Existence of solution to scalar BSDEs with weakly $L^{1+}$-integrable terminal values

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Existence of solution to scalar BSDEs with weakly L

¹⁺

-integrable terminal values

Ying Hu, Shanjian Tang

To cite this version:

Ying Hu, Shanjian Tang. Existence of solution to scalar BSDEs with weaklyL¹⁺-integrable terminal values. 2017. �hal-01507371v2�

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Existence of solution to scalar BSDEs with weakly L

¹⁺

-integrable terminal values

Ying Hu^∗ Shanjian Tang^† April 16, 2017

Abstract. In this paper, we study a scalar linearly growing BSDE with a weakly L¹⁺- integrable terminal value. We prove that the BSDE admits a solution if the terminal value satisfies some Ψ-integrability condition, which is weaker than the usual L^p (p > 1) integrability and stronger than LlogL integrability. We show by a counterexample that LlogL integrability is not sufficient for the existence of solution to a BSDE of a linearly growing generator.

AMS Subject Classification: 60H10

Key WordsBackward stochastic differential equation, weak integrability, terminal condition, dual representation.

1 Introduction

Consider the following Backward Stochastic Differential Equation (BSDE):

Y_t=ξ+ Z T

t

f(s, Y_s, Z_s)ds− Z T

t

Z_sdW_s, (1.1)

where the function f : [0, T]×Ω×R×R¹^×^d→Rsatisfies

|f(s, y, z)| ≤α_s+β|y|+γ|z|, withα ∈L¹(0, T), β ≥0 and γ >0.

It is well known that ifξ ∈L^p, with p > 1 , then there exists a solution to BSDE (1.1), see e.g. [5, 4, 1]. The aim of this paper is to find a weaker integrability condition for the terminal valueξ, under which the solution still exists .

Set

Ψ_λ(x) =xe⁽^λ²^log(x+1))^1/2, (λ, x) ∈(0,∞)×[0,∞).

Our sufficient condition is: there existsλ∈(0,_γ2¹T) such that E[Ψ_λ(|ξ|)]<+∞.

∗IRMAR, Universit´e Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France ([email protected]) and School of Mathematical Sciences, Fudan University, Shanghai 200433, China. Partially supported by Lebesgue center of mathematics “Investissements d’avenir” program - ANR-11-LABX-0020-01, by ANR CAESARS - ANR- 15-CE05-0024 and by ANR MFG - ANR-16-CE40-0015-01.

†Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China (e-mail: [email protected]). Partially supported by National Science Foundation of China (Grant No. 11631004) and Science and Technology Commission of Shanghai Municipality (Grant No. 14XD1400400).

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Remark 1.1 Note that the preceding Ψ_λ-integrability is stronger than L¹, weaker than L^p for any p >1, because for anyε >0, we have,

x≤xe⁽²^λ^log(x+1))^1/2 ≤xe^ε^log(x+1)+^2ελ¹ ≤e^2ελ¹ x(x+ 1)^ε, x≥0.

Moreover, for any p≥1, there exists a constant C_p>0 such that xe⁽^λ²^log(x+1))^1/2 ≥C_pxlog^p(x+ 1).

We will see that even the condition

E[Ψ_λ(|ξ|)]<+∞

for a certain λ > _γ2¹T (which implies that |ξ|log^p(|ξ|+ 1) ∈ L¹) is still too weak to ensure the existence of solution by giving a simple example in Example 2.3.

Note that if the generatorf is of sublinear growth with respect toz, i.e. there existsq ∈[0,1),

|f(t, y, z)| ≤α+β|y|+γ|z|^q, then there exists a solution forξ ∈L¹, see [1].

Our method applies the dual representation of solution to BSDE with convex generator (see, e.g. [4, 6, 3]) in order to establish some a priori estimate and then the localization procedure of real-valued BSDE [2].

Let us close this introduction by giving the notations that we will use in all the paper. For the remaining of the paper, let us fix a nonnegative real number T >0. First of all, (W_t)_t_∈_[0,T_] is a standard Brownian motion with values in R^d defined on some complete probability space (Ω,F,P). (Ft)_t_≥₀ is the natural filtration of the Brownian motion W augmented by the P-null sets of F. The sigma-field of predictable subsets of [0, T]×Ω is denoted by P.

