Non LinearFBSDEs Perturbation

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a r X i v : 1 1 0 6 . 0 1 2 3 v 2 [ q - f i n . C P ]

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Analytical Approximation for Non-linear FBSDEswith Perturbation Scheme

Masaaki Fujii , Akihiko Takahashi

First version: June 1, 2011This version: June 13, 2011

Abstract

In this work, we have presented a simple analytical approximation scheme for thegeneric non-linear FBSDEs. By treating the interested system as the linear decou-pled FBSDE perturbed with non-linear generator and feedback terms, we have shownthat it is possible to carry out recursive approximation to an arbitrarily higher order,where the required calculations in each order are equivalent to those for the standardEuropean contingent claims. We have also applied the perturbative method to thePDE framework following the so-called Four Step Scheme. The method was found torender the original non-linear PDE into a series of standard parabolic linear PDEs.Due to the equivalence of the two approaches, it is also possible to derive approxi-

mate analytic solution for the non-linear PDE by applying the asymptotic expansionto the corresponding probabilistic model. Two simple examples were provided todemonstrate how the perturbation works and show its accuracy relative to the knownnumerical techniques. The method presented in this paper may be useful for variousimportant problems which have prevented analytical treatment so far.

Keywords : BSDE, FBSDE, Four Step Scheme, Asymptotic Expansion, Malliavin Deriva-tive, Non-linear PDE, CVA

This research is supported by CARF (Center for Advanced Research in Finance) and the globalCOE program The research and training center for new development in mathematics. All the contentsexpressed in this research are solely those of the authors and do not represent any views or opinions of any institutions. The authors are not responsible or liable in any manner for any losses and/or damagescaused by the use of any contents in this research.

Graduate School of Economics, The University of Tokyo Graduate School of Economics, The University of Tokyo

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1 Introduction

In this paper, we have proposed a simple analytical approximation for backward stochasticdifferential equations (BSDEs). These equations were introduced by Bismut (1973) [ 1] forthe linear case and later by Pardoux and Peng (1990) [13] for the general case, and haveearned strong academic interests since then. They are particularly relevant for the pricingof contingent claims under the constrained or incomplete markets, and for the study of recursive utilities as presented by Duffie and Epstein (1992) [ 2]. For a recent comprehensivestudy with nancial applications, one may consult with Ma and Yong (2000) [12] andreferences therein, for example.

The importance of BSDEs, or more specically non-linear FBSDEs which have non-linear generators coupled with some state processes satisfying the forward SDEs, has risengreatly in recent years also among practitioners. Collapse of major nancial institutionsfollowed by the drastic reform of regulations make them well aware of the importance of

counter party risk management, the credit value adjustments (CVA) in particular. Evenin a very simple setup, if there exists asymmetry in the credit risk between the two parties,the relevant dynamics of portfolio value follows a non-linear FBSDE as clearly shown byDuffie and Huang (1996) [ 3]. We have recently found that the asymmetric treatment of collateral between the two parties also leads to a non-linear FBSDE [5]. Furthermore, inMay 2010, the regulators were forced to realize the importance of mutual interactions andfeedback loops in the trading activities among nancial rms, shocked by the astonishinginstant crash of Dow Jones index by almost 1 , 000 points. Once we take the feedbackeffects from the behavior of major players into account, it would naturally end up withcomplicated coupled FBSDEs.

Unfortunately, however, an explicit solution for a FBSDE has been only known for

a simple linear example. In the last decade, several techniques have been introduced byresearchers, but they tend to be quite complicated for practical applications. They eitherrequire to solve non-linear PDEs, which are very difficult in general, or resort to quitetime-consuming simulation. Although regression based Monte Carlo simulation has beenrather popular among practitioners for the pricing of callable products, the appropriatechoice of regressors and attaining the numerical stability becomes a more subtle issue fora general FBSDE. In fact, it is in a clear contrast to the pricing of callable products, onecannot tell if the price goes up or down when he or she improves the regressors, whichmakes it particularly hard to select the appropriate basis functions.

In this paper, we have presented a simple analytical approximation scheme for the non-linear FBSDEs coupled with generic Markovian state processes. We have perturbativelyexpanded the non-linear terms around the linearized FBSDE, where the expansion can bemade recursively to an arbitrary higher order. In each order of approximation, the requiredcalculations are equivalent to those for the standard European contingent claims. Inorder to carry out the perturbation scheme, we need to express the backward componentsexplicitly in terms of the forward components for each order of approximation. For thatpurpose, we have proposed to use the asymptotic expansion of volatility for the forwardcomponents, which is now widely adopted to price various European contingent claimsand compute optimal portfolios (See, for examples [ 7, 8, 9, 10, 11] and references thereinfor the recent developments and review.). In the case when the underlying processes haveknown distributions, of course, we can directly proceed to higher order approximation

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without resorting to the asymptotic expansion.We have also studied the perturbation scheme in the PDE framework, or in the so-

called Four Step Scheme [ 12], for the generic fully-coupled non-linear FBSDEs. We haveshown that our perturbation method renders the original non-linear PDE into the series of classical linear parabolic PDEs, which are easy to handle with the standard techniques. Wethen provided the corresponding probabilistic framework by using the equivalence betweenthe two approaches. We have shown that, also in this case, the required calculations in agiven order are equivalent to those for the classical European contingent claims. As a by-product, by applying the asymptotic expansion method to the corresponding probabilisticmodel, it was actually found possible to derive analytic expression for the solution of the non-linear PDE up to a given order of perturbation. Therefore, our method can beinterpreted as a practical implementation of the Four Step Scheme in the perturbativeapproach.

The organization of of the paper is as follows: In the next section, we will explain

our new approximation scheme with perturbative expansion for the generic decoupledFBSDEs. Then, in the following section, we will apply it to the two concrete examples todemonstrate how it works and test its numerical performance. One of them allows a directnumerical treatment by a simple PDE and hence easy to compare the two methods. Inthe second example, we will consider a slightly more complicated model. We compare ourapproximation result to the detailed numerical study recently carried out by Gobet et.al.(2005) [6] using a regression-based Monte Carlo simulation. Thirdly, we will consider thegeneric situation where the distribution of underlying processes are not known in a closedform. We will explain how the asymptotic expansion for the forward components cansolve the issue. Fourthly, we will give an extension of our method to the fully coupled non-linear FBSDEs under the PDE framework, and then formulate the equivalent probabilistic

approach in the following section. After concluding the paper, we have attached a brief appendix to present another equivalent expansion method.

2 Approximation Scheme

2.1 Setup

Let us briey describe the basic setup. The probability space is taken as ( , F , P ) andT (0, ) denotes some xed time horizon. W t = ( W 1t , , W rt ), 0 t T is R r -valuedBrownian motion dened on ( , F , P ), and ( F t ){0tT } stands for P -augmented naturalltration generated by the Brownian motion.We consider the following forward-backward stochastic differential equation (FBSDE)

dV t = f (X t , V t , Z t )dt + Z t dW t (2.1)V T = ( X T ) (2.2)

where V takes the value in R , and X t R d is assumed to follow a generic Markovian

forward SDEdX t = 0(X t )dt + (X t ) dW t . (2.3)

Here, we absorbed an explicit dependence on time to X by allowing some of its componentscan be a time itself. ( X T ) denotes the terminal payoff where ( x) is a deterministic

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function of x. The following approximation procedures can be applied in the same wayalso in the presence of coupon payments. Z and take values in R r and R dr respectively,and

in front of the dW t represents the summation for the components of r -dimensional

Brownian motion. Throughout this paper, we are going to assume that the appropriateregularity conditions are satised for the necessary treatments.

2.2 Perturbative Expansion for Non-linear Generator

In order to solve the pair of ( V t , Z t ) in terms of X t , we extract the linear term fromthe generator f and treat the residual non-linear term as the perturbation to the linearFBSDE. We introduce the perturbation parameter , and then write the equation as

dV ()t = c(X t )V ()

t dt g(X t , V ()

t , Z ()t )dt + Z

()t dW t (2.4)

V ()T = ( X T ) , (2.5)

where = 1 corresponds to the original model by 1

f (X t , V t , Z t ) = c(X t )V t + g(X t , V t , Z t ) . (2.6)Usually, c(X t ) corresponds to the risk-free interest rate at time t, but it is not a necessarycondition. One should choose the linear term in such a way that the residual non-linearterm becomes as small as possible to achieve better convergence.

