Weak Solutions of Forward-Backward SDEs and ... - Semantic Scholar

Weak Solutions of Forward-Backward SDEs and Their Uniqueness Jin Ma,∗ Jianfeng Zhang,† and Ziyu Zheng‡ February 15, 2006

Abstract In this paper we propose a new notion of Forward-Backward Martingale Problem (FBMP), and study its relationship with the weak solution to the backward stochastic differential equations. The FBMP extends the idea of the well-known (forward) martingale problem of Stroock and Varadhan, but it is structured specifically to fit the nature of a forward-backward stochastic differential equation (FBSDE). Assuming that the FBSDE is Markovian with H¨older continuous coefficients we show that the solution to the FBMP does exist. Furthermore, we prove that the uniqueness of the FBMP (whence the uniqueness of the weak solution) is determined by the uniqueness of the viscosity solution of the corresponding quasilinear PDE.

Keywords: forward-backward stochastic differential equations, weak solution, forwardbackward martingale problems, viscosity solutions. 1991 Mathematics Subject Classification. Primary: 60H10, Secondary: 34F05, 90A12 ∗

Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395.

E-mail:

ma-

[email protected]. This author is supported in part by NSF grants #0204332 and #0505427. † Department of Mathematics, University of Southern California, Los Angeles, CA 90089. E-mail: [email protected]. This author is supported in part by NSF grant #0403575. ‡ Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413. E-mail:[email protected]. This author is supported in part by NSF grant #0306233.

1

1

Introduction

In a recent paper Antonelli and Ma (2003) [1] introduced the notion of weak solution to a class of FBSDEs, and proved that some standard results such as principle of causality, and some Tanaka-Watanabe type of results still hold. However, the results in that paper were a far cry from a systematic study for weak solutions. In particular, the authors were not able to address the core issue regarding the uniqueness. The similar topics were studied recently also by Buckdahn, Engelbert, and Rascanu (2004) [2], with a more general definition and more extended investigation. But the issue of uniqueness remains. In fact, to our best knowledge, so far there has not been any work trying to address the issue of uniqueness in law for BSDEs/FBSDEs; and it is our hope that this paper could be the first step in that direction. The main idea of this paper is to introduce a “backward” version of the martingale problem associated to an FBSDE. Following the forward martingale problem initiated by Stroock and Varadhan (cf. e.g., [15]), we recast the FBSDE in terms of some fundamental martingales, from which the notion of the Forward-Backward Martingale Problem (FBMP) is formulated. Such a notion extends the usual martingale problem and it is equivalent to the weak solution defined in [1]. We then would like to study the existence and especially the uniqueness of the solution to FBMP, whence those of weak solution. As a first step, in this paper we shall study only the special case where the FBSDE is Markovian with H¨older continuous coefficients. We should note that although there is a large amount of recent studies on the existence of (strong) solutions to BSDEs/FBSDEs with less-regular coefficients, notably the works of [6] for BSDEs and [3] for FBSDEs, but the case of multidimensional, fully coupled FBSDEs has still been largely restricted to the “Lipschitz” paradigm. So even in the Markovian case, our (weak) existence result under H¨older conditions is new. The general case, including those with possible path-dependent FBMPs and Markovian case with more general coefficients will be pursued further in our forthcoming paper [11], as well as other future publications. The principle issue now is of course to prove the uniqueness of the solution to the FBMP (whence that of weak solution). We should note that one of the main difficulties in proving of the uniqueness in law of the weak solutions has been the lack of a workable canonical space for the martingale integrand in the BSDE (the process Z). In fact, although in many cases the process Z is c`adl`ag or even continuous (see, e.g., [10]), such path regularity is by no means clear a priori. However, by assuming that all the coefficients are H¨older continuous, we show that the martingale integrand is a function of the other two (continuous)

2

components of the solution, owing to the idea of the “Four Step Scheme” (cf. [7]), which partially eliminated the subtlety caused by the canonical spaces. Our uniqueness proof is originated from the idea of “method of optimal control” for solving an FBSDE (see, [8], [9]), particularly the notion of “nodal set” in [8]. We show that the uniqueness of the viscosity solutions to the corresponding quasilinear implies the uniqueness of the solution to FBMP, at least among those solutions whose martingale integrands are “bounded” in a certain sense. The rest of the paper is organized as follows. In section 2 we give the preliminaries and introduce the notions of an FBMP. In section 3 we prove the existence of the FBMP to FBSDE, assuming that all coefficients are H¨older continuous. Finally in section 4 we prove the uniqueness of the solution to FBMP, under the same assumptions on the coefficients.

2

Preliminaries

Throughout this paper, for a given time horizon [0, T ] we say that a quintuple (Ω, F, P, F, W ) 4

is a standard set-up if (Ω, F, P ) is a complete probability space; F = {Ft }t∈[0,T ] is a filtration satisfying the usual hypothesis (see, e.g., [14]); and W is an {Ft }-Brownian motion. In particular, if Ft = FtW , the natural filtration generated by the Brownian motion W , augmented by all the P -null sets of F and satisfying the usual hypothesis, then we say that the standard set-up is Brownian. Let us consider the following (Markovian) forward-backward SDE:  Z t Z t   b(s, Xs , Ys , Zs )ds + σ(s, Xs , Ys , Zs )dWs   Xt = x + 0

    Yt

= g(XT ) +

Z t

0

T

h(s, Xs , Ys , Zs )ds −

Z t

(2.1)

T

h Zs , dWs i,

Here the coefficients b, σ, h, and g are functions with appropriate dimensions. An adapted (strong) solution to the FBSDE (2.1) is understood as a triplet of processes (X, Y, Z) defined on any given Brownian set-up such that (2.1) holds P -almost surely (cf. e.g., [9]). The following definition of weak solution is proposed in [1]. Definition 2.1. A standard set-up (Ω, F, P, {Ft }, W ) along with a triplet of processes (X, Y, Z) defined on this set-up is called a weak solution of (2.1) if (i) the processes X, Y are continuous, and all processes X, Y , Z are Ft −adapted; 4