Consider real valued BSDEs which are equations of type (1.1), wheref (hereafter called the generator) is a random function [0, T]×Ω×R×R¹^×^d → R and measurable with respect to P ⊗ B(R)⊗ B(R¹^×^d), and ξ (hereafter called the terminal condition or terminal value) is a real FT-measurable random variable.

Definition 1.2 By a solution to BSDE (1.1), we mean a pair (Y_t, Z_t)_t_∈_[0,T_] of predictable pro- cesses with values in R×R¹^×^d such that P-a.s.,t7→Y_t is continuous, t7→Z_t belongs toL²(0, T) and t7→g(t, Y_t, Z_t) belongs to L¹(0, T), and P-a.s. (Y, Z) verifies (1.1).

By BSDE (ξ,f), we mean the BSDE of generatorf and terminal condition ξ.

For any realp≥1,S^p denotes the set of real-valued, adapted and c`adl`ag processes (Y_t)_t_∈_[0,T_] such that

||Y||S^p :=E

"

sup

0≤t≤T|Y_t|^p

#1/p

<+∞,

and M^p denotes the set of (equivalent class of) predictable processes (Z_t)_t_∈_[0,T] with values in R¹^×^d such that

||Z||M^p :=E

"

Z T

0 |Z_s|²ds

^p/2#1/p

<+∞.

The rest of the paper is organized as follows. Section 2 establishes a necessary and sufficient condition for the existence of solution to BSDE (1.1) for the typical form of generatorf(t, y, z) = α_t+βy+γ|z|. Section 3 gives the Ψ_λintegrability condition for the existence of solution to BSDE (1.1) for f(t, y, z) =α_t+βy+γ|z|. Section 4 is devoted to the sufficiency of the Ψ_λ-integrability condition for the existence of solution to BSDE (1.1) of the general linearly growing generator.

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2 Typical Case

Let us first consider the following BSDE:

Y_t=ξ+ Z _T

t

(α_s+βY_s+γ|Z_s|)ds− Z _T

t

Z_sdW_s, (2.2)

where α∈L¹(0, T), and β ≥0 and γ >0 are some real constants. We suppose further that the terminal condition ξ is nonnegative. Note that if Y is a solution belonging to class D, then as e^βtY_t is a local supermartingale, it is a supermatingale, from which we deduce that Y ≥0. In this subsection, we restrict ourselves to nonnegative solution.

Forξ∈L^p (p >1), BSDE (2.2) has a unique solution. It has a dual representation as follows (see, e.g. [4, 3])

Y_t= ess sup

q∈A {E_q[e^β(T⁻^t)ξ|Ft]}+ Z T

t

e^β(s⁻^t)α_sds, (2.3) where Ais the set of progressively measurable processesq such that|q| ≤γ,

dQ^q

dP =M_T^q, with

M_t^q= exp{ Z t

0

q_sdW_s−1 2

Z t

0 |q_s|²ds}, t∈[0, T], and E_q is the expectation with respect toQ^q.

Theorem 2.1 Let us suppose that ξ ≥0. Then BSDE (2.2) admits a solution (Y, Z) such that Y ≥0 if and only if there exists a locally bounded process Y¯ such that

ess sup

q∈A {E_q[e^β(T⁻^t)ξ|Ft]}+ Z _T

t

e^β(s⁻^t)α_sds≤Y¯_t.

Proof. If BSDE (2.2) admits a solution (Y, Z) such that Y ≥ 0, then we define a sequence of stopping times

σ_n=T ∧inf{t≥0 :|Y_t|> n}, with the convention that inf∅= +∞.

AsW_s^q=W_s−Rs

0 q_rdris a Brownian motion underQ^q, we have Y_t_∧_σ_n =Y_σ_n+

Z _σ_n

t∧σn

(α_s+βY_s+γ|Z_s| −Z_sq_s)ds− Z _σ_n

t∧σn

Z_sdW_s^q.