Now, we are going to expand the solution of BSDE ( 2.4) and ( 2.5) in terms of : thatis, suppose V ()t and Z

()t are expanded as

V ()t = V (0)

t + V (1)

t + 2V (2)t + (2.7)

Z ()t = Z (0)t + Z

(1)t +

2Z (2)t +

. (2.8)

Once we obtain the solution up to the certain order, say k for example, then by putting= 1,

V t =k

i=0

V (i)t , Z t =k

i=0

Z (i)t (2.9)

is expected to provide a reasonable approximation for the original model as long as theresidual term is small enough to allow the perturbative treatment. As we will see, V (i)tand Z (i )t , the corrections to each order can be calculated recursively using the results of the lower order approximations.

2.3 Recursive Approximation for Perturbed linear FBSDE

2.3.1 Zero-th Order

For the zero-th order of , one can easily see the following equation should be satised:

dV (0)t = c(X t )V (0)

t dt + Z (0)t dW t (2.10)

V (0)T = ( X T ) . (2.11)1 Or, one can consider = 1 as simply a parameter convenient to count the approximation order. The

actual quantity that should be small for the approximation is the residual part g.

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It can be integrated asV (0)t = E e

T t c(X s )ds (X T ) F t (2.12)

which is equivalent to the pricing of a standard European contingent claim.Since we have

e T

0 c(X s )ds (X T ) = V (0)

0 + T

0e

u

0 c(X s )ds Z (0)u dW u (2.13)it can be shown that, by applying Malliavin derivative Dt ,

Dt e T

0 c(X s )ds (X T ) = T

t Dt e u0 c(X s )ds Z (0)u dW u + e

t

0 c(X s )ds Z (0)t . (2.14)

Thus, by taking conditional expectation E [|F t ], we obtainZ (0)t = E Dt e

T t c(X s )ds (X T ) F t . (2.15)

2.3.2 First Order

Now, let us consider the process V () V (0) . One can see that its dynamics is governedbyd V ()t V

(0)t = c(X t ) V

()t V

(0)t g(X t , V

()t , Z

()t )dt + Z

()t Z

(0)t dW t

V ()T V (0)

T = 0 . (2.16)

Now, by extracting the -rst order terms, we can once again recover the liner FBSDE

dV (1)t = c(X t )V (1)

t dt g(X t , V (0)

t , Z (0)t )dt + Z

(1)t dW t (2.17)

V (1)T = 0 , (2.18)

which leads to

V (1)t = E T

te

ut c(X s )ds g(X u , V (0)u , Z

(0)u )du F t (2.19)

straightforwardly. By the same arguments in the zero-th order example, we can expressthe volatility term as

Z (1)t = E Dt

T

t e ut c(X s )ds

g(X u , V (0)

u , Z (0)u )du F t . (2.20)

From these results, we can see that the required calculation is nothing more difficult thanthe zero-th order case as long as we have explicit expression for V (0) and Z (0) .

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2.3.3 Second and Higher Order Corrections

We can proceed the same way to the second order correction. By extracting the -second

order terms from V ()

t (V (0)

t + V (1)

t ), one can show that

dV (2)t = c(X t )V (2)

t dt

vg(X t , V

(0)t , Z

(0)t )V

(1)t + z g(X t , V

(0)t , Z

(0)t ) Z

(1)t dt + Z

(2)t dW t

V (2)T = 0 (2.21)

is a relevant FBSDE, which is once again liner in V (2)t . As before, it leads to the followingexpression straightforwardly:

V (2)t = E T

te

ut c(X s )ds

vg(X u , V (0)u , Z (0)u )V (1)u + z g(X u , V

(0)u , Z (0)u ) Z (1)u du F t

(2.22)

Z (2)t = E Dt T

te

ut c(X s )ds

vg(X u , V (0)u , Z

(0)u )V

(1)u + z g(X u , V

(0)u , Z

(0)u ) Z (1)u du F t .

(2.23)

In exactly the same way, one can derive an arbitrarily higher order correction. Dueto the in front of the non-linear term g, the system remains to be linear in the everyorder of approximation. However, in order to carry out explicit evaluation, we need togive Malliavin derivative explicitly in terms of the forward components. We will discussthis issue in the next.

2.4 Evaluation of Malliavin Derivative

Firstly, let us introduce a d d matrix process Y t,u , for u[t, T ], as the solution for thefollowing forward SDE:

d(Y t,u )i j =d

k=1

k i0(X u )(Y t,u )k j du + k

i (X u )(Y t,u )k j dW u (2.24)(Y t,t )i j =

i j , (2.25)

where k denotes the differential with respect to the k-th component of X , and i j denotesKronecker delta.

Now, for Malliavin derivative, we want to express, for u[t, T ],

E Dt e u

t c(X s )ds G(X u ) F t (2.26)in terms of X t , where G is a some deterministic function of X , in general. Thank to theknown chain rule of Malliavin derivative, we have

Dt e u

t c(X s )ds G(X u ) =d

i=1e

ut c(X s )ds i G(X u )(Dt X iu )

e u

t c(X s )ds G(X u ) u

t i c(X s )(Dt X is )ds . (2.27)

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Thus, it is enough for our purpose to evaluate ( Dt X u ). Since we have

(Dt X iu ) =

d

k=1 u

t k i0(X s )(Dt X

ks )ds +

u

t k i

(X s )(Dt X ks ) dW s +

i

(X t ) (2.28)

it can be shown that Dt X u follows the next SDE:

d(Dt X iu ) =d

k=1

k i0(X u )(Dt X ku )du + k i (X u )(Dt X ku ) dW u (2.29)(Dt X it ) = i (X t ) . (2.30)

Thus, comparing to Eqs. ( 2.24) and ( 2.25), we can conclude that

(Dt X u ) = Y t,u (X t ) . (2.31)As a result, combining the SDE for Y t,u and the Markovian property of X , one can conrmthat the conditional expectation

E Dt e u

t c(X s )ds G(X u ) F t (2.32)is actually given by a some function of X t . Therefore, in principle, both of the backwardcomponents can be expressed in terms of X t in each approximation order.

In fact, this is an easy task when the underlying process has a known distribution. Inthe next section, we present two such models, and demonstrate how our approximationscheme works. We will also compare our approximate solution to the direct numericalresults obtained from, such as PDE and Monte Carlo simulation. However, in more generic

situations, we do not know the distribution of X . We will explain how to handle theproblem in this case using the asymptotic expansion method for the forward componentsin Sec.4.

3 Simple Examples

3.1 A forward agreement with bilateral default risk

As the rst example, we consider a toy model for a forward agreement on a stock withbilateral default risk of the contracting parties, the investor (party-1) and its counter party(party-2). The terminal payoff of the contract from the view point of the party-1 is

(S T ) = S T K (3.1)where T is the maturity of the contract, and K is a constant. We assume the underlyingstock follows a simple geometric Brownian motion:

dS t = rS t dt + S t dW t (3.2)

where the risk-free interest rate r and the volatility are assumed to be positive constants.The default intensity of party- i h i is specied as

h1 = , h 2 = + h (3.3)

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where and h are also positive constants. In this setup, the pre-default value of thecontract at time t, V t , follows 2

dV t = rV t dt h1 max(V t , 0)dt + h2 max( V t , 0)dt + Z t dW t= ( r + )V t dt + h max( V t , 0)dt + Z t dW t (3.4)V T = ( S T ) . (3.5)

Now, following the previous arguments, let us introduce the expansion parameter ,and consider the following FBSDE:

dV ()t = V ()

t dt g(V ()

t )dt + Z ()t dW t (3.6)

V ()T = ( S T ) (3.7)dS t = S t (rdt + dW t ) , (3.8)

where we have dened = r + and g(v) = hv1{v0}.3.1.1 Zero-th order

In the zero-th order, we have

dV (0)t = V (0)

t dt + Z (0)t dW t (3.9)

V (0)T = ( S T ) . (3.10)

Hence we simply obtain

V (0)t = E e(T t ) (S T ) F t= e(T t ) S t er (T t ) K (3.11)

andZ (0)t = e (T t ) S t . (3.12)

3.1.2 First order

In the rst order, we have

dV (1)t = V (1)

t dt g(V (0)

t )dt + Z (1)t dW t (3.13)

V (1)T = 0 . (3.14)

Thus, we obtain

V (1)t = E T

te(T u )g(V (0)u )du F t (3.15)

= e(T t ) h T

tE max( S u er (T u ) K, 0) F t du (3.16)

= e(T t ) h T

tC (u; t, S t )du , (3.17)

2 See, for example, [ 3, 5].

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where

C (u; t, S t ) = S t er (T t )N d1(u; t, S t ) KN d2(u; t, S t ) (3.18)d1(2) (u; t, S t ) =

1u t

ln S t er (T t )K

12

2(u t) , (3.19)and N denotes the cumulative distribution function for the standard normal distribution.We can also derive

Z (1)t = e (T t )hS t T

tN (d1(u; t, S t ))du . (3.20)

3.1.3 Second order

Finally, let us consider the second order value adjustment. In this case, the relevant

dynamics is given bydV (2)t = V

(2)t dt

v

g(V (0)t )V (1)

t dt + Z (2)t dW t (3.21)

V (2)T = 0 . (3.22)

As a result, we have

V (2)t = E T

te(ut )

vg(V (0)u )V

(1)u du F t (3.23)

= e(T t ) h2 T

t T

uE 1{S u e r ( T u )K 0}C (s; u, S u ) F t dsdu (3.24)

which can be evaluated asV (2)t = e(T t ) h2

T

t T

u d2 (u ;t,S t ) (z)C s; u, S u (z, S t ) dzdsdu , (3.25)where we have dened

S u (z, S t ) = S t e(r 12

2 )(ut )+ utz (3.26)and

(z) =1

2 e12 z

2. (3.27)

3.1.4 Numerical comparison to PDE

For this simple model, we can directly evaluate the contract value V t by numerically solvingthe PDE: v

V (t, S ) + rS s

V (t, S ) +12

2S 2 2

s 2V (t, S ) + h1{V (t,S )0} V (t, S ) = 0

with the boundary conditions

V (T, S ) = S K V (t, M ) = e(+ h )( T t ) Me r (T t ) K , M K V (t, m ) = e(T t ) me r (T t ) K , mK .

