,𝑡,⋅,⋅) is continuous.
Then, if 𝜉∈𝐿²(Ω,ℱ_𝑇,𝑃), the BSDE(1) has a unique minimal solution (𝑌_⋅,𝑍_⋅) and a unique maximal solution (𝑌_⋅,𝑍_⋅), which means that both (𝑌_⋅,𝑍_⋅) and (𝑌_⋅,𝑍_⋅) are the solution of (1.2), and for any other solution (𝑌_⋅,𝑍_⋅) of (1.2) one has 𝑌_⋅≤𝑌_⋅≤𝑌_⋅ For convenience, for each 𝑡∈[0,𝑇], one denoted 𝑌_𝑡 by ℰ^𝑔_𝑡,𝑇[𝜉], and 𝑌_𝑡 by ℰ^𝑔_𝑡,𝑇[𝜉].

This paper will work on the assumptions (H1) and (H2) and investigate the monotonic limit properties on the operators ℰ^𝑔_𝑡,𝑇[⋅] and ℰ^𝑔_𝑡,𝑇[⋅].

2. Main Results

In this section, we always assume that the generator 𝑔 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 generator 𝑔 satisfies assumptions (H1) and (H2). Let 𝑡∈[0,𝑇], 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), 𝑛∈𝐍, and 𝐸|𝜉|²<+∞.
If 𝜉^𝑛↑𝜉,𝑃−𝑎.𝑠., then for all 𝑠∈[0,𝑡], lim_𝑛→∞↑ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]=ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]=ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.1)
If 𝜉^𝑛↓𝜉,𝑃−𝑎.𝑠., then for all 𝑠∈[0,𝑡], lim_𝑛→∞↓ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]=ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]=ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.2)

Remark 2.2. If the condition “𝜉^𝑛↑𝜉” in Theorem 2.1 is replaced by “𝜉^𝑛↓𝜉”, the conclusion of the first part of Theorem 2.1 does not hold in general. Similarly, the condition “𝜉^𝑛↓𝜉” in Theorem 2.1 cannot be replaced by “𝜉^𝑛↑𝜉” in general. For example, we consider the BSDE with 𝑔(𝑢,𝑦,𝑧)=7𝑦^6/7,𝜉=0. (2.3) It is easy to see that both (𝑦_𝑟,𝑧_𝑟)_{𝑟∈[0,𝑡]}=(0,0)_{𝑟∈[0,𝑡]},(𝑌_𝑟,𝑍_𝑟)_{𝑟∈[0,𝑡]}=((𝑡−𝑟)⁷,0)_{𝑟∈[0,𝑡]} (2.4) are solutions of BSDE 𝑦_𝑟=^𝑡_𝑟7𝑦_𝑢^6/7d𝑢−^𝑡_𝑟𝑧_𝑢d𝐵_𝑢,𝑟∈[0,𝑡]. (2.5) For each 𝑛∈𝐍, set 𝜉^𝑛=1/𝑛, then 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), 𝐸|𝜉|²<+∞ and 𝜉^𝑛↓𝜉,𝑃−𝑎.𝑠. However, one can verify that for each 𝑠∈[0,𝑡], ℰ^𝑔_𝑠,𝑡[𝜉^𝑛1]=(𝑡−𝑠+₇√𝑛)⁷. (2.6) Consequently, lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]=𝑌_𝑠≠ℰ^𝑔_𝑠,𝑡[𝜉]≤0. (2.7) So, the conclusion of the first part of Theorem 2.1 does not hold.

In order to prove Theorem 2.1, 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 generator 𝑔 satisfies assumptions (H1) and (H2). Let 𝑡∈[0,𝑇] and 𝜉,𝜉^′∈𝐿²(Ω,ℱ_𝑡,𝑃). If 𝜉≤𝜉^′,𝑃−𝑎.𝑠., then ∀𝑠∈[0,𝑡],ℰ^𝑔_𝑠,𝑡[𝜉]≤ℰ^𝑔_𝑠,𝑡[𝜉^′],𝑃−𝑎.𝑠.,∀𝑠∈[0,𝑡],ℰ^𝑔_𝑠,𝑡[𝜉]≤ℰ^𝑔_𝑠,𝑡[𝜉^′],𝑃−𝑎.𝑠. (2.8)

From the procedure of the proof of Theorem 2.1 in [4], we can obtain the following Lemma 2.4.