Applying Itˆo’s formula to e^βsY_s, we deduce e^β(t^∧^σⁿ⁾Y_t_∧_σ_n =e^βσⁿY_σ_n+

Z _σ_n

t∧σn

e^βs(α_s+γ|Z_s| −Z_sq_s)ds− Z _σ_n

t∧σn

e^βsZ_sdW_s^q,

from which we obtain

E_q[e^β(σⁿ⁻^t^∧^σⁿ⁾Y_σ_n+ Z _σ_n

t∧σn

e^β(s⁻^t^∧^σⁿ⁾α_sds|Ft]≤Y_t_∧_σ_n. Fatou’s lemma yields that

E_q[e^β(T⁻^t)ξ|Ft] + Z T

t

e^β(s⁻^t)α_sds≤Y_t.

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On the other hand, if there exists a locally bounded process ¯Y such that ess sup

t

e^β(s⁻^t)α_sds≤Y¯_t,

then we construct the solution by use of a localization method (see e.g. [2]). We describe this method here for completeness. Let (Yⁿ, Zⁿ) be the unique solution in S²× M² of the following BSDE

Y_tⁿ=ξ1_{_ξ_≤_n_}+ Z _T

t

(α_s+βY_sⁿ+γ|Z_sⁿ|)ds− Z _T

t

Z_sⁿdW_s.

By comparison theorem,Yⁿis nondecreasing with respect ton. Moreover, settingq_sⁿ=γsgn(Z_sⁿ), we obtain

Y_tⁿ = E_qn[e^β(T⁻^t)ξ1_{_ξ_≤_n_}|Ft] + Z _T

t

e^β(s⁻^t)α_sds

≤ E_qn[e^β(T⁻^t)ξ|Ft]}+ Z T

t

e^β(s⁻^t)α_sds

≤ Y¯_t. Set

τ_k =T∧inf{t≥0 : ¯Y_t> k}, and

Y_kⁿ(t) =Y_tⁿ_∧_τ_k, Z_kⁿ(t) =Z_tⁿ1_t_≤_τ_k. Then (Y_kⁿ, Z_kⁿ) satisfies

Y_kⁿ(t) =Y_kⁿ(T) + Z _T

t

1_s_≤_τ_k(α_s+βY_kⁿ(s) +γ|Z_kⁿ(s)|)ds− Z _T

t

Z_kⁿ(s)dW_s. (2.4) For fixedk, Y_kⁿ is nondecreasing with respect to nand remains bounded by k. We can now apply the stability property of BSDE with bounded terminal data (see e.g. Lemma 3, page 611 in [2]). Setting Y_k(t) = sup_nY_kⁿ(t), there exists Z_k such that limnZ_kⁿ=Z_k inM² and

Y_k(t) = sup

n

Y_τⁿ_k+ Z τk

t

(α_s+βY_k(s) +γ|Z_k(s)|)ds− Z τk

t

Z_k(s)dW_s. (2.5) Finally, noting that

Y_k+1(t∧τ_k) =Y_k(t∧τ_k), Z_k+11_t_≤_τ_k =Z_k1_t_≤_τ_k, we conclude the existence of solution (Y, Z).

Remark 2.2 Consider the case d= 1. If BSDE (2.2) admits a solution(Y, Z) such that Y ≥0, by taking q=γ and q=−γ, we deduce that both ξe^γW^T and ξe⁻^γW^T are in L¹(Ω), which implies that ξe^γ^|^W^T^|∈L¹(Ω), as

ξe^γ^|^W^T^|≤ξe^γW^T +ξe⁻^γW^T. Example 2.3 Let us set d= 1, T = 1, β = 0, γ = 1, µ∈(0,1), and

ξ=e¹²^W¹²⁻^µ^|^W¹^|⁺¹²^µ²−1.

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In this case, BSDE (2.2) does not admit a solution (Y, Z) such that Y ≥0, as ξe^|^W¹^| does not belong to L¹(Ω) by the following direct calculus:

E[ξe^|^W¹^|] = 1

√2π Z +∞

−∞

(e¹²^|^x^|²⁻^µ^|^x^|⁺¹²^µ² −1)e^|^x^|e⁻¹²^|^x^|²dx= +∞. Whereas it is straightforward to see that for any p≥1, ξlog^p(ξ+ 1)∈L¹(Ω).