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Figure 1: Numerical Comparison to PDE

In Fig. 1, we have plot the numerical results of the forward contract with bilateraldefault risk with various maturities with the direct solution from the PDE. We have used

r = 0 .02, = 0 .01, h = 0 .03, (3.28) = 0 .2, S 0 = 100 , (3.29)

where the strike K is chosen to make V (0)0 = 0 for each maturity. We have plot V (1)

for the rst order, and V (1)

+ V (2)

for the second order. Note that we have put = 1 tocompare the original model. One can observe how the higher order correction improves theaccuracy of approximation. In this example, the counter party is signicantly riskier thanthe investor, and the underlying contract is quite volatile 3. Even in this situation, thesimple approximation to the second order works quite well up to the very long maturity.

3.2 A self-nancing portfolio with differential interest rates

In this subsection, we consider the valuation of self-nancing portfolio under the situationwhere there exists a difference between the lending and borrowing interest rates. Here, weconsider the problem under the physical measure.

The dynamics of the self-nancing portfolio is governed by [ 4]

dV t = rV t dt (R r )maxZ t V t , 0 Z t dt + Z t dW t (3.30)

V T = ( S T ) (3.31)

dS t = S t dt + dW t (3.32)

where r and R are the lending and the borrowing rate, respectively. = ( r )/ denotesthe risk premium. For simplicity, we assume all of the r , R, and are positive constants.3 Of course, people rarely make such a risky contract to the counter party in the real market.

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Here, Z t / represents the amount invested in the risky asset, i.e. stock S t . Let us choosethe terminal wealth function as

(S T ) = max( S T K 1, 0) 2max( S T K 2, 0) . (3.33)This spread introduces both of the lending and borrowing activities, which makes theproblem more interesting. The setup explained here is in fact exactly the same as that of adopted by Gobet et.al. (2005) [ 6]. They have carried our detailed numerical studies forthe above problem and evaluate V 0 by regression-based Monte Carlo simulation. In thefollowing, we will apply our perturbative approximation scheme to the same problem andtest its accuracy.

As usual, let us introduce the expansion parameter as

dV ()t = rV ()

t dt g(V ()

t , Z ()t )dt + Z

()t dW t (3.34)

V ()

T = ( S T ) , (3.35)

where we have dened the non-linear perturbation function as

g(v, z ) = ( R r )maxz v, 0 z . (3.36)

Now, we are going to expand V ()t in terms of .

3.2.1 Zero-th order

In the zero-th order, the BSDE reduces to

dV (0)t = rV (0)t dt + Z (0)t dW t (3.37)V (0)T = ( S T ) , (3.38)

which allows us to obtain

V (0)t = E er (T t ) (S T K 1)+ 2(S T K 2)+ F t (3.39)= er (T t ) (C (K 1 , S t ) 2C (K 2, S t )) , (3.40)

where we have dened

C (K i , S t ) = S t e(T t ) N d1(K i , S t )

K iN d2(K i , S t ) (3.41)

d1(2) (K i , S t ) =1

T tln

S t e(T t )K i

12

2(T t ) (3.42)

for i {1, 2}. The volatility term is given byZ (0)t = e

(r )( T t ) S t N d1(K 1, S t ) 2N d2(K 2 , S t ) . (3.43)

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3.2.2 First order

Now, in the rst order, we have

dV (1)t = rV (1)

t dt g(V (0)t , Z (0)t )dt + Z (1)t dW t (3.44)V (1)T = 0 . (3.45)

As before, we can easily integrate it to obtain

V (1)t = E T

ter (ut ) g(V (0)u , Z (0)u )du F t . (3.46)

Now, using the zero-th order results, one can show

g(V (0)t , Z (0)t ) = er (T t ) (R r ) K 1N (d2(K 1, S t )) 2K 2N (d2(K 2 , S t ))

+

er (T

t )

( r )S t e(T

t )

N (d1(K 1 , S t )) 2N (d1(K 2 , S t )) ,(3.47)which leads to

V (1)t = er (T t ) T

tdu E (R r ) K 1N (d2(K 1 , S u )) 2K 2N (d2(K 2, S u ))

+

( r )S u e(T u ) N (d1(K 1, S u )) 2N (d1(K 2, S u )) F t .(3.48)

By setting

S u (S t , z) = S t e(ut ) exp 12

2(u t) + u tz (3.49)we can write the rst order correction as

V (1)t = er (T t ) T

tdu R dz(z)

(R r ) K 1N (d2(K 1, S u (S t , z))) 2K 2N (d2(K 2 , S u (S t , z)))+

( r )S u (S t , z)e(T u ) N (d1(K 1, S u (S t , z))) 2N (d1(K 2, S u (S t , z))) .(3.50)

The volatility term can also be derived easily as

Z (1)t = S t S t V (1)t (S t )

= er (T t ) T

tdu R dz(z)

(u, z )1

T uK 1(d2(K 1; S u (S t , z))) 2K 2(d2(K 2, S u (S t , z)))

( r )S u (S t , z)e(T u ) N (d1(K 1, S u (S t , z))) 2N (d1(K 2, S u (S t , z)))+

1T u

(d1(K 1 , S u (S t , z))) 2(d1(K 2, S u (S t , z))) , (3.51)

12


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where we have dened

(u, z ) = 1 if K 1N (d2(K 1, S u (S t , z)))

2K 2N (d2(K 2, S u (S t , z)))

0

= 0 otherwise . (3.52)

3.2.3 Second order

Finally, in the second order, the relevant FBSDE is given by

dV (2)t = rV (2)

t dt v

g(V (0)t , Z (0)t )V

(1)t +

z

g(V (0)t , Z (0)t )Z

(1)t dt + Z

(2)t dW t

V (2)T = 0 . (3.53)

Using the fact that

v

g(v, z ) = (R r )1{z v0} (3.54) z

g(v, z ) =R r

1{

z v0} (3.55)

we obtain

V (2)t = er (T t ) T

tdu

T

uds R dz1 R dz2(z1)(z2)

(u, z1) (R r )2 K 1N d2(K 1 , S s (u, z)) 2K 2N d2(K 2 , S s (u, z))+

+( R

r )(

r )S s (u, z)e(T s ) N (d1(K 1; S s (u, z)))

2N (d1(K 2 , S s (u, z)))

+R r

(u, z1)

(R r ) (s, u, z )1

T sK 1(d2(K 1, S s (u, z))) 2K 2(d2(K 2 , S s (u, z)))

( r )S s (u, z)e(T s ) [N (d1(K 1 , S s (u, z))) 2N (d2(K 2 , S s (u, z)))]+

1T s

[(d1(K 1 , S s (u, z))) 2(d1(K 2 , S s (u, z)))] . (3.56)Here, we have dened

S s (u, z) = S s (S u (S t , z1), z2) (3.57)

and also

(s, u, z ) = 1 if K 1N (d2(K 1 , S s (u, z))) 2K 2N (d2(K 2, S s (u, z))) 0= 0 otherwise . (3.58)

If one needs, it is also straightforward to derive the volatility component.

13


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3.2.4 Numerical comparison to the result of Gobet et.al.

Gobet et.al. (2005) [6] have carried out the detailed numerical study for the above problem

using the regression-based Monte Carlo simulation. They have used = 0 .05, = 0 .2, r = 0 .01, R = 0 .06T = 0 .25, S 0 = 100 , K 1 = 95 , K 2 = 105 . (3.59)

After trying various sets of basis functions, they have obtained the price as V 0 = 2 .95 withstandard deviation 0 .01.