Lemma 2.4. If the function 𝑔 satisfies (H1) and (H2), and one sets 𝑔_𝑚(𝑡,𝑦,𝑧)∶=inf_{(𝑢,𝑣)∈𝐐1+𝑑}{𝑔(𝑡,𝑢,𝑣)+𝑚(|𝑦−𝑢|+|𝑧−𝑣|)},𝑔_𝑚(𝑡,𝑦,𝑧)∶=sup_{(𝑢,𝑣)∈𝐐1+𝑑}{𝑔(𝑡,𝑢,𝑣)−𝑚(|𝑦−𝑢|+|𝑧−𝑣|)}, (2.9) then for any 𝑚>𝐾, 𝑔_𝑚 and 𝑔_𝑚 are Lipschitz functions with constant 𝑚, that is, for any 𝑦₁,𝑦₂∈𝐑,𝑧₁,𝑧₂∈𝐑^𝑑 and 𝑡∈[0,𝑇], |𝑔_𝑚(𝑡,𝑦₁,𝑧₁)−𝑔_𝑚(𝑡,𝑦₂,𝑧₂)|≤𝑚(|𝑦₁−𝑦₂|+|𝑧₁−𝑧₂||),𝑔_𝑚(𝑡,𝑦₁,𝑧₁)−𝑔_𝑚(𝑡,𝑦₂,𝑧₂)|≤𝑚(|𝑦₁−𝑦₂|+|𝑧₁−𝑧₂|). (2.10)

Moreover, let 𝑡∈[0,𝑇] and 𝜉∈𝐿²(Ω,ℱ_𝑡,𝑃), and let (𝑌^𝑚_𝑟,𝑍^𝑚_𝑟)_{𝑟∈[0,𝑡]} and (𝑌^𝑚_𝑟,𝑍^𝑚_𝑟)_{𝑟∈[0,𝑡]} be the unique solutions of the BSDEs with parameters (𝜉,𝑡,𝑔_𝑚) and (𝜉,𝑡,𝑔_𝑚), respectively. For convenience, from now on, we denoted 𝑌^𝑚_𝑠 by ℰ^𝑔𝑚_𝑠,𝑡[𝜉], and 𝑌^𝑚_𝑠 by ℰ^𝑔𝑚_𝑠,𝑡[𝜉] for each 𝑠∈[0,𝑡], then for each 𝑠∈[0,𝑡], we have

lim_𝑚→∞↑ℰ^𝑔𝑚_𝑠,𝑡[𝜉]=ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠.,lim_𝑚→∞↓ℰ^𝑔𝑚_𝑠,𝑡[𝜉]=ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.11)

Finally, the following Lemma 2.5 can be easily obtained by [6,Lemma 1].

Lemma 2.5. Let 𝑡∈[0,𝑇], 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), 𝑛∈𝐍, and 𝐸|𝜉|²<+∞, and let the generators 𝑔,𝑔_𝑚 and 𝑔_𝑚 be defined as that in Lemma 2.4.
If 𝜉^𝑛↑𝜉,𝑃−𝑎.𝑠., then for all 𝑠∈[0,𝑡] and each 𝑚>𝐾, lim_𝑛→∞↑ℰ^𝑔𝑚_𝑠,𝑡[𝜉^𝑛]=ℰ^𝑔𝑚_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]=ℰ^𝑔𝑚_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.12)
If 𝜉^𝑛↓𝜉,𝑃−𝑎.𝑠., then for all 𝑠∈[0,𝑡] and each 𝑚>𝐾, lim_𝑛→∞↓ℰ^𝑔𝑚_𝑠,𝑡[𝜉^𝑛]=ℰ^𝑔𝑚_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]=ℰ^𝑔𝑚_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.13)

Now, we are in the position to prove Theorem 2.1.

