BSDE – Existence and Uniqueness
Tino Böhme
Berlin Mathematical School
02.12.2008
Abstract
In this paper we are concerned with existence and uniqueness of the solution of
Backward Stochastic Differential Equations (BSDEs). First, we give some a priori
estimates of the spread between the solutions of two BSDEs from which we deduce
the results of existence and uniqueness.
1 Introduction
In the first part of this paper, we will recall the definiton of Backward Stochastic Differential Equations (BSDEs). After that, we shall see in the second part that a BSDE
associated with a set of standard parameters (f, ξ) has a unique square-integrable solution. The philosophy behind the proof to this result is indeed comparable to the
approach of proving uniqueness and existence in the case of classical forward SDEs. By
using suitable a priori estimates, one shows that a Picard iteration yields a unique fixed
point which is the unique solution of the given forward SDE.
In our case of BSDEs, we will at first deduce a priori estimates on the spread between
the solutions of two BSDEs which will eventually ensure a Picard iteration to converge.
2 Notation
Let us first fix some notation. For x ∈ Rd , we denote its Euclidean norm by |x| and
for any x, x̄ ∈ Rd , hx, x̄i denotes their inner product. A real-valued n × d matrix is
denoted by y ∈ p
Rn×d and its transpose by y ∗ ∈ Rd×n . Note that the Euclidean norm
reads as |y| =
trace (yy ∗ ) and that the inner product of y, z ∈ Rn×d is given by
∗
hy, zi = trace (yz ).
Given a probability space (Ω, F, P) endowed with an Rn -valued Standard Brownian
motion W , we introduce the following notations:
• T ∈ R, T > 0, the so-called terminal time,
1
• (Ft )0≤t≤T , the filtration generated by the Brownian motion W and augmented,
• P, the predictable σ-field on Ω × [0, T ],
• B = B(R); the Lebesgue measure on B is denoted by λ,
• L2T (Rd ), the space
of all FT -measurable random variables X : Ω 7→ Rd satisfying
2
2
kXk = E |X| < ∞,
d
• H2T (Rd ), the
R space of all predictable processes ϕ : Ω × [0, T ] 7→ R such that
T
kϕk2 = E 0 |ϕs |2 ds < ∞,
1
d
d
• H
), the space
T (R
of all predictable processes ϕ : Ω × [0, T ] 7→ R such that
qR
T
|ϕs |2 ds < ∞,
E
0
R
T
• for fixed β ∈ R, β > 0 and ϕ ∈ H2T (Rd ), denote kϕk2β := E 0 eβs |ϕs |2 ds and
denote H2T,β (Rd ) to be the space H2T (Rd ) endowed with the norm k·kβ .
3 Existence and Uniqueness
For the sake of convenience, let us initially recall the ingredients of a BSDE.
Definition 1. Let ξ : Ω 7→ Rd be an FT -measurable random variable and let f :
Ω × [0, T ] × Rd × Rn×d 7→ Rd be a P ⊗ B d ⊗ B n×d -measurable mapping. Then, for
0 ≤ t ≤ T , the equation
−dYt = f (t, Yt , Zt )dt − Zt∗ dWt , YT = ξ
(1)
or, equivalently,
Z
T
Z
f (s, Ys , Zs )ds −
Yt = ξ +
t
T
Zs∗ dWs
(2)
t
is called a Backward Stochastic Differential Equation (BSDE) with terminal value ξ and
generator f .
A solution of the BSDE (1) is a pair (Y, Z) such that (Yt )0≤t≤T is a continuous Rd valued, adapted process and (Zt )0≤t≤T is an Rn×d -valued, predictable process satisfying
RT
|Zs |2 ds < ∞ P-a. s..
0
Remark. A solution (Y, Z) ∈ H2T (Rd ) × H2T (Rn×d ) is often referred to as a squareintegrable solution.
The theory on the solutions of BSDEs is based on the conditions which we introduce
in the following definition.
2
Definition 2. Suppose that ξ ∈ L2T (Rd ), f (·, 0, 0) ∈ H2T (Rd ) and assume that f is
uniformly Lipschitz continuous, i. e. there exists a constant C > 0 such that P ⊗ λ-a. s.
the inequality
f (ω, t, y1 , z1 ) − f (ω, t, y2 , z2 ) ≤ C |y1 − y2 | + |z1 − z2 |
holds true for all (y1 , z1 ), (y2 , z2 ) ∈ Rd × Rn×d . Then the pair (f, ξ) is said to be a set of
standard parameters of the BSDE (1).
In order to the prove the following Proposition, we need two useful Lemmas.