Now, let us provide the results from our perturbative expansion. We have obtained

V (0)0 = 2 .7863

V (1)0 = 0 .1814

V (2)0

=

0.0149

using the same model inputs. Thus, up to the rst order, we have V (0)0 + V (1)

0 = 2 .968,which is already fairly close, and once we include the second order correction, we have

2i=0 V

(i)0 = 2 .953, which is perfectly consistent with their result of Monte Carlo simula-

tion. Note that, we have derived analytic formulas with explicit expressions both for thecontract value and its volatility.

4 Application of Asymptotic Expansion to Generic Marko-vian Forward Processes

In this section, we consider the situation where the forward components {X t}consist of the generic Markovian processes. In this case, we cannot express V (i)t and Z (i)t in terms of {X t}exactly, which prohibits us from obtaining the higher order corrections in a simplefashion as we have done in the previous section.

However, notice the fact that what we have to do in each order of expansion is equiv-alent to the pricing of generic European contingent claims and hence we can borrow theknown techniques adopted there. In the following, we will explain the use of asymptoticexpansion method, now for the forward components. Although it is impossible to obtainthe exact result, we can still obtain analytic expression for ( V (i)t , Z

(i)t ) up to a certain order

of the volatilities of {X t}. For the details of asymptotic expansion for volatility, pleaseconsult with the works [ 7, 8, 9, 10, 11], for example.Let us introduce a new expansion parameter , which is now for the asymptotic ex-pansion for the forward components. We express the relevant SDE of generic Markovian

process X ()R d as

dX ()u = 0(X ()

u , )du + a (X ()

u , )dW au . (4.1)

Here, we have used Einstein notation which assumes the summation of all the pairedindexes. For example, in the above equation, the second term means

a (X ()u , )dW au =

r

a =1 a (X ()u , )dW

au . (4.2)

14


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We assume a (x, 0) = 0 (4.3)

for a = {1, , r }. Intuitively speaking, it suggests that counts the order of volatility.Suppose that, in the ( i 1)-th order of , we succeeded to express V (i1)t and Z (i1)t

in terms of X ()t . Then, in the next order, we can express the backward components as

V (i)t = E T

te

ut c(X

( )s )ds G(X ()u , )du F t (4.4)

Z (i)t = E Dt T

te

ut c(X

( )s )ds G(X ()u , )du F t (4.5)

with some function G. If there is no need to obtain ( V (i+1)t , Z (i+1)t ), we can just run Monte

Carlo simulation for X () to evaluate these quantities in the standard way. However, if

there is a need to obtain higher order corrections, one is required somehow to express the(V (i)t , Z (i )t ) in terms of X

()t .

What we are going to propose is to expand the backward components around = 0:

V (i )t = V (i, 0)

t + V (i, 1)

t + 2V (i, 2)t + o(

2) (4.6)

Z (i)t = Z (i, 0)t + Z

(i, 1)t +

2Z (i, 2)t + o(2) (4.7)

and express each V (i,j )t and Z (i,j )t in terms of X

()t up to a certain order j of . Although

we can proceed to arbitrarily higher order of , we will present explicit expressions upto the second order in this paper. For the interested readers, the work [10] provides thesystematic methods to obtain higher order corrections.

Thank to the well-known chain rule for Malliavin derivative, what we have to do isonly expanding the two fundamental quantities, X u and Dt X u for u[t, T ], in terms of . Firstly, let us introduce a simpler notation,

dX ()u = 0(X ()u , )du + a (X

()u , )dW

au

:= (X ()u , )dwu , (4.8)

where runs through 0 to r with the convention w0u = u and wau = W au for a {1, , r }.We set the time t-value of X () as x. Thus our goal is to express V (i,j ) and Z (i,j ) asfunctions of x. We rst introduce a d d matrix process Y () dened as

d(Y ()t,u )i j = k

i (X

()u , )(Y

()t,u )

k j dw

u (4.9)

(Y ()t,t )i j =

i j , (4.10)

where k denotes the differential with respect to the k-th component of X . Since we have

(X ()u )i = x i +

u

t i0(X

()s , )ds +

u

t ia (X

()s , )dW

as (4.11)

applying a Malliavin derivative Dt, with {1, , r }gives

Dt, (X ()u )i = u

t k i0(X

()s , )Dt, (X ()s )k ds +

u

t k ia (X

()s , )Dt, (X ()s )k dW as + i (x, ) .

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Thus one can show that

d

Dt, (X ()u )

i = k i (X ()

u , )

Dt, (X ()u )

k dwu (4.12)

Dt, (X ()t )i = i (x, ) . (4.13)Therefore, for a {1, , r }, we conclude that

Dt,a (X ()u )i = ( Y ()

t,u )i j

ja (x, ) , (4.14)

which implies that the asymptotic expansion of Dt X ()u can be obtained from that of Y () .

Therefore, in the following, we rst carry out the asymptotic expansion for X and Y .

4.1 Asymptotic Expansion for X ()u and Y ()

u

We are now going to expand for u[t, T ] as

X ()u = X (0)u + D t,u +

12

2E t,u + o(2) , (4.15)

andY ()t,u = Y t,u + H t,u + o() , (4.16)

where

D t,u =X ()u

=0

, E t,u = 2X ()u

2=0

, (4.17)

andY t,u = Y

(0)t,u , H t,u =

Y ()t,u

=0

. (4.18)

4.1.1 Zero-th order

Since a (, 0) = 0 for a {1, , r }, we havedX (0)u = 0(X

(0)u , 0)du (4.19)

d(Y t,u )i j = k i0(X

(0)u , 0)(Y t,u )

k j du (4.20)

with the initial conditions X (0)t = x and ( Y t,t )

i j =

i j , which allows us to express X

(0)u andY t,u as deterministic functions of x. It is also convenient for later calculations to notice

that Y 1 is the solution of

d(Y 1t,u )i j = (Y 1t,u )ik j k0 (X (0)u , 0)du (4.21)with ( Y 1t,t )i j = i j .

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4.1.2 First order

By applying , we can easily obtain

d( X ()u )i = ( (X ()u , )) i j (X ()u ) j dwu + i (X ()u , )dwu (4.22)

d( Y ()

t,u )i j = k

i (X

()u , )( Y

()t,u )

k j + kl

i (X

()u , )( X

()u )

l(Y ()t,u )k j

+ k i (X ()u , )(Y

()t,u )

k j dw

u .

(4.23)

Putting = 0, they leads to

dD it,u = j i0(X

(0)u , 0)D

jt,u du +

i (X

(0)u , 0)dw

u (4.24)

d(H t,u )i j = ( k i0(X

(0)u , 0))( H t,u )

k j du + kl

i0(X

(0)u , 0))D

lt,u (Y t,u )

k j du

+( k i (X (0)u , 0))( Y t,u )

k j dw

u . (4.25)

Now, by using Eq.( 4.21), one can show that

D it,u = ( Y t,u )i j

u

t(Y 1t,s ) jk k (s)dws (4.26)

(H t,u )i j = ( Y t,u )ik

u

t(Y 1t,s )kl ( mn l0(s))D nt,s (Y t,s )m j ds + ( m l (s))( Y t,s )m j dws

(4.27)

where we have dened the shorthand notation that

i j (s) := i

j (X (0)s , 0), (4.28)

which will be used in the following calculations, too.

4.1.3 Second order

Applying 2 to the SDE of X () gives us

d( 2 X ()u )

i = ( (X ()u , ))i j 2 (X

()u )

j + jk i (X ()

u , ) (X ()

u ) j (X ()u )

k

+2 j i (X ()u , ) (X

()u )

j + 2 i (X

()u , ) dw

u . (4.29)

Thus, putting = 0, we obtain

dE it,u = ( 0(X (0)u , 0)) i j E jt,u du + jk i0(X (0)u , 0)D jt,u D kt,u du

+2 j i (X (0)u , 0)D jt,u dw

u + 2

i (X (0)u , 0)dw

u . (4.30)

Now we can integrate it as

E it,u = ( Y t,u )i j

u

t(Y 1t,s ) jk lm k0 (s)D lt,s D mt,s ds + 2 l k (s)D lt,s + 2 k (s) dws .

(4.31)

We do not need the second order terms for Y () .