(ii) denoting ηs = η(s, Xs , Ys , Zs ) for η = b, σ, h, it holds that ½Z T ¾ ¡ ¢ 2 2 2 2 P |bs | + |σs | + |hs | + |Zs | ds + |g(XT )| < ∞ = 1; 0

3

(iii) (X, Y, Z) verifies (2.1) P −a.s. We remark here that, unlike a “strong solution”, a weak solution does not require that the underlying set-up be Brownian. But instead, it requires the flexibility of the set-up for each solution, similar to the forward SDE case. We should point out that in [1] and [2] it is shown that and that there does exist weak solution that is not “strong”. We now describe the notion of a “Forward-Backward Martingale Problem”. To begin with, let us give a detailed description of a “canonical set-up” on which our discussion will be carried out. We denote, for a generic Euclidean space E , C([0, T ]; E ) to be the space of all E -valued continuous functions endowed with the sup-norm. Define, for 0 ≤ s ≤ T , 4

4

Ω1 = C([0, T ]; IRn );

Ω2 = C([0, T ]; IRm ),

(2.2)

where Ω1 denotes the path space of the forward component X and Ω2 the path space of the backward component Y of the FBSDE, respectively. We define the canonical space by 4

4

4

Ω = Ω1 × Ω2 , and the canonical filtration by Ft = Ft1 ⊗ Ft2 , where Fti = σ{ω i (r ∧ t) : r ≥ 0}, 4

4

i = 1, 2 and t ≥ 0. Without loss of generality, we denote F = FT = FT1 ⊗ FT2 . Furthermore, denoting the the generic element of Ω by ω = (ω 1 , ω 2 ), we define the canonical processes on (Ω, F) as 4

xt (ω) = ω 1 (t),

and

4

yt (ω) = ω 2 (t),

t ∈ [0, T ].

Finally, let P(Ω) be all the probability measures defined on (Ω, F), endowed with the Prohorov metric. To simplify presentation, in the paper we consider only the case when σ = σ(t, x, y). (The case σ = σ(t, x, y, z) is a little more complicated, but can be treated along the lines of the “Four Step Scheme” (see, [9] for a more detailed discussion. Further, for f = b, h, we denote fˆ(t, x, y, z) = f (t, x, y, zσ(t, x, y)), and let a = σσ ∗ . We give the following defintion for a Forward-Backward Martingale Problem. Definition 2.2. Let coefficients a, b, h, and g be given. For any 0 ≤ s < T , x ∈ IRn , a solution to the forward-backward martingale problem with coefficients (a, b, h, g) (FBMPs,x,T (a, b, h, g) for short) is a pair (P , z), where P ∈ P(Ω), and {zt } is a IRm×n valued predictable process defined on the canonical space (Ω, Ft ), such that following properties hold: (i) the processes 4

Mx (t) = xt −

Z s

t

ˆb(r, xr , yr , zr )dr

and

4

4

My (t) = yt +

Z s

t

ˆ xr , yr , zr )dr h(r,

(2.3)

are both P -martingalesZfor t ≥ s; t

(ii) [Mxi , Mxj ](t) = aij (r, xr , yr )dr, t ≥ s, i, j = 1, · · · n; Z t s (iii) My (t) = zr dMx (r), t ≥ s. s

(iv) P {xt = x, zt = 0, 0 ≤ t ≤ s} = 1 and P {yT = g(x)T } = 1. Remark 2.3. (i) The process {zt } in Definition 2.2 is different from the martingale integrand {Zt } in FBSDE (2.1). In fact one has the relation: Zt = zt σ(t, xt , yt ). Note that in the strong solution case the process {zt } is actually associated directly to the gradient of the solutions to a quasilinear parabolic PDE (see, e.g., [9]). (ii) With a simple application of Itˆo’s formula one can check that this formulation of FBMP is a natural generalization of the traditional martingale problem, given the special structure of an FBSDE. (iii) It can be easily verified that (2.1) has a weak solution if and only if the FBMP has a solution (with a = σσ ∗ ). We refer to [11] for more detailed discussions on all these issues.

3

Existence

In this section we prove the existence of the solution to the forward-backward martingale problem FBMPs,x,T (a, b, h, g). As we pointed out in Remark 2.3-(iii), we need only show that the FBSDE (2.1) has a weak solution on [s, T ], in the sense of Definition 2.1. To be more precise, we shall consider the following FBSDE: for (s, x) ∈ [0, T ] × IR,  Z t Z t   b(r, Xr , Yr , Zr )dr + σ(r, Xr , Yr )dWr ;  Xt = x + s Z s Z T t ∈ [s, T ]. T    Yt = g(XT ) + h(r, Xr , Yr , Zr )dr − Zr dWr , t

(3.1)

t

We denote the solution by (X s,x , Y s,x , Z s,x ) when necessary. For simplicity, we shall assume b ≡ 0 and that all processes are 1-dimensional. We note that all the results in this section should hold true for higher dimensions case, as long as dim(X) = dim(W ) (see the assumption (H2) below). In particular, this means that Y can be high dimensional as well. In the rest of the paper we will make use of the following Standing Assumptions: (H1) The functions σ, h, g are bounded and uniformly H¨older continuous on (x, y, z); (H2) There exists a constant K > 0, such that 1 ≤ σ 2 (t, x, y) ≤ K, K

∀(t, x, y) ∈ [0, T ] × R × R.

