Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Backward Stochastic Differential Equations
Existence, Uniqueness and Comparison
Christoph Belak
Stochastic Control and Financial Mathematics Group
University of Kaiserslautern
Seminar on
Continuous-Time Contract Theory
JProf. Dr. Frank Seifried and Dr. Sascha Desmettre
May 15, 2012
Christoph Belak
Backward Stochastic Differential Equations
1 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Overview
1
Introduction: Reversing the Time of an SDE
2
BSDEs: Existence and Uniqueness
3
Explicit Solutions for Linear BSDEs
4
A Comparison Principle for BSDEs
Christoph Belak
Backward Stochastic Differential Equations
2 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Overview
1
Introduction: Reversing the Time of an SDE
2
BSDEs: Existence and Uniqueness
3
Explicit Solutions for Linear BSDEs
4
A Comparison Principle for BSDEs
Christoph Belak
Backward Stochastic Differential Equations
3 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
An Example
Consider the following differential equation:
dYt = 0,
t ∈ [0, T ],
(1)
with terminal condition YT = ξ.
If (1) is considered as an ordinary differential equation (in particular
ξ ∈ R), it possesses a unique solution Yt ≡ ξ.
However, if we consider (1) as a stochastic differential equation (ξ is
square-integrable here), we run into trouble, since solutions are required
to be (Ft )t∈[0,T ] -adapted (we assume here that (Ft )t∈[0,T ] is the natural
filtration generated by some Brownian motion Wt ). Unless ξ is constant,
(1) does not have a solution!
Christoph Belak
Backward Stochastic Differential Equations
4 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Example continued
We therefore need to reformulate the differential equation so that it
makes sense in a stochastic setting. Set
Yt = E[ξ|Ft ],
t ∈ [0, T ].
By the martingale representation theorem, there exists a unique
square-integrable, progressively-measurable process Zt , such that
Z t
Yt = Y0 +
Zs dWs ,
for all t ∈ [0, T ], a.s.
0
In differential form, this becomes
dYt = Zt dWt ,
t ∈ [0, T ],
YT = ξ.
(2)
The solution of (2) is given by the pair (Yt , Zt ).
Christoph Belak
Backward Stochastic Differential Equations
5 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Overview
1
Introduction: Reversing the Time of an SDE
2
BSDEs: Existence and Uniqueness
3
Explicit Solutions for Linear BSDEs
4
A Comparison Principle for BSDEs
Christoph Belak
Backward Stochastic Differential Equations
6 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Setting and Notation
Let Wt be a d-dimensional Brownian motion defined on the complete
filtered probability space (Ω, F, F, P), where F = (Ft )t∈[0,T ] is the
augmented natural filtration of Wt .
We denote by S2 (0, T ) the set of real-valued progressively measurable
processes Yt such that
"
#
E
sup |Yt |2 < ∞.
t∈[0,T ]
We denote by H2 (0, T )d the set of Rd -valued progressively measurable
processes Zt such that
"Z
#
T
|Zt |2 dt < ∞.
E
0
Christoph Belak
Backward Stochastic Differential Equations
7 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Backward Stochastic Differential Equations (BSDEs)
Let ξ ∈ L2 (Ω, FT , P; R) and let f : Ω × [0, T ] × R × Rd → R be such
that
f (t, y , z) is progressively measurable for all y , z,
f (t, 0, 0) ∈ H2 (0, T ),
f satisfies a uniform Lipschitz condition in (y , z).
We consider the following BSDE:
− dYt = f (t, Yt , Zt )dt − Zt .dWt ,
YT = ξ.
(3)
A solution of (3) is given by a pair (Yt , Zt ) ∈ S2 (0, T ) × H2 (0, T )d
satisfying
Z
T
Z
f (s, Ys , Zs ) ds −
Yt = ξ +
t
T
Zs . dWs ,
t ∈ [0, T ].
t
Christoph Belak
Backward Stochastic Differential Equations
8 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Existence and Uniqueness
Existence and Uniqueness of BSDEs
Let ξ ∈ L2 (Ω, FT , P; R) and let f : Ω × [0, T ] × R × Rd → R be such
that
f (t, y , z) is progressively measurable for all y , z,
f (t, 0, 0) ∈ H2 (0, T ),
f satisfies a uniform Lipschitz condition in (y , z).
Then, there exists a unique solution (Yt , Zt ) ∈ S2 (0, T ) × H2 (0, T )d
to the BSDE
−dYt = f (t, Yt , Zt )dt − Zt .dWt ,
YT = ξ.
Note: f and ξ as above are sometimes called standard parameters.
Christoph Belak
Backward Stochastic Differential Equations
9 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Overview
1
Introduction: Reversing the Time of an SDE
2
BSDEs: Existence and Uniqueness
3
Explicit Solutions for Linear BSDEs
4
A Comparison Principle for BSDEs
Christoph Belak
Backward Stochastic Differential Equations
10 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Linear BSDEs
Consider a linear BSDE of the form
− dYt = (At Yt + Zt .Bt + Ct )dt − Zt .dWt ,
YT = ξ,
(4)
where At , Bt are bounded progressively measurable processes valued in R
and Rd , resp., and Ct is a process in H2 (0, T ).
Explicit Solution of the Linear BSDE
The unique solution (Yt , Zt ) to (4) is given by
#
"
Z T
Γt Yt = E ΓT ξ +
Γs Cs ds Ft ,
t
where Γt is the adjoint process given by the solution of
dΓt = Γt (At dt + Bt .dWt ),
Christoph Belak
Γ0 = 1.
Backward Stochastic Differential Equations
11 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Overview
1
Introduction: Reversing the Time of an SDE
2
BSDEs: Existence and Uniqueness
3
Explicit Solutions for Linear BSDEs
4
A Comparison Principle for BSDEs
Christoph Belak
Backward Stochastic Differential Equations
12 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
A Comparison Theorem
Comparison Principle
Let (ξ 1 , f 1 ) and (ξ 2 , f 2 ) be standard parameters and let (Yt1 , Zt1 ) and
(Yt2 , Zt2 ) be the solutions to their corresponding BSDEs. Suppose that
ξ 1 ≤ ξ 2 a.s.
f 1 (t, Yt1 , Zt1 ) ≤ f 2 (t, Yt1 , Zt1 ) dt ⊗ dP a.e.
f 2 (t, Yt1 , Zt1 ) ∈ H2 (0, T ).
Then Yt1 ≤ Yt2 for all t ∈ [0, T ], a.s.
Furthermore, if Y01 ≥ Y02 a.s., then Yt1 = Yt2 , t ∈ [0, T ]. In particular, if
P(ξ 1 < ξ 2 ) > 0 or f 1 (t, ·, ·) < f 2 (t, ·, ·) on a set of strictly positive
measure dt ⊗ dP, then Y01 < Y02 .
Christoph Belak
Backward Stochastic Differential Equations
13 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Any Questions??
Christoph Belak
Backward Stochastic Differential Equations
14 / 15
Introduction: Reversing the Time of an SDE
BSDEs: Existence and Uniqueness
Explicit Solutions for Linear BSDEs
A Comparison Principle for BSDEs
Thank you for your attention!!!
Christoph Belak
Backward Stochastic Differential Equations
15 / 15