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4.2 Asymptotic Expansion for Malliavin Derivative: Dt X ()u

For convenience, let us dene

(X ia )()t,u = ( Dt X ()u )ia (4.32)

and its expansion as

(X ia )()t,u = (X ia )

(1)t,u +

12

2(X ia )(2)t,u + o(

2) , (4.33)

where

(X ia )(1)t,u =

(X ia )()t,u

=0, (X ia )

(2)t,u =

2

2(X ia )

()t,u

=0. (4.34)

Note that, the zero-th order term ( X ia )(0) vanishes, due to the assumption ( 4.3).From ( 4.14), we can easily show that

(X ia )(1)t,u = ( Y t,u )

i j (

ja (x, 0)) (4.35)

(X ia )(2)t,u = ( Y t,u )

i j (

2

ja (x, 0)) + 2( H t,u )

i j (

ja (x, 0)) . (4.36)

4.3 Asymptotic Expansion for V (i, )

Now, we try to express

V (i, )t = T

tE e

ut c(X

( )s )ds G(X ()u , ) F t du (4.37)

as a function of x = X ()t using the previous results. For that purpose, we rst need tocarry out asymptotic expansion for

R ()t,u := e u

t c(X( )s )ds G(X ()u , ) (4.38)

to obtain

R ()t,u = R(0)t,u + R

(1)t,u +

12

2R (2)t,u + o(2) , (4.39)

where

R (1)t,u = R()t,u

=0, R(2)t,u =

2

2 R()t,u

=0. (4.40)

Then, we can take the conditional expectation straightforwardly.

4.3.1 Zero-th order

We haveR (0)t,u = e

ut c(X

(0)s )ds G(X (0)u , 0) (4.41)

which is a deterministic function of x.

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4.3.2 First order

It is easy to obtain

R()t,u = e

ut c(X

( )s )ds G(X ()u , )

u

t i c(X ()s )( X

()s )

i ds

+( iG(X ()u , ))( X ()u )

i + G(X ()u , ) , (4.42)

which leads to

R (1)t,u = e u

t c(X(0)s )ds ( i G(X (0)u , 0))D

it,u + G(X

(0)u , 0) G(X (0)u , 0)

u

t i c(X (0)s )D

it,s ds .

(4.43)

4.3.3 Second order

In the same way, we can show that

R (2)t,u = e u

t c(X(0)s )ds ( ij G(X (0)u , 0))D

it,u D

jt,u + 2( i G(X

(0)u , 0))D

it,u

+( i G(X (0)u , 0))E it,u +

2 G(X

(0)u , 0)

+ u

t i c(X (0)s )D

it,s ds

2

G(X (0)u , 0)

2 u

t i c(X (0)s )D

it,s ds j G(X

(0)u , 0)D

jt,u + G(X

(0)u , 0)

u

t ij c(X (0)s )D

it,s D

jt,s + ic(X

(0)s )E

it,s ds G(X

(0)u , 0) . (4.44)

4.3.4 Expression for V (i, )t

Evaluation of the conditional expectation can be easily done by simply applying Ito-isometry. Let us rst dene


u

t(Y 1t,s ) jk ( k0 (s))ds (4.45)


u

t(Y 1t,s ) jk ( ka (s))dW as (4.46)

and then we have D it,u = Dit,u + D it,u . Since the rst one is a deterministic function, we

have, for u, s t ,D it,u D

jt,s := E D

it,u D

jt,s F t

= D it,u D jt,s + D it,u D

jt,s , (4.47)

where

D it,u D jt,s = ( Y t,u )

ik (Y t,s )

jl

us

t(Y 1t,v )km (Y 1t,v )ln ( ma (v))( na (v))dv . (4.48)

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Then, similarly, we can express

E it,u := E E it,u

F t

= ( Y t,u )i j ut (Y 1t,s ) jk lm k0 (s)D lt,s D mt,s + 2 l k0 (s)D lt,s + 2 k0 (s) ds . (4.49)Using these results, we have

R (0)t,u := E R(0)t,u F t = e

ut c(X

(0)s )ds G(X (0)u , 0) (4.50)

R (1)t,u := E R(1)t,u F t

= e u

t c(X(0)s )ds ( i G(X (0)u , 0))D

it,u + G(X

(0)u , 0) G(X (0)u , 0)

u

t i c(X (0)s )D

it,s ds ,

(4.51)

and also

R (2)t,u := E R(2)t,u F t = e

ut c(X

(0)s )ds

ij G(X (0)u , 0)D it,u D jt,u + 2( i G(X

(0)u , 0))D

it,u + i G(X

(0)u , 0)E

it,u +

2 G(X

(0)u , 0)

+ G(X (0)u , 0) u

t u

t i c(X (0)s ) j c(X

(0)v )D it,s D

jt,v dsdv

2 j G(X (0)u , 0) u

t i c(X (0)s )D it,s D

jt,u ds

2 G(X (0)u , 0) u

t ic(X (0)s )D it,s ds

G(X (0)u , 0) u

t ij c(X (0)s )D it,s D

jt,s + i c(X

(0)s )E

it,s ds . (4.52)

Now, we are able to express V (i, )t as a function of x up to the second order of as desired:

V (i, )t = T

tR (0)t,u + R

(1)t,u +

12

2 R (2)t,u du + o(2) . (4.53)

4.4 Asymptotic Expansion for Z (i, )

Finally, we are going to express

Z (i, )t = T

tE Dt e

ut c(X

( )s )ds G(X ()u , ) F t du (4.54)

as a function of x = X ()t . Let us introduce the two quantities:

(()a )t,u = e u

t c(X( )s )ds i G(X ()u , )(Dt X ()u )ia (4.55)

(()a )t,u = e u

t c(X( )s )ds G(X ()u , )

u

t i c(X ()s )(Dt X ()s )ia ds . (4.56)

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Then, we have

Dt e u

t c(X( )s )ds G(X ()u , ) = ( (

)a )t,u + ( (

)a )t,u . (4.57)

Similarly to the previous section, we try to obtain the expressions as

(()a )t,u = ((1)a )t,u +

12

2((2)a )t,u + o(2) (4.58)

(()a )t,u = ((1)a )t,u +

12

2((2)a )t,u + o(2) (4.59)

where both of the zero-th order terms vanish.We have

(()a )t,u = e u

t c(X( )s )ds

u

t i c(X ()s )( X

()s )

i ds j G(X ()u , )(Dt X ()u ) ja+ ( ij G(X ()u , ))( X

()u )

j (

Dt X ()u )

ia + ( i G(X

()u , ))(

Dt X ()u )

ia + ( i G(X

()u , ))(

Dt X ()u )

ia .

(4.60)

Thus, we obtain

((1)a )t,u :=

(()a )t,u=0

= e u

t c(X(0)s )ds i G(X (0)u , 0)(X ia )

(1)t,u . (4.61)

Similarly we can show that

((2)a )t,u := 2

2(()a )t,u

=0

= 2 u

t i c(X (0)s )D

it,s ds ((1)a )t,u

+ e u

t c(X(0)s )ds ( i G(X (0)u , 0))(X ia )

(2)t,u + 2( ij G(X

(0)u , 0))D

jt,u (X ia )

(1)t,u + 2( i G(X

(0)u , 0))(X ia )

(1)t,u .

(4.62)

In the same way, for () , we have

(a )t,u = e u

t c(X( )s )ds G(X ()u , )

u

t( i c(X ()s ))( X

()s )

i ds u

t( j c(X ()s ))(Dt X ()s ) ja ds

e u

t c(X( )s )ds G(X ()u , ) + ( i G(X

()u , ))( X

()u )

i

u

t

ic(X ()s )(Dt X ()s )ia ds

e u

t c(X( )s )ds G(X ()u , )

u

t( ij c(X ()s ))( X

()s )

j (Dt X ()s )ia + ( i c(X ()s ))( Dt X ()s )ia ds .(4.63)

Thus we can show that

((1)a )t,u :=

(a )t,u=0

= e u

t c(X 0s )ds G(X (0)u , 0) u

t( i c(X (0)s ))(X ia )

(1)t,s ds (4.64)

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and similarly

((2)a )t,u = 2

2(a )t,u

=0

= 2 u

t( i c(X (0)s ))D

it,s ds (

(1)a )t,u

e u

t c(X(0)s )ds 2 G(X (0)u , 0) + ( i G(X

(0)u , 0))D

it,u

u

t( j c(X (0)s ))(X ja )

(1)t,s ds

+ G(X (0)u , 0) u

t( i c(X (0)s ))(X ia )

(2)t,s + 2( ij c(X

(0)s ))D

jt,s (X ia )

(1)t,s ds . (4.65)

For the evaluation of the conditional expectation, let us dene, for u t ,(H t,u )i j := E (H t,u )

i j F t

= ( Y t,u )ik u

t (Y 1t,s )kl ( mn l0(s))Dnt,s + ( m l0(s)) (Y t,s )m j ds (4.66)

(X ia )(1)t,u := E (X ia )

(1)t,u F t = ( Y t,u )i j ( ja (x, 0)) (4.67)

(X ia )

(2)t,u := E (X ia )

(2)t,u F t = ( Y t,u )i j ( 2 ja (x, 0)) + 2( H t,u )i j ( ja (x, 0)) . (4.68)