Lemma 3. For a, b ∈ Rd it holds
|a + b|2 ≤ 2 |a|2 + 2 |b|2 .
Proof. This result is a direct conclusion from Young’s inequality:
|a + b|2 ≤ |a|2 + 2 |a| |b| + |b|2 .
| {z }
≤|a|2 +|b|2
Lemma 4. For positive, real numbers λ, µ > 0 and C, y, z, t ≥ 0, the inequality
z2
t2
2y(Cz + t) ≤ C 2 + 2 + y 2 µ2 + Cλ2
λ
µ
is valid.
Proof. We immediately get the claim by applying the binomial formula:
2
√ z
√ 2
t
C − λy C +
− yµ
0≤
λ
µ
z2
t2
= C 2 − 2yCz + λ2 y 2 C + 2 − 2yt + y 2 µ2
λ
µ
2
2
t
z
= C 2 + 2 + y 2 µ2 + Cλ2 − 2y(Cz + t).
λ
µ
Proposition 5 (A priori estimates). Let (f 1 , ξ 1 ), (f 2 , ξ 2 ) be two standard parameters
of the BSDE (1) and let (Y 1 , Z 1 ), (Y 2 , Z 2 ) be two square-integrable solutions to the
corresponding standard parameters. Let C > 0 be the Lipschitz constant of f 1 and for
0 ≤ t ≤ T put
δYt := Yt1 − Yt2
δZt := Zt1 − Zt2
δ2 ft := f 1 (t, Yt2 , Zt2 ) − f 2 (t, Yt2 , Zt2 ).
3
Then for any (λ, µ, β) ∈ R3 such that µ > 0, λ2 > C and β ≥ C (2 + λ2 ) + µ2 it holds
1
2
2
2
βT
kδY kβ ≤ T e E |δYT | + 2 kδ2 f kβ
(3)
µ
1
λ2
2
2
2
βT
e E |δYT | + 2 kδ2 f kβ .
(4)
kδZkβ ≤ 2
λ −C
µ
Proof. We divide the proof into five steps.
1. Let (f, ξ) be a pair of standard parameters and let (Y, Z) ∈ H2T (Rd ) × H2T (Rn×d )
be a solution of the associated BSDE
Z T
Z T
Yt = ξ +
f (s, Ys , Zs )ds −
Zs∗ dWs , 0 ≤ t ≤ T .
t
t
In this step, we want to show that sup0≤t≤T |Yt | ∈ L2T (R1 ). To this end, note that
Z T
Z T
∗
Zs dWs .
f (s, Ys , Zs ) ds + sup sup |Yt | ≤ |ξ| +
0≤t≤T
0≤t≤T
t
0
We need to show that every single term on the right hand side is in L2T (R1 ).
By assumption, ξ ∈ L2T (Rd ), i. e. E |ξ|2 < ∞ and thus, |ξ| ∈ L2T (R1 ).
Using Lipschitz continuity of f with Lipschitz constant L > 0 we get
f (s, Ys , Zs ) ≤ f (s, Ys , Zs ) − f (s, 0, 0) + f (s, 0, 0)
≤ L |Ys | + |Zs | + f (s, 0, 0)
and by Cauchy-Schwarz inequality and Lemma 3 we have that
Z T
2
Z T
2
f (s, Ys , Zs ) ds
f (s, Ys , Zs ) ds ≤ T E
E
0
0
Z T
Z T
2
2
2
2
|Zs | ds
|Ys | ds + 4T L E
≤ 4T L E
(5)
0
0
Z T
f (s, 0, 0)2 ds
+ 2T E
0
< ∞.
Hence,
R T
f (s, Ys , Zs )ds ∈ L2 (R1 ).
T
0
Now turn to the third term.
Z T
2 !
∗
E sup Zs dWs = E
0≤t≤T
t
Z
sup 0≤t≤T
0
Z
≤ 2E T
0
4
T
Zs∗ dWs
Z
−
0
2 !
Zs∗ dWs + 2E
t
2 !
Zs∗ dWs Z t
2 !
sup Zs∗ dWs .
0≤t≤T
0
Due to Burkholder–Davis–Gundy inequalities1 , there exists some constant K > 0
such that
2 !
Z t
Z T
2
∗
|Zs | ds
E sup Zs dWs ≤ KE
0≤t≤T
0
0
and thus
2 !
Z T
2
∗
|Zs | ds < ∞.
Zs dWs ≤ 2(1 + K)E
0
t
R
T
Consequently, sup0≤t≤T t Zs∗ dWs ∈ L2T (R1 ) as well.