5

(3.2)

Our arguments below, including the proof of uniqueness, will rely heavily on the idea of the Four Step Scheme initiated in [7]. That is, we shall first look for solutions to (3.1) for which the relation Yt = u(t, Xt ) holds, where u is a viscosity solution to the PDE   u + 1 σ 2 (t, x, u)u + h(t, x, u, u σ) = 0; t xx x 2  u(T, x) = g(x).

(3.3)

We first establish some estimates for the solution to PDE (3.3). We note that these results are not surprising from PDE point of view, but we provide a probabilistic proof here for completeness. Theorem 3.1. Assume (H1) and (H2), and assume further that σ, h, g are smooth. Let u ∈ C 1,2 be the classical solution to PDE (3.3). Then (i) there exist constants C > 0 and α ∈ (0, 1), depending only on T and the uniform parameters in (H1) and (H2), such that |u(t, x)| ≤ C;

|ux (t, x)| ≤ C(T − t)

α−1 2

α

|ut (t, x)| + |uxx (t, x)| ≤ C(T − t) 2 −1 .

;

(3.4)

(ii) for any ε > 0, there exists a constant Cε , such that √ |ux (t1 , x) − ux (t2 , x)| ≤ Cε t2 − t1 ,

∀0 ≤ t1 < t2 ≤ T − ε.

(3.5)

Proof. First, it is known that under our assumptions u is uniformly H¨older continuous on (t, x) (see, for example, [4]). For the given solution u we denote 4

σ ˜ (t, x) = σ(t, x, u(t, x));

˜ x, z) = h(t, x, u(t, x), z). h(t,

˜ satisfy (H1) and (H2), and u is the solution to the PDE Then, clearly σ ˜ and h 1 2 ˜ x, ux σ ut + σ ˜ (t, x)uxx + h(t, ˜ (t, x)) = 0. 2 By a slight abuse of notation let us drop the “˜” from the functions σ and h above. That is, from now on we reduce the equation (3.3) to the case where σ and h are independent of 4

y. Denote σ ¯ 2 (t, x) = σ 2 (t, x) − σ 2 (t, 0) and 4 1 ¯ x) = uxx σ ¯ 2 (t, x) + h(t, x, ux σ(t, x)). h(t, 2

Then one may rewrite (3.3) as 1 ¯ x) = 0. ut + σ 2 (t, 0)uxx + h(t, 2 6

(3.6)

Applying the Feynman-Kac formula with the underlying diffusion being the Gaussian marRs tingale Xst,x = x + t σ(r, 0)dWr , s ∈ [t, T ], we obtain that Z T n o ¯ Xs )ds u(t, x) = E t,x g(XT ) + h(s, t Z t = g(y)p(ΓT ; y − x)dy (3.7) R Z TZ h i 1 + uxx (s, y)¯ σ 2 (s, y) + h(s, y, ux σ(s, y)) p(Γts ; y − x)dyds, t R 2 4

where Γts = 4

p(Γ; x) =

Rs t

σ 2 (r, 0)dr is the variance process of X at time s, starting from (t, x); and

√1 2πΓ

x exp(− 2Γ ) is the probability density function of centered normal distribution

2

with variance Γ. Furthermore, note that

R

t IR p(ΓT , y

− x)dy ≡ 1, for all (t, x). Denoting

4

hu (s, y) = h(s, y, ux σ(s, y)) − h(s, 0, ux σ(s, 0)),

(3.8)

and differentiating (3.7) with respect to x we have Z ³ Z T ´ i h h(s, 0, ux σ(s, 0))ds p(ΓtT , y − x)dy ux (t, x) = ∂x u(t, x) − g(0) + t IR Z y−x = [g(y) − g(0)]p(ΓtT ; y − x) t dy ΓT R Z TZ 1 y−x + (3.9) uxx (s, y)¯ σ 2 (s, y)p(Γts ; y − x) t dyds 2 t R Γs Z TZ y−x + hu (s, y)p(Γts ; y − x) t dyds; Γs t R and Z

uxx (t, x) =

h (y − x)2 1 i [g(y) − g(0)]p(ΓtT ; y − x) − dy |ΓtT |2 ΓtT R Z Z h (y − x)2 1 T 1i uxx (s, y)¯ σ 2 (s, y)p(Γts ; y − x) + − dyds 2 t R |Γts |2 Γts Z TZ h (y − x)2 1i + hu (s, y)p(Γts ; y − x) − dyds. |Γts |2 Γts t R

(3.10)

(i) We now prove (3.4). First, the boundedness of u(t, x) is a direct consequence of the boundedness of both h and g and the fact that u(t, x) = Ytt,x . Namely, we have |u(t, x)| ≤ C for any (t, x), where C > 0 is a generic constant which may very from line to line.

7

To estimate the derivatives we let α > 0 be some small number such that all the coefficients are at least H¨older-α continuous. Denote 4

4

α

Bt = sup (T − s)1− 2 As ;

At = sup{|uxx (t, x)|}, x

t≤s≤T

4 A˜t = sup{|ux (t, x)|},

1−α 4 ˜t = B sup (T − s) 2 A˜s .

x

(3.11)

t≤s≤T

Note that h i |hu (s, y)| ≤ C |y|α + |ux (s, y)σ(s, y) − ux (s, 0)σ(s, 0)|α h i 2 ≤ C |y|α + |As |α |y|α + |A˜s |α |y|α . Setting x = 0 in (3.10) we have Z y2 1 |uxx (t, 0)| ≤ C |y|α p(ΓtT ; y)[ t 2 + t ]dy |Γ | Γ R T T Z TZ y2 1 +C As |y|α p(Γts ; y)[ t 2 + t ]dyds (3.12) |Γs | Γs t R Z TZ y2 1 2 +C [|y|α + |As |α |y|α + |A˜s |α |y|α ]p(Γts ; y)[ t 2 + t ]dyds |Γ Γ | t R s s = I1 + I2 + I3 , where I1 , I2 , I3 are the three integrals on the right side of (3.12). Note that by (H2) we have