Using these expressions, one show that

((1)a + (1)a )t,u := E (

(1)a +

(1)a )t,u F t

= e u

t c(X(0)s )ds ( i G(X (0)u , 0))(X

ia )

(1)t,u G(X (0)u , 0)

u

t( i c(X (0)s ))(X

ia )

(1)t,s ds ,

(4.69)

and in the same way that

((2)a + (2)a )t,u := E (

(2)a +

(2)a )t,u F t

= 2 u

t( i c(X (0)s ))D

it,s ds ((1)a +

(1)a )t,u

+ e u

t c(X(0)s )ds ( i G(X (0)u , 0))(X

ia )

(2)t,u + 2( ij G(X

(0)u , 0))D

jt,u (X

ia )

(1)t,u + 2( i G(X

(0)u , 0))(X

ia )

(1)t,u

2 G(X (0)u , 0) + ( i G(X (0)u , 0))Dit,u

u

t( j c(X (0)s ))(X

ja )

(1)t,s ds

G(X (0)u , 0)

u

t ( ic(X (0)s ))(X

i

a )(2)t,s + 2( ij c(X

(0)s ))D

j

t,s (X i

a )(1)t,s ds . (4.70)

Now that we are able to express Z (i, )t as a function of x = X ()t as

(Z (i, )a )t = T

t((1)a +

(1)a )t,u +

12

2((2)a + (2)a )t,u du + o(

2) . (4.71)

This completes the goal of asymptotic expansion for V (i, ) and Z (i, ) , which are nowexpressed as functions of x as desired.

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5 Perturbation in PDE Framework

In this section, we will study the perturbation scheme under the PDE (partial differentialequation) framework following the so-called four step scheme [12]. We will see that ourperturbative method makes the four step scheme tractable for the generic situations, whichonly requires the standard techniques for the classical parabolic linear PDE. In the nextsection, we will explain the equivalent perturbation method in the probabilistic framework.

5.1 PDE Formulation based on Four Step Scheme

Let us consider the following generic coupled non-linear FBSDE:

dV t = f (t, X t , V t , Z t )dt + Z t dW tV T = ( X T )

dX t = 0(t, X t , V t , Z t )dt + (t, X t , V t , Z t ) dW tX 0 = x . (5.1)Here, we made the dependence on t explicitly to clearly distinguish it from the stochastic X components. As before, we assume that V , Z , X take value in R , R r and R d respectively,and W denotes a r dimensional standard Brownian motion.

Following the arguments of the four step scheme of Ma and Yong [12], let us postulatethat V t is given by the function of t and X t as

V t = v(t, X t ) (5.2)

almost surely for

t

[0, T ]. Then, applying Itos formula, we obtain

dV t = t v(t, X t )dt

+ i v(t, X t ) i0(t, X t , v(t, X t ), Z t ) +12

ij v(t, X t )( i j )( t, X t , v(t, X t ), Z t ) dt+ i v(t, X t ) i (t, X t , v(t, X t ), Z t ) dW t . (5.3)

Thus, in order that v is the right choice, it should satisfy

v(T, x) = ( x) (5.4)

t v(t, x ) + i v(t, x ) i0(t ,x,v (t, x ), z(t, x )) +12

ij v(t, x )( i j )( t ,x,v (t, x ), z(t, x ))+ f (t ,x,v (t, x ), z(t, x )) = 0 (5.5)

z(t, x ) = iv(t, x ) i (t ,x,v (t, x ), z(t, x )) , (5.6)

where the last equation arises to match the volatility term.

In the four step scheme, one rst needs to nd the solution z(t, x ) satisfying theEq.( 5.6). And secondly, one has to solve the PDE ( 5.5) to obtain v(t, x ), which then al-lows one to run X as a standalone Markovian process in the third step. And then nally,one will obtain the backward components by setting V t = v(t, X t ) and Z t = z(t, X t ). Thecrucial point in the above four step scheme is whether one can nish the step 1 and 2

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successfully. Even if one nds the solution for z, the second step requires to solve thenon-linear PDE ( 5.5), which is very difficult in general. In the remainder of this section,let us study how our perturbation method works to achieve this goal.

We consider, as before, the original system Eq ( 5.1) as a linear decoupled FBSDE withperturbations of non-linear generator and feedbacks of the order of . We write it as

dV ()t = c(t, X ()t )V

()t dt g(t, X

()t , V

()t , Z

()t )dt + Z

()t dW t

V ()T = ( X ()T )

dX ()t = r (t, X ()t ) + (t, X

()t , V

()t , Z

()t ) dt

+ (t, X ()t ) + (t, X ()t , V

()t , Z

()t ) dW t

X ()0 = x (5.7)

and the corresponding PDE:

v()(T, x) = ( x)

t v() (t, x ) + i v() (t, x ) i0(t ,x,v() , z() ) +

12

ij v() (t, x )( i j )( t ,x,v () , z () )+ f (t ,x,v () , z () ) = 0

z()(t, x ) = iv() (t, x ) i (t ,x,v () (t, x ), z () (t, x )) , (5.8)

where

f (t ,x,v () , z() ) = c(t, x )v() (t, x ) + g(t,x,v () (t, x ), z () (t, x )) (5.9) 0(t ,x,v () , z() ) = r (t, x ) + (t,x,v () (t, x ), z () (t, x )) (5.10) (t ,x,v () , z() ) = (t, x ) + (t,x,v () (t, x ), z() (t, x )) . (5.11)

We suppose that the solution of the above PDE can be expanded perturbatively in sucha way that

v()(t, x ) = v(0) (t, x ) + v(1) (t, x ) + 2v(2) (t, x ) + (5.12)z()(t, x ) = z(0) (t, x ) + z(1) (t, x ) + 2z(2) (t, x ) + , (5.13)

and then try to solve v(i) , z(i) order by order. If the non-linear terms are small enough,we can expect to obtain a good approximation by putting = 1 in the above expansionto a certain order.

5.2 Zero-th order

In the zero-th order, the PDE ( 5.8) reduces to

t + L(t, x ) v(0) (t, x ) = 0v(0) (T, x) = ( x) (5.14)

andz(0) (t, x ) = i v(0) (t, x )i (t, x ) . (5.15)

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Here, we have dened the operator Las

L(t, x ) = r i (t, x ) i +

1

2( i

j )( t, x ) ij

c(t, x ) . (5.16)

This is a standard parabolic PDE and can be handled in the usual way. One can easilycheck that V t = v(0) (t, X t ) and Z t = z(0) (t, X t ) solves the FBSDE ( 5.7) when = 0.

5.3 First order

By extracting -rst order terms from the PDE, we obtain

t + L(t, x ) v(1) (t, x ) + G(1) (t, x ) = 0v(1) (T, x) = 0 (5.17)

andz(1) (t, x ) =

iv(1) (t, x ) i (t, x ) +

iv(0) (t, x )i(0) (t, x ) . (5.18)

Here, we have dened

G(1) (t, x ) = i v(0) (t, x )i (0) (t, x ) + ij v(0) (t, x )(i j (0) )( t, x ) + g(0) (t, x ) (5.19)and the following notations:

(0) (t, x ) = (t,x,v (0) (t, x ), z(0) (t, x )) (5.20)(0) (t, x ) = (t,x,v (0) (t, x ), z (0) (t, x )) (5.21)g(0) (t, x ) = g(t,x,v (0) (t, x ), z(0) (t, x )) . (5.22)

As a result, we once again obtained a linear parabolic PDE. Hence ( v(1) , z (1) ) can also besolved, at least numerically, in a standard fashion.

5.4 Second order

In the second order, one can show v(2) and z(2) should satisfy

t + L(t, x ) v(2) (t, x ) + G(2) (t, x ) = 0v(2) (T, x) = 0 (5.23)

and

z(2) (t, x ) = i v(2) (t, x )i (t, x ) + i v(1) (t, x )i(0) (t, x )+ iv(0) (t, x ) v(1) (t, x ) v + z(1) (t, x ) z i(0) (t, x ) . (5.24)

Here, G(2) is given byG(2) (t, x ) = i v(1) (t, x )i(0) (t, x ) + i v(0) (t, x ) v(1) (t, x ) v + z(1) (t, x ) z i(0) (t, x )

+ ij v(1) (t, x )( i j (0) )( t, x ) +12

ij v(0) (t, x )(i(0) j (0) )( t, x )+ ij v(0) (t, x ) i (t, x ) v(1) (t, x ) v + z(1) (t, x ) z j (0) (t, x )+ v(1) (t, x ) v + z(1) (t, x ) z g(0) (t, x ) , (5.25)

where the partial differentials with respect to v and z are taken by considering , and gas functions of ( t ,x,v,z ). It is still a linear parabolic PDE.