T
Z
E sup 0≤t≤T
2. Now consider (Y 1 , Z 1 ) and (Y 2 , Z 2 ), the two square-integrable solutions associated
with the standard parameters (f 1 , ξ 1 ) and (f 2 , ξ 2 ), respectively. Then, applying
Itô’s Formula to the semimartingale eβt |δYt |2 from s = t to s = T , 0 ≤ t ≤ T ,
yields
Z T
2
2
βT
βt
e |δYT | = e |δYt | − 2
eβs δYs , f 1 (s, Ys1 , Zs1 ) − f 2 (s, Ys2 , Zs2 ) ds
t
Z T
Z T
Z T
2
βs
∗
βs
e hδYs , δZs dWs i +
e |δZs | ds + β
eβs |δYs |2 ds
+2
t
t
t
which rearranges to
βt
T
eβs δYs , f 1 (s, Ys1 , Zs1 ) − f 2 (s, Ys2 , Zs2 ) ds
|δYT | + 2
t
Z T
Z T
Z T
2
βs
∗
βs
eβs |δYs |2 ds.
e hδYs , δZs dWs i −
e |δZs | ds − β
−2
2
2
βT
e |δYt | = e
Z
t
t
t
3. Our next goal is to show that
βt
2
E e |δYt |
βT
≤E e
2
|δYT |
1
+ 2E
µ
Z
T
2
βs
e |δ2 fs | ds
t
holds true for 0 ≤ t ≤ T . For sup0≤t≤T |δYt | ∈ L2T (R1 ) due to step 1, we conclude
that
s

s

Z T
Z T
E
|eβs δZs δYs |2 ds ≤ E 
e2βs |δZs |2 |δYs |2 ds
0
0

s
Z
≤ E  sup |δYt |
0≤t≤T

T
e2βs |δZs |2 ds
0
s s Z
≤ E sup |δYt |2
E
0≤t≤T
< ∞.
1
cf. [KS88], Chapter 3.3, Theorem 3.28, page 166
5
0
T
e2βs
2
|δZs | ds
Thus, eβs δZs δYs belongs to H1T (Rn ) which implies that the stochastic integral
R T βs
e hδYs , δZs∗ dWs i becomes P-integrable and has zero expectation. With this
t
result at hand, we find from the last step that
Z T
2
2
βt
βT
βs
1
1
1
2
2
2
E e |δYt | = E e |δYT | + E
2e δYs , f (s, Ys , Zs ) − f (s, Ys , Zs ) ds
t
Z T
Z T
2
2
βs
βs
−E
e |δZs | ds − E β
e |δYs | ds .
t
t
Once again, using Lipschitz condition, now on f 1 , we obtain
1
f (s, Ys1 , Zs1 ) − f 2 (s, Ys2 , Zs2 )
≤ f 1 (s, Ys1 , Zs1 ) − f 1 (s, Ys2 , Zs2 ) + f 1 (s, Ys2 , Zs2 ) − f 2 (s, Ys2 , Zs2 )
≤ C |δYs | + |δZs | + |δ2 fs |
which implies together with Lemma 4
Z T
βs
1
1
1
2
2
2
2e δYs , f (s, Ys , Zs ) − f (s, Ys , Zs ) ds
E
t
Z T
1
βs
1
1
2
2
2
2e |δYs | f (s, Ys , Zs ) − f (s, Ys , Zs )ds
≤E
t
Z T
2
βs
βs
2Ce |δYs | + e 2 |δYs | C |δZs | + |δ2 fs | ds
≤E
t
!
|δZ |2 |δ f |2
s
2
s
2
≤E
2Ceβs |δYs | + eβs C 2 +
+ |δYs | µ2 + Cλ2 ds
2
λ
µ
t
Z T
Z T
C
2
2
βs
βs
2
2
e |δYs | ds + 2 E
e |δZs | ds
= C 2+λ +µ E
λ
t
t
Z T
1
2
βs
+ 2E
e |δ2 fs | ds .
µ
t
Z
T
2
Finally, putting things together and recalling the conditions on λ and β, we deduce
the estimation
Z T
2
2
2
βt
βT
2
2
βs
e |δYs | ds
E e |δYt | ≤ E e |δYT | + C 2 + λ + µ − β E
t
Z T
Z T
C
1
2
2
βs
βs
+
−1 E
e |δZs | ds + 2 E
e |δ2 fs | ds (6)
λ2
µ
t
t
Z T
1
2
2
βs
βT
e |δ2 fs | ds .