Forλ >1, consider the following terminal condition

ξ=e¹²^W¹²⁻^µ^|^W¹^|⁺¹²^µ² −1, µ∈( 1

√λ,1).

We have Ψ_λ(ξ)∈L¹ by the following straightforward calculus:

E[Ψ_λ(ξ)] = 1

√2π Z +∞

−∞

(e¹²^|^x^|²⁻^µ^|^x^|⁺¹²^µ²−1)e^√¹^λ _|x|−µ

e⁻¹²^|^x^|²dx <+∞,

while BSDE (2.2) has no solution, in view of the fact that ξe^|^W¹^| does not belong to L¹(Ω).

3 Sufficient Condition

Let us now look for a sufficient condition for the existence of a locally bounded process ¯Y such that

ess sup

q∈A {E_q[e^β(T⁻^t)ξ|Ft]}+ Z _T

t

e^β(s⁻^t)α_sds≤Y¯_t. Forλ >0, define the functions Φ_λ and Ψ_λ:

Φ_λ(x) =e¹²^λ^log²^(x), x >0, Ψ_λ(y) =ye⁽^λ²^log(y+1))^1/2, y≥0.

Then we have

Proposition 3.1 For any x∈R and y≥0, we have

e^xy ≤Φ_λ(e^x) +e²^λΨ_λ(y).

Proof. Set

z= (2

λlog(y+ 1))^1/2≥0, then

y =e^λ²^z² −1.

It is sufficient to prove that for any x∈Rand z≥0,

e¹²^λx²⁻^x+ (e^λ²^z² −1)(e^z+²^λ⁻^x−1)≥0.

It is evident to see that the above inequality holds when z+²_λ −x≥0.

Consider the casez+²_λ −x <0. Then x > z+_λ² >0. Hence e¹²^λx²⁻^x+ (e^λ²^z² −1)(e^z+²^λ⁻^x−1)

= e¹²^λ(x⁻^λ¹⁾²⁻^2λ¹ +e^λ²^z²^+z+^λ²⁻^x+ 1−e^z+^λ²⁻^x−e^λ²^z²

≥ e¹²^λ(z+^λ¹⁾²⁻^2λ¹ −e^λ²^z²

≥ 0.

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Proposition 3.2 Let 0< λ < _γ2¹T. For any q∈ A,

E[Φ_λ(e^R^t^T^q^s^dW^s)|Ft]≤ 1

p1−λγ²(T−t). Proof. Firstly, by use of Girsanov’s lemma, for θ∈R,

E[e^θ^R^t^T^q^s^dW^s|Ft]

= E[e^θ^R^t^T^q^s^dW^s⁻^θ

2 2

R_T

t |qs|²dse^θ

2 2

R_T

t |qs|²ds

|Ft]

≤ e^θ

2γ2 2 (T−t). Then we apply

e^λ^x

2

2 = 1

√2π Z ₊_∞

−∞

e^√^λyx⁻^y

2 2 dy to deduce that

E

Φ_λ(e^R^t^T^q^s^dW^s) Ft

= E e^λ²⁽

R_T

t qsdWs)² Ft

= 1

√2π Z +∞

−∞

E[e^√^λy^R^t^T^q^s^dW^s⁻^y

2 2 |Ft]dy

≤ 1

√2π Z +∞

−∞

e⁽

√λyγ)2

2 (T−t)−^y₂²dy

= 1

p1−λγ²(T−t).

Applying the above two propositions, we deduce the following sufficient condition.

Theorem 3.3 Let us suppose that there exists λ∈(0,_γ2¹T) such that E[Ψ_λ(ξ)]<+∞.