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5.5 Higher orders and an equivalent simpler formulation

Although we can proceed to higher orders in the same way and solve ( v(i) , z (i ) ), there is

another way with a clearer representation. Let us dene

v[i](t, x ) =i

j =0

j v( j ) (t, x ), z[i](t, x ) =

i

j =0

j z( j ) (t, x ) . (5.26)

and the operator

L[k ](t, x ) = l0 t ,x,v [k ](t, x ), z[k ](t, x ) l +12

( l m ) t ,x,v [k ](t, x ), z[k ](t, x ) lm+ f t ,x,v [k ](t, x ), z [k ](t, x ) .

(5.27)

Then, one can easily check that the PDE for v[i] with ( i 1) can be expressed as t + L[i1](t, x ) v[i](t, x ) = 0

v[i](T, x) = ( x) (5.28)

andz[i](t, x ) = lv[i](t, x ) l t ,x,v [i1](t, x ), z [i1](t, x ) . (5.29)

It is straightforward to conrm the consistency with the summation of each ( v(k ) , z (k ) ) for(0 k i) up to the error terms of o(i ), which is due to the additional in front of the non-linear terms. Note that, in an arbitrary order, the PDE has a linear parabolic form.

The above formulation clearly shows that the perturbative treatment of non-lineareffects of the original system allows us to obtain a series of linear parabolic PDEs withthe same structure. Solving the PDE for the zero-th order, and then recursively replacingthe backward components by the solution of the previous expansion order, we can obtainan arbitrary higher order of the approximation.

6 Perturbation in Probabilistic Framework for the GenericCoupled Non-linear FBSDEs

We have now seen the perturbation method in the PDE framework can work even forthe fully-coupled non-linear FBSDEs. In this section, we will provide a correspondingperturbation scheme under the probabilistic framework. As we will see, it is nothing moredifficult than the decoupled case studied in Sec. 2, and reduces to the standard calculationsfor the European contingent claims. As a by-product, applying the asymptotic expansionmethod explained in Sec. 4, we can also show that it is possible to obtain an analyticexpression for the non-linear PDE in the Four Step Scheme up to the given order of expansion.

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6.1 Generic Formulation

We try to solve the same FBSDE ( 5.1) treated in the PDE framework. Suppose that we

have somehow obtained a solution of ( v[i

1]

(t, x ), z[i

1]

(t, x )). Then, let us consider thefollowing FBSDE:

dV [i]t = f t, X [i]t , v

[i1](t, X [i ]t ), z [i1](t, X [i]t ) dt + Z [i]t dW tV [i]T = ( X

[i]T )

dX [i ]t = 0 t, X [i]t , v

[i1](t, X [i]t ), z[i1](t, X [i]t ) dt

+ t, X [i]t , v[i1](t, X [i]t ), z[i1](t, X [i]t ) dW t

X [i ]0 = x . (6.1)

Here, one can immediately check that the solution of the above FBSDE ( V [i]t , Z [i]t ), as afunction of ( t, X [i]t ), actually satises the PDE in the Four Step Scheme given in ( 5.28)and ( 5.29) by setting

v[i](t, x ) = V [i](t, x ), z[i](t, x ) = Z [i](t, x ). (6.2)

Hence the solution of the above FBSDE can be interpreted as the i -th order approximationof the original FBSDE in ( 5.1). Therefore, if we can solve the above FBSDE in probabilisticway, we can proceed to an arbitrarily higher order of approximation by simply updatingthe backward components of the non-linear terms recursively. We can also say that it is aprobabilistic way to solve the non-linear PDE ( 5.1) order by order of .

One can check that the above FBSDE is actually decoupled and linear by writing itexplicitly as

dV [i]t = c(t, X [i]t )V

[i ]t dt g(t, X

[i]t , v

[i1](t, X [i]t ), z [i1](t, X [i ]t ))dt + Z [i]t dW tV [i]T = ( X

[i]T )

dX [i]t = r (t, X [i]t ) + t, X

[i]t , v

[i1](t, X [i]t ), z [i1](t, X [i ]t ) dt

+ (t, X [i]t ) + t, X [i]t , v

[i1](t, X [i]t ), z [i1](t, X [i]t ) dW tX [i]0 = x , (6.3)

and hence, we can straightforwardly integrate it as

V [i]t = E e T

t c(s,X[i ]s )ds (X [i]T )

+ T

te

ut c(s,X

[i ]s )ds g u, X [i]u , v

[i1](u, X [i]u ), z [i1](u, X [i]u ) du F t (6.4)Z [i]t = E Dt e

T t c(s,X

[i ]s )ds (X [i]T )

+ T

te

ut c(s,X

[i ]s )ds g u, X [i]u , v

[i1](u, X [i]u ), z [i1](u, X [i]u ) du F t . (6.5)

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The result is equivalent to the pricing of the standard European contingent claims, andalso has the same form appeared in Sec. 2. Thus, we can apply the asymptotic expansionmethod given in Sec. 4 to the forward components X [i] in the same way. This will give usthe analytical result of ( V [i ]t , Z [i]t ) as a function of ( t, X [i]t ), up to a given order of volatilityparameter, say k . Then we can set

v[i](t, x ) = V [i]t (t, x ), z[i ](t, x ) = Z [i]t (t, x ) . (6.6)

up to the error terms of o(k ), and can move on to the higher order of approximations 4.

6.2 Summary of Recursive Procedures

Here, let us summarize the procedures of our perturbation method. Firstly, in the zero-thorder, the corresponding FBSDE is given by

dV [0]t = c(t, X [0]t )V

[0]t dt + Z

[0]t

dW t (6.7)

V [0]T = ( X [0]T ) (6.8)

dX [0]t = r (t, X [0]t )dt + (t, X

[0]t ) dW t (6.9)

X [0]0 = x . (6.10)

This can be integrated as

V [0]t = E e T

t c(s,X[0]s )ds (X [0]T ) F t (6.11)

Z [0]t = E Dt e T

t c(s,X[0]s )ds (X [0]T ) F t (6.12)

which can be solved either exactly, or analytically up to the certain order of volatility by

the asymptotic expansion method. Then we setv[0](t, x ) = V [0](t, x ) , z[0](t, x ) = Z [0](t, x ), (6.13)

and then put them back in the backward components of ( 6.1) with i = 1. We then obtain(6.4) and ( 6.5) with i = 1. We can express V [1]t and Z

[1]t in terms of t and X

[1]t by using the

asymptotic expansion method, and use them to dene ( v[1](t, x ), z[1](t, x )) in turn. Now,we can move to ( 6.4) and ( 6.5) with i = 2. We repeat the same procedures to the desiredorder of approximation.

The above procedures can be interpreted as the perturbative implementation of thefour step scheme in [ 12] under the very general setup. In particular, due to the equiva-lence between the PDE and probabilistic approaches, we can actually derive the analyticalsolution of the corresponding non-linear PDE to the same order of approximation.

Remark : Although we have considered one-dimensional process for V , it is straightforward to extend the method for higher dimensional cases. Once we take the basis of X in such a way that the linear drift term V is diagonal, we can proceed without anydifficulty. The mixing from the other components of V always appears in the lower orderof , which keeps the diagonal form of drift term intact in an arbitrary order.

4 Since we nally put = 1 (and also = 1), the actual order of error terms are of o(( + )k ) in thisexample.

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7 Conclusion and Discussions

In this paper, we have presented a simple perturbation scheme for non-linear decoupledas well as coupled FBSDEs. By considering the interested system as a decoupled linearFBSDE with non-linear perturbation terms, we succeeded to provide the analytic ap-proximation method to an arbitrarily higher order of expansion. We have shown thatthe required calculations in each order are equivalent to those for the standard Europeancontingent claims. We have applied the method to the two simple models and comparedthem with the numerical results directly obtained from the PDE and regression-basedMonte Carlo simulation. Both of the examples clearly demonstrated the strength of ourmethod. We have also shown that the use of the asymptotic expansion method for forwardcomponents allows us to proceed to the higher order of perturbation even if the forwardcomponents do not have known distributions.

In the last part of the paper, we have studied the perturbative method in the PDE

framework based on the so-called Four Step Scheme. We have shown that our perturba-tive treatment renders the original non-linear FBSDE into the series of linear parabolicPDEs, which are straightforward to handle. Furthermore, by the equivalence of the twoapproaches, we were also able to provide the corresponding perturbative method in prob-abilistic framework which is explicitly consistent with the Four Step Scheme up to a givenorder of expansion.

The perturbation theory presented in this paper may turn out to be crucial to inves-tigate various interesting problems, such as those given in the introduction, which havebeen preventing analytical treatment so far. The application of the new method to theimportant nancial problems is one of our ongoing research topics.