≤ E e |δYT | + 2 E
µ
t
4. We obtain the a priori estimate on the difference of the processes Y 1 and Y 2 by
6
integrating (6) from 0 to T and using Fubini’s theorem
Z T
Z T Z T
1
2
2
2
βt
βT
βs
E
e |δYt | dt ≤ T E e |δYT | + 2
E
e |δ2 fs | ds dt
µ 0
0
t
Z T
!
1
≤ T eβT E |δYT |2 + 2 E
eβs |δ2 fs |2 ds
.
µ
0
5. Finally, note that (6) also gives
Z T
Z T
1
C
2
2
2
βs
βT
βs
e |δZs | ds ≤ E e |δYT | + 2 E
e |δ2 fs | ds
1− 2 E
λ
µ
t
t
for all 0 ≤ t ≤ T . Now taking t = 0 gives the estimate on the spread of the
processes Z 1 and Z 2 and finishes the proof.
The preceeding Proposition paves the way for our Main Result.
Theorem 6 (Pardoux–Peng, 1990). Given standard parameters (f, ξ), there exists a
unique pair (Y, Z) ∈ H2T (Rd ) × H2T (Rn×d ) which solves the BSDE (1).
Proof. For H2T,β (Rd ) is a Banach space, we will use the Banach fixed point theorem for
the mapping
Φ : H2T,β (Rd ) × H2T,β (Rn×d ) → H2T,β (Rd ) × H2T,β (Rn×d ),
(y, z) 7→ (Y, Z),
where (Y, Z) solves the BSDE with generator f (t, yt , zt ), i. e.
Z T
Z T
Yt = ξ +
f (s, ys , zs )ds −
Zs∗ dWs , 0 ≤ t ≤ T .
t
(7)
t
Again, we divide the proof into steps.
1. In this step we will verify that the mapping Φ is indeed well-defined, i. e. there does
exist a solution (Y, Z) to (7). Note that the assumption that (f, ξ) are standard
parameters implies2 that f (·, y, z) ∈ H2T (Rd )
Z T
Z T
Z T
2
2
2
2
2
f (s, ys , zs ) ds ≤ 4C E
E
|ys | ds + 4C E
|zs | ds
0
0
0
Z T
2
+ 2E
f (s, 0, 0) ds
0
< ∞.
2
cf. (5) with C the Lipschitz constant of f
7
Moreover, we have for 0 ≤ t ≤ T
2 !
Z T
Z T
2
f (s, ys , zs ) ds
E f (s, ys , zs )ds ≤ (T − t)E
t
t
Z
≤ TE
T
f (s, ys , zs )2 ds
0
< ∞,
showing that
conclude that
RT
t
f (s, ys , zs )ds ∈ L2T (Rd ) for every 0 ≤ t ≤ T . For this reason, we
Z T
Mt := E
f (s, ys , zs )ds + ξ Ft
0
3
is a square-integrable martingale . Due to the martingale representation theorem
for the Brownian motion4 there exists the P ⊗ λ-a. s. unique representation
Z t
Zs∗ dWs , 0 ≤ t ≤ T ,
Mt = M0 +
0
with a unique integrable process Z ∈ H2T (Rn×d ). Now define the adapted and
continuous process Y via
Z t
Yt := Mt −
f (s, ys , zs )ds.
0
Note that we also have that
Z T
Z t
f (s, ys , zs )ds + ξ Ft −
f (s, ys , zs )ds
Yt = E
0
0
Z T
f (s, ys , zs )ds + ξ Ft .
=E
t
Now it is straightforward to check that Y ∈ H2T (Rd )
Z
E
T
T
Z
2
|Yt | dt = E
0
0
T
Z
E
t
0
Z
T
≤
≤2
0
3
T
t
< ∞.
4
!
2
f (s, ys , zs )ds + ξ Ft dt
2 !
f (s, ys , zs )ds + ξ dt
t
!
Z T
2
E f (s, ys , zs )ds + |ξ|2 dt
Z
E Z
T
choose M as the continuous version
cf. [KS88], Chapter 3.4, Theorem 4.15, page 182
8
In order to conclude this step, we need to clarify that the uniquely determined
pair (Y, Z) solves the equation (7). But
Z T
ξ = YT = M T −
f (s, ys , zs )ds
0
Z T
Z T
∗
f (s, ys , zs )ds
Zs dWs −
= M0 +
0
0
implies
Z
t
f (s, ys , zs )ds
Yt = M t −
0
Z t
Z t
Z T
Z T
∗
∗
f (s, ys , zs )ds
f (s, ys , zs )ds +
Zs dWs −
Zs dWs +
= ξ−
0
0
0
0
Z t
Z T
Zs∗ dWs
f (s, ys , zs )ds −
=ξ+
t
t
and hence we get the claim.