1 K (s

− t) ≤ Γts ≤ K(s − t) for all s ∈ [t, T ]. Recall the

definition of p. Thus a simple change of variable yields Z α C I1 ≤ |y|α p(1; y)[|y|2 + 1]dy ≤ C(T − t) 2 −1 . 1− α t |ΓT | 2 R

(3.13)

Similarly, by changing variable s = t + (T − t)r, Z

Z

α −1 2

T

α

As ds ≤ C [(s − t)(T − s)] 2 −1 dsBt t t Z 1 α = C(T − t)α−1 Bt [r(1 − r)] 2 −1 dr = C(T − t)α−1 Bt ; (3.14) 0 Z Th i α α2 ≤ C (s − t) 2 −1 [1 + |As |α ] + (s − t) 2 −1 |A˜s |α ds t Z Th i α(α−1) α α α2 ˜t |α ds ≤ C (s − t) 2 −1 [1 + (T − s)α( 2 −1) |Bt |α ] + (s − t) 2 −1 (T − s) 2 |B

I2 ≤ C

I3

T

(s − t)

t

α

≤ C(T − t) 2 + C(T − t) ≤ C(T − t)

α(α−1) 2

α2 −α 2

+ C(T − t)

|Bt |α + C(T − t)α α2 −α 2

2− α 2

Bt + C(T − t)α 8

˜t |α |B

2− α 2

˜t , B

(3.15)

where the last inequality is due to the fact that |x|α ≤ 1 + |x|. Thus α

α

(T − t)1− 2 |uxx (t, 0)| ≤ C + C(T − t) 2 ∧(1−α+

α2 ) 2

2 ˜t . Bt + C(T − t)1−α+α B

(3.16)

Similarly, setting x = 0 in (3.9) we get (T − t)

1−α 2

α

|ux (t, 0)| ≤ C + C(T − t) 2 ∧(1−α+

α2 ) 2

2 ˜t . Bt + C(T − t)1−α+α B

(3.17)

We note that (3.16) and (3.17) hold true for any uxx (t, x) and ux (t, x). In fact, for any x = x0 , we may rewrite (3.6) as 1 ˆ x) = 0, ut + σ 2 (t, x0 )uxx + h(t, 2 where

4 1 ˆ x) = h(t, uxx [σ 2 (t, x) − σ 2 (t, x0 )] + h(t, x, ux σ(t, x)). 2

Then follow exactly the same arguments at above we prove the estimates for ux (t, x0 ) and uxx (t, x0 ). Therefore, α

(T − t)1− 2 At + (T − t)

1−α 2

2

α α 2 ˜t , A˜t ≤ C + C(T − t) 2 ∧(1−α+ 2 ) Bt + C(T − t)1−α+α B

˜t are decreasing, implies that which, together with the fact that Bt and B α

˜t ≤ C + C(T − t) 2 ∧(1−α+ Bt + B

α2 ) 2

2 ˜t . Bt + C(T − t)1−α+α B

(3.18)

Choose δ > 0 small enough such that α

Cδ 2 ∧(1−α+

α2 ) 2

1 ≤ ; 2

1 2 Cδ 1−α+α ≤ , 2

for the constant C in (3.18). Then we get immediately that ˜t ≤ C, Bt + B

∀t ∈ [T − δ, T ).

(3.19)

Finally, for t ≤ T − δ, consider u as the solution to PDE (3.3) with terminal condition u(T − δ, x). Note again that by [4] u is uniformly H¨older continuous. Then repeat the same arguments we know (3.19) holds true for t ∈ [T − 2δ, T − δ). Repeat the arguments for finite many times we prove that (3.19) holds true for any t ∈ [0, T ). The estimate for ut follows from (3.3) immediately. (ii) We now prove (3.5). Without loss of generality we again prove it only for x = 0. In the following arguments let us denote Cε > 0 to be a generic constant which may depend

9

on ε and may vary from line to line. For 0 ≤ t1 < t2 ≤ T − ε, by (3.9) and (3.8) we have |ux (t1 , 0) − ux (t2 , 0)| Z y y ≤ |g(y) − g(0)||p(ΓtT1 ; y) t1 − p(ΓTt2 ; y) t2 |dy (3.20) ΓT ΓT R Z Z h i y y 1 T 1 + |uxx (s, y)¯ σ 2 (s, y)| + |hu (s, y)| |p(Γts1 ; y) t1 − p(Γts2 ; y) t2 |dyds 2 t2 R 2 Γs Γs Z t2 Z h i |y| 1 + |uxx (s, y)¯ σ 2 (s, y)| + |hu (s, y)| p(Γts1 ; y) t1 dyds 2 Γs t1 R = I4 + I5 + I6 , where I4 , I5 , I6 are the three integrals on the right side of (3.20). Note that p(Γts1 ; y)

1 1 − p(Γts2 ; y) t2 Γts1 Γs

− yt − yt i 1 h 1 1 2Γs1 − 2Γs2 √ e e 3 3 t t 1 2 2π |Γs | 2 |Γs | 2 Z Γts1 1 3 1 − y2 y 2 2θ [ √ − ]dθ. 5 e t2 2θ 2 2π Γs θ 2 2

= =

2

Then, by a change of variable, we have Z Γts1 Z Z 2 2 1 y 3 y t2 t1 − y2θ y + ]dydθ |p(Γs ; y) t1 − p(Γs ; y) t2 |dy ≤ C t |y|e [ 5 2θ 2 Γs Γs Γs2 θ 2 R R Z Γts1 Z 2 y2 y 1 3 1 1 =C t −p ] |y|e− 2 [ + ]dydθ = C[ p 3 t 2 2 2 2 Γs θ2 R Γs Γts1 =p

