Backward stochastic dynamics on a filtered probability space

Abstract

We demonstrate that backward stochastic differential equations (BSDE) may be
reformulated as ordinary functional differential equations on certain path
spaces. In this framework, neither It\^{o}'s integrals nor martingale
representation formulate are needed. This approach provides new tools for the
study of BSDE, and is particularly useful for the study of BSDE with partial
information. The approach allows us to study the following type of backward
stochastic differential equations: \[dY_t^j=-f_0^j(t,Y_t,L(M)_t)
dt-\sum_{i=1}^df_i^j(t,Y_t), dB_t^i+dM_t^j\] with $Y_T=\xi$, on a general
filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$, where $B$ is
a $d$-dimensional Brownian motion, $L$ is a prescribed (nonlinear) mapping
which sends a square-integrable $M$ to an adapted process $L(M)$ and $M$, a
correction term, is a square-integrable martingale to be determined. Under
certain technical conditions, we prove that the system admits a unique solution
$(Y,M)$. In general, the associated partial differential equations are not only
nonlinear, but also may be nonlocal and involve integral operators.

Full text

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arXiv:0904.0377v3 [math.PR] 18 Mar 2010
Backward stochastic dynamics
on a filtered probability space
By G. LiangT. Lyons andZ. Qian
Oxford University, England
Abstract. We demonstrate that backward stochastic differential equations
(BSDE) may be reformulated as ordinary functional differential equations on
certain path spaces. In this framework, neither Itˆ o’s integrals nor martingale
representation formulate are needed. This approach provides new tools for
the study of BSDE, and is particularly useful for the study of BSDE with
partial information. The approach allows us to study the following type of
backward stochastic differential equations
dYj
t= −fj
0(t,Yt,L(M)t)dt −
d
?
i=1
fj
i(t,Yt)dBi
t+ dMj
t
with YT = ξ, on a general filtered probability space (Ω,F,Ft,P), where B
is a d-dimensional Brownian motion, L is a prescribed (non-linear) mapping
which sends a square-integrable M to an adapted process L(M), and M, a
correction term, is a square-integrable martingale to be determined. Under
certain technical conditions, we prove that the system admits a unique solu-
tion (Y,M). In general, the associated partial differential equations are not
only non-linear, but also may be non-local and involve integral operators.
Key words. Brownian motion, backward SDE, SDE, semimartingale
AMS Classification. 60H10, 60H30, 60J45
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1Introduction
Stochastic differential equations (SDE) may be considered as dynamical systems
perturbed by random signals which are often modelled by Brownian motion.
The important class of stochastic differential equations considered in literature
are Itˆ o’s type equations such as
dXj
t= fj
0(t,Xt)dt +
d
?
i=1
fj
i(t,Xt)dBi
t,(1.1)
where B = (B1,··· ,Bd) is Brownian motion in Rdon a completed probability
space (Ω,F,P), fi =?d′
solutions to (1.1) by developing a theory of stochastic integration against Brow-
nian motion (called Itˆ o’s calculus), and obtained strong solutions by specifying
an initial data at a starting time T.
SDE (1.1) has to be interpreted as an integral equation
j=1fj
i
∂
∂xj are bounded, smooth vector fields in Rd′,
j = 1,··· ,d′, where d, d′are two positive integers. K. Itˆ o gave the meaning of
Xj
t− Xj
0=
?t
0
fj
0(s,Xs)ds +
d
?
i=1
?t
0
fj
i(s,Xs)dBi
s,
which can be solved forward (i.e. for t > 0). Itˆ o’s calculus requires that a
solution X = (Xt) has to be adapted to Brownian motion B = (B1,··· ,Bd),
it is thus not necessarily possible to solve (1.1) backward from a certain time T
to t < T.
There are interesting applications on the other hand to be able to solve
(1.1) backwards. Suppose u is a smooth solution to the Cauchy problem of the
quasi-linear parabolic equation
?1
2∆ −∂
∂t
?
u + f(u,∇u) = 0on [0,∞) × Rd
with the initial data u(x,0) = u0(x). Let T > 0 and h(t,x) = u(T − t,x) for
t ∈ [0,T]. Then h solves the backward parabolic equation
?1
and h(x,T) = u0(x). Let Yt = h(t,Bt) where B is Brownian motion in Rd.