2. Now let (y 1 , z 1 ) and (y 2 , z 2 ) be two elements of H2T,β (Rd )×H2T,β (Rn×d ) and consider
their images Φ (y 1 , z 1 ) = (Y 1 , Z 1 ) and Φ (y 2 , z 2 ) = (Y 2 , Z 2 ), respectively. The term
δ2 fs now reads as
δ2 fs = f 1 (s, Ys2 , Zs2 ) − f 2 (s, Ys2 , Zs2 )
= f (s, ys1 , zs1 ) − f (s, ys2 , zs2 ).
Since
1
f (s, Ys1 , Zs1 ) − f 1 (s, Ys2 , Zs2 ) = f (s, ys1 , zs1 ) − f (s, ys1 , zs1 )
the Lipschitz constant C from Proposition 5 is equal to zero in this case. Applying
Proposition 5, plugging C = 0 and β = µ2 into the estimates (3) and (4) and using
Lipschitz continuity5 of of the generator f (s, ·, ·) yields
Z T
T
2
1 1
2 2 2
βs kδY kβ ≤ E
e f (s, ys , zs ) − f (s, ys , zs ) ds
β
0
Z T
2
T 2
βs
≤ C E
e |δys | + |δzs | ds
β
0
T
≤ 2 C 2 kδyk2β + kδzk2β
β
and
kδZk2β
5
Z T
1
βs 1 1
2 2 2
e f (s, ys , zs ) − f (s, ys , zs ) ds
≤ E
β
0
1 ≤ 2 C 2 kδyk2β + kδzk2β .
β
by a slight abuse of notation, we denote the Lipschitz constant of the generator also by C
9
Hence,
2(T + 1)C 2 kδyk2β + kδzk2β .
(8)
β
Choosing β > 2(T + 1)C 2 , we achieve that
Φ is a contractive self-mapping. Thus,
there exists a unique fixed point Ȳ , Z̄ ∈ H2T,β (Rd ) × H2T,β (Rn×d ) satisfying
Z T
Z T
∗
f (s, Ȳs , Z̄s )ds −
Ȳt = ξ +
Z̄s dWs , 0 ≤ t ≤ T .
kδY k2β + kδZk2β ≤
t
t
Note that we can choose the continuous version Y defined by
Z T
f (s, Ȳs , Z̄s )ds + ξ Ft ,
Yt = E
t
i. e. Y, Z̄ is the uniquely determined solution of the BSDE (1) which finishes the
proof.
Corollary 7. Let β ∈ R, β > 0 be such that β > 2(T + 1)C 2 . Further let Y k , Z k k∈N
be the sequence defined recursively by (Y 0 , Z 0 ) = (0, 0) and
Z T
Z T
∗
k+1
k
k
Yt
=ξ+
f (s, Ys , Zs )ds −
Zsk+1 dWs , 0 ≤ t ≤ T .
(9)
t
k
t
k
Then the sequence Y , Z converges to the solution (Y, Z) of (2) P ⊗ λ-a. s. and in
H2T (Rd ) × H2T (Rn×d ) as k goes to ∞.
Proof. Let Y k , Z k be the sequence defined recursively by (9). Then, by (8)
k+1
k+1
2(T + 1)C 2 k 2
k 2
Y
Y k − Y k−1 2 + Z k − Z k−1 2
−Y β + Z
−Z β ≤
β
β
β
≤ ...
!k
1
2(T + 1)C 2
Y − Y 0 2 + Z 1 − Z 0 2
≤
β
β
β
|
{z
}
<1
which implies
∞
∞
X
X
k+1
k+1
2
k 2
Y
Z
−Y β +
− Z k β < ∞.
k=0
k
k=0
converges in
× H2T,β (Rn×d ) to (Y, Z). Note that, since
Z T
Z T
Z T
2
2
βs
βT
|ϕs |2 ds ,
1·E
|ϕs | ds ≤ E
e |ϕs | ds ≤ e · E
Hence, Y , Z
k
H2T,β (Rd )
0
0
0
the norms k·kβ and k·k are equivalent. Therefore, the sequence Y k , Z k converges in
H2T (Rd ) × H2T (Rn×d ) as well and working along a subsequence, we also obtain P ⊗ λ-a. s.
covergence.
10
References
[KPQ97] N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential
equations in finance, Mathematical Finance 7 (1997), no. 1, 17–21.
[KS88]
I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus (graduate texts in mathematics), Springer, New York, 1988.
11