Γts1

CΓtt12 C(t2 − t1 ) p p √ √ √ ≤√ . s − t1 s − t2 [ s − t1 + s − t2 ] Γts2 [ Γts1 + Γts2 ]

p

Thus, noting that t1 < t2 ≤ T − ε, we see that C(t2 − t1 ) √ √ √ I4 ≤ √ ≤ Cε (t2 − t1 ); T − t1 T − t2 [ T − t1 + T − t2 ]

(3.21)

and, by (3.4) and noting that hu is bounded, Z T α (t2 − t1 ) √ √ √ I5 ≤ C (T − s) 2 −1 √ ds s − t1 s − t2 [ s − t1 + s − t2 ] t2 h Z T − 2ε Z T i α (t2 − t1 ) √ √ √ = C + (T − s) 2 −1 √ ds (3.22) s − t1 s − t2 [ s − t1 + s − t2 ] t2 T − 2ε Z T−ε Z T 2 α (t2 − t1 ) √ √ √ √ ≤ Cε ds + Cε (t2 − t1 ) (T − s) 2 −1 ds ε s − t s − t [ s − t + s − t ] 1 2 1 2 t2 T−2 Z ∞ √ 1 p ≤ Cε t2 − t1 dr + Cε (t2 − t1 ) √ √ r(1 + r)[ r + 1 + r] 0 √ ≤ Cε t2 − t1 , 10

where we used the change of variable: s = t2 + (t2 − t1 )r. Finally, note that Z I6 ≤ C

t2

α −1 2

(T − s) t1

Z ≤ Cε

Z R

t2

t1

ds p Γts1

Z

|y| 3

|Γts1 | 2

e

−

y2 t 2Γs1

Z

2

|y|e

− y2

R

dyds

dy ≤ Cε

t2

t1

(3.23)

√ ds √ = Cε t2 − t1 . s − t1

Combining (3.21)-(3.23) we prove (3.5). Theorem 3.2. Assume (H1) and (H2). Then for any (s, x) ∈ [0, T ] × IR, the martingale problem FBMPs,x,T (a, 0, h, g) has a solution. Moreover, the solution can be constructed via a weak solution to FBSDE (3.1) on [s, T ]. In particular, denoting the weak solution of (3.1) to be (Ω, F, P, F, X s,x , Y s,x , Z s,x , W ), then one has P = P ◦ (X s,x , Y s,x )−1 , yt = u(s, xt ), and zt = ux (t, xt ), t ≥ s, where u(·, ·) is a viscosity solution to PDE (3.3). Proof. For simplicity let us assume that s = 0. Let (σn , hn , gn ) be a standard molifier of (σ, f, g), and un be the classical solution to PDE: ( unt + 12 σn2 (t, x, un )unxx + hn (t, x, un , unx σn (t, x, un )) = 0; un (T, x) = gn (x).

(3.24)

Now Theorem 3.1 tells us that (3.4) and (3.5) hold true, uniformly for all un . Thus by applying Arzela-Ascoli’s Theorem as well as a standard diagonalizaton argument we deduce that, possibly along a subsequence, un converges to some function u ∈ C([0, T ] × IR) and unx converges to ux ∈ C([0, T ) × IR), both locally uniformly. Furthermore, it holds that |ux (t, x)| ≤ C(T − t)

α−1 2

.

(3.25)

Moreover, by the stability of viscosity solutions, u is a viscosity solution to PDE (3.3). Now let us consider the SDE Z t Z t Xt = x + σ(s, Xs , u(s, Xs ))dWs = σ ˜ (t, Xs )dWs . 0

(3.26)

0

Since σ ˜ is bounded and continuous, by [15] (3.26) has a weak solution, denote it by (Ω, F, P, F, X, W ). Now define, on this probability space: 4

Ytn = un (t, Xt );

4

Yt = u(t, Xt );

11

4

Zt = ux (t, Xt )σ(t, Xt , u(t, Xt )).

Then clearly, Y n converges to Y in ucp (uniformly on compacts, in probability, cf. [14]). Furthermore, applying Ito’s formula and then by (3.24) we have Z

Ytn

T

1 [unt (r, Xr ) + unxx σ 2 (r, Xr , u(r, Xr ))]dr + unx σ(r, Xr , u(r, Xr ))dWr 2 t Z T 1 n 2 = gn (XT ) + uxx [σn (r, Xr , un (r, Xr )) − σ 2 (r, Xr , u(r, Xr ))]dr t 2 Z T + h(r, Xr , un (r, Xr ), unx (r, Xr )σn (r, Xr , un (r, Xr )))dr t Z T − unx σ(r, Xr , u(r, Xr ))dWr .

= gn (XT ) −

t

Letting n → ∞ and applying the Dominated Convergence Theorem we see that (X, Y, Z) satisfies (3.1). Finally, defining P = P ◦ (X, Y )−1 , and noting the relation between the processes Z and z (Remark 2.3-(i)), we see that (P , z) solves the FBMP0,x,T (a, 0, h, g), proving the theorem.