According to Itˆ o’s formula
2∆ +∂
∂t
?
h + f(h,∇h) = 0 on [0,T] × Rd
YT− Yt=
?T
0∇h(s,Bs).dBsis a martingale. Substituting?∂
?T
t
?∂
∂s+1
2∆
?
h(s,Bs)ds + MT− Mt
(1.2)
for t ≤ T, where Mt=?t
∂s+1
2∆?h
by −f0(h,∇h) in (1.2) to obtain
YT− Yt= −
t
f(Ys,∇h(s,Bs))ds + MT− Mt.(1.3)
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According to Itˆ o’s martingale representation theorem, the density process Zt=
∇h(t,Bt) of M with respect to Brownian motion is uniquely determined as the
unique predictable process Ztsuch that
MT− M0=
d
?
j=1
?T
0
Zj
tdBj
t.
In terms of the pair (Y,Z) (1.3) may be written as
YT− Yt= −
?T
t
f(Ys,Zs)ds +
d
?
j=1
?T
t
Zj
sdBj
s
with the terminal data YT = u0(BT), which is the integral form of the following
backward stochastic differential equation
dYt= −f(Yt,Zt)dt + Zt.dBt, YT = ξ
(1.4)
introduced and studied in Pardoux-Peng [33].
In the past twenty years, there is a large number of articles devoted to the
theory of BSDE and its applications in various research areas. Our references
listed at the end of the paper is by no means complete, and the reader should
refer to excellent surveys such as articles in the book [19] edited by El Karoui
and Mazliak, the recent paper by El Karoui et al [17] and the references therein
for a guide to the BSDE literature.
To the knowledge of the present authors, it was Bismut [6] (see [7], [8]) who
first formulated terminal problems for a class of stochastic differential equations
in order to study stochastic optimal control problems by means of Pontrya-
gin’s maximum principal. His equations, called backward stochastic differential
equations, have been extended and developed to a non-linear case in the seminal
paper [33] by Pardoux and Peng. A lot of efforts have been made to generalize
the class of BSDE considered in [33]. For example, Lepeltier and San Martin
[28] relaxed the Lipschitz continuous condition on the driver and studied BSDEs
with coefficients of linear growth. Yong [42] employed the continuity method
to prove the existence of solution with arbitrary time horizon. In [9] Briand et
al. considered Lp-solutions for BSDE. It is also natural to consider BSDE cou-
pled with a forward stochastic differential equation, called a forward-backward
stochastic differential equations. Antonelli [1] first studied such FBSDE, his
equation doesn’t involve a density process Z in the driver. A definite account
about FBSDE may be found in Ma et la [29], Hu and Peng [25], Peng and
Wu [37], the recent book [30] and the literature therein. Most authors con-
sider BSDE on a probability space with Brownian filtration, and there are a
few papers dealing with BSDE with jumps or with reflecting boundary condi-
tions. Tang and Li [41] have studied BSDE with random jumps, and Barles et al
[4] have explored the connection between BSDE with random jumps and some
parabolic integro-differential equations.
uniqueness under non-Lipschitz continuous coefficients for this class of BSDE.
Rong [39] proved the existence and
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Analogous to free-boundary PDE problems, El Karoui et al [18] introduced an
obstacle to BSDE such that the solution always stays above such obstacle. This
so-called reflected BSDE is further developed to double reflected barriers by
Cvitani´ c and Karatzas [15], and Hamadene et al [23]. Furthermore, Barles et
al [4] have considered BSDE on the probability space associated with Dirichlet
processes.
If the driver of BSDE is with quadratic growth of Z, the nature of equations
is completely changed. This problem is firstly solved by Kobylanski [27] by using
the monotonicity method adopted from the PDE theory. Her results have been
substantially developed and generalized by Briand and Hu [10], [11], where they
extend to equations with convex drivers subject to unbounded terminal values.
Most of the existing literature concentrates on solutions of BSDEs in a strong
sense, i.e. the underlying filtered probability space is given. One of the first
attempts to introduce weak solutions for BSDEs was presented in Buckdahn et
al [12], and Buckdahn and Engelbert [13] further proved the uniqueness of their
weak solutions, while the coefficients of their BSDEs do not evolve a density
process Z. On the other hand, the notion of weak solution for FBSDEs was
introduced by Antonelli and Ma [2] and further developed by Ma et al [31] by
employing the martingale problem approach.