Then

ess sup

t

e^β(s⁻^t)α_sds≤Y¯_t, (3.6) with

Y¯_t=e^β(T⁻^t) 1

p1−λγ²(T −t) +e^λ²E[Ψ_λ(ξ)|Ft]

! +

Z _T

t

e^β(s⁻^t)α_sds,

and (2.2) admits a solution (Y, Z) such that

Y_t≤Y¯_t. Proof. Applying the above two propositions, we deduce

E_q[ξ|Ft] = E[M_T^q(M_t^q)⁻¹ξ|Ft]≤Eh

e^R^t^T^q^s^dW^sξ Ft

i

≤ E[Φ_λ(e^R^t^T^q^s^dW^s)|Ft] +e^λ²E[Ψ_λ(ξ)|Ft]

≤ 1

p1−λγ²(T −t) +e^λ²E[Ψ_λ(ξ)|Ft].

Then we get (3.6) and the rest follows from Theorem 2.1.

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4 General Case

Consider the following BSDE:

Y_t=ξ+ Z T

t

f(s, Y_s, Z_s)ds− Z T

t

Z_sdW_s, (4.7)

where f satisfies

|f(s, y, z)| ≤α_s+β|y|+γ|z|, (4.8) withα ∈L¹(0, T), β ≥0 and γ >0.

Theorem 4.1 Let f be a generator which is continuous with respect to (y, z) and verifies (4.8), and ξ be a terminal condition. Let us suppose that there exists λ∈(0,_γ2¹T) such that

E[Ψ_λ(|ξ|)]<+∞. Then BSDE (4.7) admits a solution (Y, Z) such that

|Y_t| ≤e^β(T⁻^t) 1

p1−λγ²(T −t) +e^λ²E[Ψ_λ(|ξ|)|Ft]

! +

Z T t

e^β(s⁻^t)α_sds.

Proof. Let us fixn∈N^∗ andp∈N^∗ and setξ^n,p=ξ⁺∧n−ξ⁻∧p. Let (Y^n,p, Z^n,p) be the unique solution inS²× M² of the BSDE (|ξ^n,p|, f). Set

f¯(s, y, z) =α_s+βy+γ|z|,

and ( ¯Y^n,p,Z¯^n,p) be the unique solution inS²× M² of the BSDE (|ξ^n,p|,f¯).

By comparison theorem,

|Y_t^n,p| ≤ |Y¯_t^n,p|. Setting q^n,p_s =γ sgn(Zs^n,p), we obtain,

|Y_t^n,p| ≤ |Y¯_t^n,p|

= E_q^n,p h

e^β(T⁻^t)|ξ^n,p| Ft

i +

Z T t

e^β(s⁻^t)α_sds.

From inequality (3.6),

|Y_t^n,p| ≤Y¯_t, with

Y¯_t=e^β(T⁻^t) 1

p1−λγ²(T−t) +e²^λEh

Ψ_λ(|ξ|) Ft

i

! +

Z T t

e^β(s⁻^t)α_sds.

Moreover, Y^n,p is nondecreasing with respect to n, and nonincreasing with respect to p. Once again, we apply the localization method as follows to conclude the existence of solution.

Set

τ_k =T∧inf{t≥0 : ¯Y_t> k}, and

Y_k^n,p(t) =Y_t^n,p_∧_τ_k, Z_k^n,p(t) =Z_t^n,p1_t_≤_τ_k. Then (Y_k^n,p, Z_k^n,p) satisfies

Y_k^n,p(t) =Y_k^n,p(T) + Z T

t

1_s_≤_τ_kf(s, Y_k^n,p(s), Z_k^n,p(s))ds− Z T

t

Z_k^n,p(s)dWs. (4.9)

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For fixedk,Y_k^n,pis nondecreasing with respect tonand nonincreasing with respect to p, and remains bounded byk. We can now apply the stability property of BSDEs with bounded terminal data. SettingY_k(t) = inf_psup_nY_k^n,p, there exists Z_k inM² such that lim_plim_nZ_k^n,p=Z_k inM² and

Y_k(t) = inf

p sup

n Y_τ^n,p_k + Z _τ_k

t

f(s, Y_k(s), Z_k(s))ds− Z _τ_k

t

Z_k(s)dW_s. (4.10) Finally, noting that

Y_k+1(t∧τ_k) =Y_k(t∧τ_k), Z_k+11_t_≤_τ_k =Z_k1_t_≤_τ_k, we conclude the existence of solution (Y, Z).

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