4

Uniqueness

We now turn our attention to the key issue of the paper: the uniqueness of the solution to FBMP. Again, we shall consider only the special case (3.1) with b ≡ 0, and assume all processes are 1-dimensional. We should note that for the uniqueness dim(Y ) = 1 is critical. 4

Recall from §2 the canonical space Ω = C([0, T ]; R) × C([0, T ]; R). Let F = {Ft }t≥0 denote the filtration generated by the canonical processes, which we shall denote by (x, y). Note that if (P , z) is a solution to the FBMPs,x,T (σ 2 , 0, h, g), and if (H2) holds, then it is 4 Rt clear that Wt = s σ −1 (r, xr , yr )dr, t ≥ s, is a P -Brownian motion on [s, T ]. Moreover, the 4

process (x, y, Z), with Zt = zt σ(t, xt , yt ) solves FBSDE (3.1) under P on [s, T ] (see Remark 2.3). For simplicity, in what follows we do not distinguish the term “solution to the FBMP” from “weak solution to the FBSDE”, and we often say that (P , Z) is a solution “FBMP (3.1)” when the context is clear. Also, we consider only the case s = 0. We first give the definition of the uniqueness for FBMP. Definition 4.1. We say that FBMP (3.1) has unique solution if (P i , Z i ), i = 1, 2 are two solutions to the FBMP such that P i (x0 = x) = 1, i = 1, 2, then P 1 = P 2 , and P 1 ◦ [Z 1 ]−1 = P 2 ◦ [Z 2 ]−1 on L2 ([0, T ]). We shall assume throughout this section that the standing assumptions (H1) and (H2) hold. Therefore, by Theorem 3.2 we know that there exists at least one solution to the 12

FBMP (3.1), and denote it by (P 0 , Z 0 ). We note that this special weak solution has the following feature: under P 0 , it holds that yt = u(t, xt ),

Zt0 = σ(t, xt , yt )ux (t, xt ),

∀t ∈ [0, T ],

(4.1)

where u is a viscosity solution to the PDE (3.3) satisfying (3.25). Before we go any further let us introduce an auxiliary definition, which will be essential in the uniqueness result of this paper. Let K : [0, T ] 7→ IR+ be a deterministic function RT such that 0 Kt2 dt < ∞. Definition 4.2. We say that a pair (P , Z) is a “K-weak solution” at (s, x, y) ∈ [0, T ]×R×R if the following hold: 4 Rt (i) Wt = s σ −1 (r, xr , yr )dxr is a P -Brownian Motion for t ≥ s; (ii) P {xt = x, yt = y, Zt = 0, ∀t ≤ s} = 1; Rt Rt (iii) yt = y − s h(r, Xr , Yr , Zr )dr + s Zr dWr , t ∈ [s, T ]; P -a.s. (iv) P {yT = g(xT )} = 1; (v) |Zt | ≤ Kt , for all t ∈ (s, T ), P -a.s. Remark 4.3. Without loss of generality we shall assume Kt ≥ C(T − t)

α−1 2

, where α is

the constant in (3.25) and C is large enough. Therefore, any weak solution constructed via solution to the PDE with initial time s will be a K-weak solution at (s, xs , u(s, xs )). In particular, the solution (P 0 , Z 0 ) is a K-weak solution at (0, x, u(0, x)). Our argument for uniqueness will be as follows. Assume that (P ∗ , Z ∗ ) is any weak solution to FBSDE (3.1) with P ∗ (x0 = x) = 1 and |Zt∗ | ≤ Kt , P ∗ -a.s., we show that it is identical to (P 0 , Z 0 ), in the sense of Definition 4.1. To this end, We shall assume without loss of generality that P ∗ (y0 = y ∗ ) = 1 (otherwise we consider the conditional probability P ∗ {·|y0 = y ∗ }, and the result will be the same). Further, for any t ∈ [0, T ] and P ∗ -a.e. ω ∈ Ω, we denote P ∗,ω (or simply P ∗ ) to be the regular conditional probability distribution t P ∗ given Ft . Our first goal is to prove P ∗ {y0 = u(0, x)} = 1. To this end, let us denote 4

O = {(t, x, y) : ∃a K-weak solution at (t, x, y)},

(4.2)

¯ denote the closure of O (we note that O is not necessarily Lebesgue measurable!). and let O Clearly, we have (t, x, u(t, x)) ∈ O for any (t, x) ∈ [0, T ] × R, and (0, x∗ , y ∗ ) ∈ O as well. 13

Now define two functions on (t, x) ∈ [0, T ] × R: 4

¯ u(t, x) = inf{y : (t, x, y) ∈ O};

4

¯ u ¯(t, x) = sup{y : (t, x, y) ∈ O}.

(4.3)

Then it is readily seen that −C0 ≤ u(t, x) ≤ u(t, x) ≤ u ¯(t, x) ≤ C0 ;

u(T, x) = u ¯(T, x) = g(x).

(4.4)

¯ is a closed set, we have (t, x, u(t, x)) ∈ O ¯ and (t, x, u ¯ Moreover, since O ¯(t, x)) ∈ O. The following result is essential to our argument. Theorem 4.4. Assume (H1) and (H2). Then, u and u ¯ are viscosity supersolution and subsolution, respectively, of the quisilinear PDE  1 4  [Lu](t, x, u) = ut + σ 2 (t, x, u)uxx + h(t, x, u, ux σ) = 0; 2  u(T, x) = g(x).

(4.5)

Proof. We first claim that u is lower semi-continuous and u ¯ is upper semi-continuous. Indeed, let us consider u (the argument for u ¯ is symmetric). Assume (tn , xn ) → (t0 , x0 ) ¯ for each n and O ¯ is closed, we see that and u(tn , xn ) → y0 . Since (tn , xn , u(tn , xn )) ∈ O ¯ hence u(t0 , x0 ) ≤ y0 . Namely u is l.s.c. (t0 , x0 , y0 ) ∈ O, We now verify that u is a viscosity supersolution. For any (t0 , x0 ) ∈ [0, T ] × R, let 4

ϕ ∈ C 1,2 ([0, T ] × R) be such that y0 = u(t0 , x0 ) = ϕ(t0 , x0 ) and u(t, x) ≥ ϕ(t, x), for all (t, x) ∈ [0, T ] × R. We shall prove that [Lϕ](t0 , x0 , ϕ(t0 , x0 )) ≤ 0.