The backward stochastic differential equations have found many connections
with other research areas: stochastic control, PDE, mathematical finance and
etc. To derive a maximum principle as necessary conditions for optimal control
problems, one can observe that the adjoint equations to the optimal control
problems satisfy certain backward equations. For stochastic control problems,
the corresponding adjoint equations are stochastic rather than deterministic.
Indeed Peng [35] established a general stochastic maximum principle by consid-
ering both first order and second order adjoint equations, and, on the other hand,
Kohlmann and Zhou [26] interpreted BSDE as equivalent to stochastic control
problems. Peng [36] derived a probabilistic representation (a Feynman-Kac
representation) for solutions of some quasi-linear PDEs, which was extended to
other cases by Ma et al [29]. The later has been summarized as a four-step
scheme of solving forward-backward stochastic differential equations (FBSDE),
see [30] by Ma and Yong for a detail. In [16] Duffie and Epstein discovered a
class of non-linear BSDE in their study of recursive utility in economics. Later
El Karoui et al [21] applied BSDE to option pricing problems and provided a
general framework for the application of BSDE in finance. In order to deal with
utility maximization problems in incomplete markets, Rouge and El Karoui [40]
introduced a class of BSDE with quadratic growth. Hu et al [24] further studied
this class of BSDE in a more general setting.
In this article, we put forward a simple approach to deal with the kind of
BSDE such as (1.4) which does not depend on any martingale representation,
thus allows to study a wide class of backward stochastic dynamics. Our main
idea and contribution in this article is to establish an ordinary functional differ-
ential equation which is equivalent to (1.4), which allows us to obtain alternative
representations for solutions of BSDE and to consider a new interesting class of
stochastic dynamical systems.
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Consider the following example of backward stochastic differential equations
dYt= −f(t,Yt,Zt)dt +
d
?
i=1
Zi
tdBi
t, YT = ξ, (1.5)
where B = (Bt)t≥0is Brownian motion in Rd, ξ ∈ L2(Ω,FT,P), and (Ft)t≥0is
the Brownian filtration. The differential equation has to be interpreted as the
integral equation
ξ − Yt= −
?T
t
f(s,Ys,Zs)ds +
d
?
i=1
?T
t
Zi
sdBi
s. (1.6)
By applying the Picard iteration to (Y,Z), one shows that if f is globally Lip-
schitz continuous, then there is a unique pair (Y,Z) which satisfies (1.6) for
all t ≤ T. This method relies on the martingale representation for Brownian
motion, and thus restricts the class of BSDE.
Our main idea is based on the following simple observation. Suppose that
Y = (Yt)t∈[τ,T]is a solution of (1.6) back to time τ < T, then Y must be a special
semimartingale whose variation part is continuous. Let Yt = Mt− Vt be the
Doob-Meyer’s decomposition into its martingale part M and its finite variation
part −V . The decomposition over [τ,T] is unique up to a random variable
measurable with respect to Fτ. Let us assume that the local martingale part
M is indeed a martingale up to T. Then, since the terminal value YT = ξ is
given, ξ = MT− VT, so that Mt= E(ξ + VT|Ft) and Yt= E(ξ + VT|Ft) − Vt
for t ∈ [τ,T]. The integral equation (1.6) thus can be written as
ξ − Mt+ Vt= −
?T
t
f(s,Ys,Zs)ds +
d
?
i=1
?T
t
Zi
sdBi
s
for every t ∈ [τ,T].
obtains
Taking expectations both sides conditional on Ft one
E(ξ|Ft) − Mt+ Vt
=−E
??T
?t
τ
f(s,Ys,Zs)ds
?????Ft
?
+
τ
f(s,Ys,Zs)ds.
By identifying the martingale parts and variational parts, we must have
Vt− Vτ=
?t
τ
f(s,Ys,Zs)ds
(1.7)
where Y and Z are considered as functionals of V , namely
Yt= E(ξ + VT|Ft) − Vt, Mt= E(ξ + VT|Ft)(1.8)
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