Proof of Theorem 2.1. We only prove the first part of this theorem, in the same way, one can complete the proof of the second part.
Since 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃) and lim_𝑛→∞↑𝜉^𝑛=𝜉,𝑃−𝑎.𝑠., one knows that 𝜉∈ℱ_𝑡. Thus, by 𝐸|𝜉|²<+∞, we have 𝜉∈𝐿²(Ω,ℱ_𝑡,𝑃), then by Proposition 1.1, for each 𝑠∈[0,𝑡], both ℰ^𝑔_𝑠,𝑡[𝜉^𝑛] and ℰ^𝑔_𝑠,𝑡[𝜉] are well defined. Moreover, according to Lemma 2.3, ℰ^𝑔_𝑠,𝑡[𝜉^𝑛] is nondecreasing with respect to 𝑛 and bounded by ℰ^𝑔_𝑠,𝑡[𝜉] from above, so in the sense of “almost surely,” the limit of the sequence ℰ^𝑔_𝑠,𝑡[𝜉^𝑛] must exist. Thus, in order to complete the proof of Theorem 2.1, we need only to prove that this limit is just ℰ^𝑔_𝑠,𝑡[𝜉].
Let functions 𝑔_𝑚 and 𝑔_𝑚 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], 0≥ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]−ℰ^𝑔_𝑠,𝑡[𝜉]=ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]−ℰ^𝑔𝑚_𝑠,𝑡[𝜉^𝑛]+ℰ^𝑔𝑚_𝑠,𝑡[𝜉^𝑛]−ℰ^𝑔𝑚_𝑠,𝑡[𝜉]+ℰ^𝑔𝑚_𝑠,𝑡[𝜉]−ℰ^𝑔_𝑠,𝑡[𝜉]≥ℰ^𝑔𝑚_𝑠,𝑡[𝜉^𝑛]−ℰ^𝑔𝑚_𝑠,𝑡[𝜉]+ℰ^𝑔𝑚_𝑠,𝑡[𝜉]−ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.14) Letting 𝑛→∞ in (2.14), from Lemma 2.5 we get that for each 𝑚>𝐾, 0≥lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]−ℰ^𝑔_𝑠,𝑡[𝜉]≥ℰ^𝑔𝑚_𝑠,𝑡[𝜉]−ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.15) Furthermore, letting 𝑚→∞ in (2.15), from Lemma 2.4 we can easily deduce that for each 𝑠∈[0,𝑡], lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝜉^𝑛]=ℰ^𝑔_𝑠,𝑡[𝜉],𝑃−𝑎.𝑠. (2.16) The proof of Theorem 2.1 is completed.

According to Theorem 2.1, we can obtain the following theorem.

Theorem 2.6. Assume that the generator 𝑔 satisfies assumptions (H1) and (H2). Let 𝑡∈[0,𝑇] and 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), 𝑛∈𝐍, Let 𝐸|𝜂|²<+∞ and 𝐸|𝜁|²<+∞.
If 𝜉^𝑛≥𝜁,𝑃−𝑎.𝑠.(𝑛∈𝐍) with 𝐸|lim_𝑛→∞𝜉^𝑛|²<+∞, then for all 𝑠∈[0,𝑡], ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]≤lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝜉^𝑛],𝑃−𝑎.𝑠. (2.17)
If 𝜉^𝑛≤𝜂,𝑃−𝑎.𝑠.(𝑛∈𝐍) with 𝐸|lim_𝑛→∞𝜉^𝑛|²<+∞, then for all 𝑠∈[0,𝑡], ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]≥lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝜉^𝑛],𝑃−𝑎.𝑠. (2.18)

Proof. We only prove the first part of this theorem, the proof of the second part is similar.
Let us fix 𝑠∈[0,𝑡]. Since 𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), then lim_𝑛→∞𝜉^𝑛∈ℱ_𝑡, and by the assumption of this theorem one knows that lim_𝑛→∞𝜉^𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), (2.19) thus by Proposition 1.1, ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛] is well defined.
We set 𝑌_𝑛=inf_𝑘≥𝑛𝜉^𝑘, then 𝑌_𝑛∈ℱ_𝑡,and𝜁≤𝑌_𝑛↑lim_𝑛→∞𝜉^𝑛,𝑃−𝑎.𝑠. So for each 𝑛∈𝐍, 𝐸|𝑌_𝑛|²≤max{𝐸|𝜁|²,𝐸|lim_𝑛→∞𝜉^𝑛|²}<+∞. (2.20) Thus, 𝑌_𝑛∈𝐿²(Ω,ℱ_𝑡,𝑃), (2.21) then ℰ^𝑔_𝑠,𝑡[𝑌_𝑛] is also well defined. Since 𝑌_𝑛≤𝜉^𝑛,𝑃−𝑎.𝑠., by Lemma 2.3 we know that ℰ^𝑔_𝑠,𝑡[𝑌_𝑛]≤ℰ^𝑔_𝑠,𝑡[𝜉^𝑛],𝑃−𝑎.𝑠., then applying Theorem 2.1 to the random variable sequence {𝑌_𝑛}, we get ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝜉^𝑛]=ℰ^𝑔_𝑠,𝑡[lim_𝑛→∞𝑌_𝑛]=lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝑌_𝑛]≤lim_𝑛→∞ℰ^𝑔_𝑠,𝑡[𝜉^𝑛],𝑃−𝑎.𝑠. (2.22) The proof is completed.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 10971220) and Youth Foundation of China University of Mining and Technology (no. 2007A029).