(4.6)

¯ so for each n there To do this, we first note that (t0 , x0 , y0 ) = (t0 , x0 , u(t0 , x0 )) ∈ O, exists a (tn , xn , yn ) ∈ O such that |tn − t0 | + |xn − x0 | + |yn − y0 | ≤

1 . n

(4.7)

Now suppose that (P n , Z n ) is a K-weak solution at (tn , xn , yn ), and let P nt be the regular conditional probability distribution of P n given Ft . Then for P n -a.e. ω = (x, y) ∈ Ω, 4

there exists a unique probability measure P n(xt ,yt ) = δ(x,y) ⊗t P nt ∈ P(Ω), such that P n(xt ,yt ) 4

coincides with P nt on F t = σ{(xr , yr ) : r ≥ t}, and P n(xt ,yt ) {x0s = xs , ys0 = ys , s ∈ [0, t]} = 1. (see [15, Chapter 6]). For notational simplicity in what follows we still denote P n(xt ,yt ) by 4 Rt P nt , if there is no danger of confusion. Moreover let us define Wtn = 0 σ −1 (s, xs , ys )dxs , t ∈ [0, T ], and 4

Wsn,t = Wsn − Wtn ,

4

Zsn,t = Zsn 1[t,T ] (s), 14

s ∈ [t, T ].

(4.8)

Then, it is readily seen that W n is a Brownian motion over [0, T ] under P n , W n,t is a Brownian motion over [t, T ] under P nt , and that (P nt , Z n,t ) is a K-weak solution at (t, xt , yt ). In other words, we must have (t, xt , yt ) ∈ O, P n -a.s, and consequently yt ≥ u(t, xt ) ≥ ϕ(t, xt ), P n -a.s., ∀t ≥ tn . Now let us denote 4

4

∆Ztn = ϕx σ(t, xt , yt ) − Ztn .

∆Yt = ϕ(t, xt ) − yt ≤ 0;

Also, for any ε > 0, let hε be a molifier of h such that khε − hk∞ ≤ ε and k∂z hε k∞ ≤ Cε , and denote 4

αtn,ε = [h(t, xt , yt , Ztn ) − h(t, xt , yt , ϕx σ(t, xt , yt ))] −[hε (t, xt , yt , Ztn ) − hε (t, xt , yt , ϕx σ(t, xt , yt ))]; 1 4 = [hε (t, xt , yt , Ztn ) − hε (t, xt , yt , ϕx σ(t, xt , yt ))]1{∆Ztn 6=0} . ∆Ztn

βtn,ε

Then, it holds that |αtn,ε | ≤ 2ε,

|βtn,ε | ≤ Cε .

(4.9)

Furthermore, applying Itˆo’s formula and using the definition of Lϕ, αn,ε , and β n,ε we have h i 1 d∆Yt = ϕt + σ 2 (t, xt , yt )ϕxx + h(t, xt , yt , Ztn ) dt + ∆Ztn dWtn 2 = {[Lϕ](t, xt , yt ) + [h(t, xt , yt , Ztn ) − h(t, xt , yt , ϕx σ(t, xt , yt ))]}dt + ∆Ztn dWtn = [Lϕ](t, xt , yt )dt + αtn,ε dt + βtn,ε ∆Ztn dt + ∆Ztn dWtn . Now let us denote 4

Γn,ε t = exp

Z

n

t

− tn

βsn,ε dWsn −

1 2

Z

t

tn

o |βsn,ε |2 ds ,

t ∈ [tn , T ]

One easily checks that Γn,ε tn = 1,

Γn,ε t > 0,

E n {Γn,ε t } = 1,

2 and E n {|Γn,ε t | } ≤ Cε ,

∀t ≥ tn .

(4.10)

n,ε n,ε n,ε n,ε n,ε n n d(Γn,ε t ∆Yt ) = Γt [Lϕ]dt + Γt αt dt + Γt [∆Zt − βt ∆Yt ]dWt ;

(4.11)

Moreover, applying Itˆo’s formula again we have, for t ∈ [tn , T ],

Now, for any δ > n1 , choose t = t0 + δ > tn (see (4.7)), we deduce from (4.11) that Z n o n n,ε n 0 ≥ E Γt0 +δ ∆Yt0 +δ = E ∆Ytn + n

t0 +δ

tn

15

o n,ε Γn,ε t {[Lϕ](t, xt , yt ) + αt }dt .

Therefore, using (4.9) and (4.10) we get Z t0 +δ Z n n En Γn,ε [Lϕ](t, x , y )dt ≤ −E ∆Y + t t tn t tn

t0 +δ

o n,ε Γn,ε α dt t t

tn t0 +δ

Z n ≤ E n |yn − y0 | + |ϕ(t0 , x0 ) − ϕ(tn , xn )| + ≤ CE n

n1 n

Z

t0 +δ

+ε tn

o

h Γn,ε dt ≤ C ε+ t

tn

o n,ε Γn,ε |α |dt t t

1 i (t0 + δ − tn ), nδ − 1

(4.12)

where C may depend on ϕ. Recall (4.4). To prove (4.6), without loss of generality we may assume that ϕ(t, x) = −C0 − 1 for x outside of a compact set. Then ϕ, ϕt , ϕx , and ϕxx are all uniformly continuous. Since σ and h are uniformly continuous, so is Lϕ. Let wϕ denote the modulus of continuity of Lϕ, and write ∆n [Lϕ](t, xt , yt ) = Lϕ(t, xt , yt ) − Lϕ(tn , xn , yn ). We see that (4.12) yields ³1´ Lϕ(t0 , x0 , y0 ) ≤ Lϕ(tn , xn , yn ) + wϕ n Z t0 +δ n o ³1´ 1 = En Γn,ε (4.13) t Lϕ(tn , xn , yn )dt + wϕ t0 + δ − tn t n n Z t0 +δ ³1´ n 1 n Γn,ε = E t {[Lϕ](t, xt , yt ) − ∆n [Lϕ](t, xt , yt )}dt + wϕ t0 + δ − tn t n n Z t0 +δ o ³ n ´ 1 C 1 |Γn,ε + wϕ En ≤ Cε + + t ∆n [Lϕ](t, xt , yt )|dt . nδ − 1 n t0 + δ − tn tn To estimate the last term on the right hand side above we first apply Cauchy-Schwartz inequality and the estimate (4.10) to get Z t0 +δ n Z n,ε n E |Γt ∆n [Lϕ]|dt ≤ E n tn

1 2

≤ Cε

t0 +δ

tn

n sup tn ≤t≤t0 +δ

2 |Γn,ε t | dt

o1 n Z 2 En

t0 +δ

tn

|∆n [Lϕ]|2 dt

o1 2 E n {|∆n [Lϕ](t, xt , yt )|2 } (t0 + δ − tn ).

o1

2

(4.14)

We now decompose ∆n [Lϕ] further as Z t n ³ ´o ∆n [Lϕ](t, xt , yt ) = [Lϕ](t, xt , yt ) − [Lϕ] tn , xt , yn + Zsn dWsn

(4.15)

tn

Z t n ³ ´ o + [Lϕ] tn , xt , yn + Zsn dWsn − [Lϕ](tn , xn , yn )} = ∆1n [Lϕ](t) + ∆2n [Lϕ](t), tn

where ∆1n [Lϕ](t) and ∆2n [Lϕ](t) are defined in an obvious way. Note that under P n , yt = Rt Rt yn + tn h(s, xs , ys , Zsn )ds + tn Zsn dWsn , for t ∈ [tn , T ], almost surely. The boundedness of 16

h then leads to that Z t ¯ ¯ ¯ ¯ Zsn dWsn ¯ ≤ C|t − tn | ≤ C(t0 + δ − tn ), ¯yt − yn − tn

∀t ∈ [tn , t0 + δ],

P n -a.s.

Thus we have n sup

E

tn ≤t≤t0 +δ

n

|∆1n [Lϕ](t)|2

o1 2

≤ wϕ (C(t0 + δ − tn )).

(4.16)

On the other hand, for any η > 0 and t ∈ [tn , t0 + δ], we apply Chebychev’s inequality to get Z t n o n o 2 2 n E |∆n [Lϕ](t)| = E |Lϕ(tn , xt , yn + Zsn dWsn ) − Lϕ(tn , xn , yn )|2 n

tn

³ ´ ´ ³ Z t 2 n n ≤ Cwϕ (η) + CP |xt − xn | ≥ η + CP | Zsn dWsn | ≥ η tn

Z t n o C ≤ Cwϕ2 (η) + 2 E n |xt − xn |2 + | Zsn dWsn |2 η tn Z t0 +δ h n i o C n 2 n 2 2 σ (s, xs , ys ) + |Zs | ds ≤ Cwϕ (η) + 2 E η tn Z t0 +δ i Ch 2 ≤ Cwϕ (η) + 2 (t0 + δ − tn ) + Ks2 ds . η tn

(4.17)

Combining (4.14)–(4.17), we see that (4.13) now becomes 1 C + wϕ ( ) + Cε wϕ (C(t0 + δ − tn )) nδ − 1 n Z t0 +δ i1 Cε h 2 +Cε wϕ (η) + t0 + δ − tn + Ks2 ds . η tn

Lϕ(t0 , x0 , y0 ) ≤ Cε +

Fixing ε and η, choosing δ =

√1 n

and letting n → ∞ we get

Lϕ(t0 , x0 , y0 ) ≤ Cε + Cε wϕ (η). Finally, letting η → 0 and then ε → 0, we obtain (4.6). That is, u is a viscosity supersolution, proving the theorem. Our main result of the paper is the following. Theorem 4.5. Assume ( H1) and (H2); and that the comparison theorem holds for viscosity solutions to the PDE (3.3). Then, FBSDE (3.1) admits a unique weak solution (P , Z) satisfying |Zt | ≤ Kt , P -a.s. for all t ∈ [0, T ].

17

Proof. Let (P ∗ , Z ∗ ) be any weak solution satisfying |Zt∗ | ≤ Kt , P ∗ -a.s., and let (P 0 , Z 0 ) be the “canonical” weak solution constructed in §3. Define u and u ¯ by (4.3). Since by Theorem 4.4 we know that u ¯ is a supersolution and u is a subsolution to (3.3), by our assumptions we must have u ¯ ≤ u, thanks to the Comparison Theorem. Thus, we must have u = u = u ¯. On the other hand, following the arguments in Theorem 4.4, one shows ¯ P ∗ -a.s. Therefore it holds that u(t, xt ) ≤ yt ≤ u that (t, xt , yt ) ∈ O ⊂ O, ¯(t, xt ). Thus yt = u(t, xt ), for all t ∈ [0, T ], P ∗ -a.s. 4

Now define dW ∗ = σ −1 (t, xt , u(t, xt )dxt , we see that W ∗ is a P ∗ -Brownian motion, and (W ∗ , x) is a weak solution to a forward SDE. Since under our assumptions the uniqueness in law holds for this forward SDE, noting the relation yt = u(t, xt ) it is easily seen that P ∗ ◦ (W ∗ , x, y)−1 = P 0 ◦ (W 0 , x, y)−1 . In particular, since (x, y) is the canonical process, we have P ∗ = P 0 . Consequently, the processes Z ∗ and Z 0 , being the integrands of the quadratic variation processes [y, W ∗ ] and [y, W 0 ], respectively, must be identical in law as well. In other words, the weak solutions (P ∗ , Z ∗ ) and (P 0 , Z 0 ) are identical by Definition 4.1, proving the theorem.

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