Finally, let us remark on the further extension to the cases including jumps. Although,

in this work, we have only considered the dynamics driven by Brownian motions, thesame approximation scheme can also be applied to more generic cases. Although it willbe more difficult to obtain explicit expressions in terms of forward components, if wechoose the specic forward processes with appropriate analytical properties, we shouldbe able to proceed in the similar way. Particularly, the separation of the original systeminto the decoupled linear FBSDE and the non-linear perturbation terms can be done in acompletely parallel fashion.

A Another expansion method for coupled FBSDEs

In the appendix, we provide another method which is more closely related to that of Sec .2

for the generic coupled FBSDE. We consider the same coupled FBSDE as in Sec. 6:dV ()t = c(X

()t )V

()t dt g(X

()t , V

()t , Z

()t )dt + Z

()t dW t (A.1)

V ()T = ( X ()T ) (A.2)

dX ()t = r (X ()t )dt + (X

()t , V

()t , Z

()t )dt + (X

()t ) + (X

()t , V

()t , Z

()t ) dW t (A.3)

X ()0 = x . (A.4)

Here, we absorbed a possible explicit time dependency to X .

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As in Sec.2, we are going to expand the backward as well as forward components interms of . Suppose that we have

V ()t = V (0)t + V (1)t + 2V (2)t + (A.5)Z ()t = Z

(0)t + Z

(1)t +

2Z (2)t + (A.6)X ()t = X

(0)t + X

(1)t +

2X (2)t + . (A.7)Now, let us derive each term separately.

A.1 Zero-th order

In the zero-th order, the original equation reduces to a linear and decoupled FBSDE,which will going to serve as the center of expansion:

dV (0)t = c(X (0)t )V (0)t dt + Z (0)t dW t (A.8)V (0)T = ( X

(0)T ) (A.9)

dX (0)t = r (X (0)t )dt + (X

(0)t ) dW t (A.10)

X (0)0 = x . (A.11)

X (0) is now the standard Markovian process completely decoupled from the backwardcomponents. We can easily solve the backward components as

V (0)t = E e T

t c(X(0)s )ds (X (0)T ) F t (A.12)

Z (0)

t = E Dt e T t c(X

(0)s )ds

(X (0)

T ) F t . (A.13)As explained in the previous section, we can express V (0)t and Z

(0)t in terms of X

(0)t by the

help of the asymptotic expansion for volatility even if the process of X (0) does not haveknown distribution.

A.2 First order correction

Now let us consider the dynamics of V () V (0) and X () X (0) . Following the samearguments in Sec .2, one can show straightforwardly thatdV (1)t = c(X

(0)t )V

(1)t dt

g(1) (t)dt + Z (1)t

dW t (A.14)

V (1)T = (1) (T ) (A.15)

dX (1)t = x r (X (0)t ) X

(1)t dt +

(0) (t)dt + (1) (t) dW t (A.16)X (1)0 = 0 , (A.17)

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where we have used shorthand notations:

(1) (T ) = (1) (X (0)T , X (1)T ) =

x (X (0)T )

X (1)T (A.18)

g(0) (t) = g(X (0)t , V (0)

t , Z (0)t ) (A.19)

g(1) (t) = g(1) (X (0)t , X (1)t , V

(0)t , Z

(0)t ) = g

(0) (t) x c(X (0)t ) X

(1)t V

(0)t (A.20)

(0) (t) = (X (0)t , V (0)

t , Z (0)t ) (A.21)

(0) (t) = (X (0)t , V (0)

t , Z (0)t ) (A.22)

(1) (t) = (1) (X (0)t , X (1)t , V

(0)t , Z

(0)t ) =

(0) (t) + x (X (0)t ) X

(1)t . (A.23)

Note that, since we have obtained V (0)t and Z (0)t as the functions of X

(0)t , the pair

(X (0) , X (1) ) consists of a Markovian process, which is indeed decoupled from ( V (1) , Z (1) ).This means that we have ended up with the decoupled linear FBSDE also for the rst

order correction. Hence, one can easily solve the backward components as

V (1)t = E e T

t c(X(0)s )ds (1) (T ) +

T

te

ut c(X

(0)s )ds g(1) (u)du F t (A.24)

Z (1)t = E Dt e T

t c(X(0)s )ds (1) (T ) +

T

te

ut c(X

(0)s )ds g(1) (u)du F t (A.25)

As a result, we have obtained V (1)t as well as Z (1)t as the functions of X

(0)t and X

(1)t .

A.3 Second and Higher order corrections

We can continue to the higher order corrections in the same way. By considering V ()

(V (0) + V (1) ) and X () (X (0) + X (1) ), and then extracting the -second order terms,one can show thatdV (2)t = c(X

(0)t )V

(2)t dt g(2) (t)dt + Z

(2)t dW t (A.26)

V (2)T = (2) (T ) (A.27)

dX (2)t = x r (X (0)t ) X

(2)t dt +

(2) (t)dt + (2) (t) dW t (A.28)X (2)0 = 0 . (A.29)

Here we have dened

(2)

(T ) = (2)

(X (0)

T , X (1)

T , X (2)

T ) (A.30)= x (X

(0)T ) X

(2)T +

12

ij (X (0)T )X

i(1)T X

j (1)T (A.31)

g(2) (t) = g(2) (X (0)t , X (1)t , X

(2)t , V

(0)t , V

(1)t , Z

(0)t , Z

(1)t ) (A.32)

= x g(0) (t) X

(1)t +

v

g(0) (t)V (1)t + z g(0) (t) Z

(1)t

x c(X (0)t ) X

(2)t +

12

ij c(X (0)t )X

i(1)t X

j (1)t V

(0)t x c(X

(0)t ) X

(1)t V

(1)t .

(A.33)

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And also for the process X (2) , we have dened

(2) (t) = (2) (X (0)t , X (1)t , X

(2)t , V

(0)t , V

(1)t , Z

(0)t , Z

(1)t ) (A.34)

= x (0) (t) X

(1)t +

v

(0) (t)V (1)t + z (0) (t) Z

(1)t (A.35)

+12

ij r (X (0)t )X

i(1)t X

j (1)t (A.36)

and

(2) (t) = (2) (X (0)t , X (1)t , X

(2)t , V

(0)t , V

(1)t , Z

(0)t , Z

(1)t ) (A.37)

= x (X (0)t ) X

(2)t +

12

ij (X (0)t )X

i(1)t X

j (1)t

+ x (0) (t)X (1)t +

v

(0) (t)V (1)t + z (0) (t) Z

(1)t . (A.38)

One can check that the pair ( X (0)t , X (1)t , X

(2)t ) consists of a Markovian process and it

is decoupled from the backward components. Hence, once again, we have obtained thedecoupled linear FBSDE, which is solvable as before:

V (2)t = E e T

t c(X(0)s )ds (2) (T ) +

T

te

ut c(X

(0)s )ds g(2) (u)du F t (A.39)

Z (2)t = E Dt e T

t c(X(0)s )ds (2) (T ) +

T

te

ut c(X

(0)s )ds g(2) (u)du F t . (A.40)

In completely the same way, we can proceed to an arbitrarily higher order correction.

In each expansion order i

, the set ( X (0)

t , , X (i)

t ) follows Markovian process decoupledfrom the backward components, and also the FBSDE continues to be linear thank to the in front of the non-linear terms.

Remark : For simplicity of the presentation, we have used the common perturbation param-eter both for the non-linearity in backward components as well as the feedback effects inthe forward components. However, as one can easily expect, it is also possible to introducemultiple perturbation parameters.

A.4 Consistency to the result of Sec. 6

For completeness, let us check the consistency to the result of Sec. 6. In the zero-th order,

the corresponding FBSDEs are exactly equal. In the rst order, we have obtained

V [1]t = E e T

t c(s,X[1]s )ds (X [1]T )

+ T

te

ut c(s,X

[1]s )ds g u, X [1]u , v(0) (u, X [1]u ), z (0) (u, X [1]u ) du F t (A.41)

and Z [1]t as its Malliavin derivative in Sec. 6. Now, in order to compare it to the currentmethod, let us consider

V [1]t V (0)

t (A.42)

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and extract rst order terms by expanding X [1]t = X (0)t + X

(1)t . Since the second term

of (A.41) has already in the front, we simply get a contribution from there as

E T t e u

t c(s,X(0)s )ds g u, X (0)u , v

(0) (u, X (0)u ), z(0) (u, X (0)u ) du F t . (A.43)

From the rst term, we have to expand X [1] in the terminal payoff and also in the discount:

E e T

t c(s,X(0)s )dsx (X

(0)T ) X

(1)T

T

te

ut c(s,X

(0)s )ds V (0)u x c(u, X

(0)u ) X (1)u du F t

where we have used the Gateaux derivative 5. Second term can be interpreted that thereis a change in value by

x c(u, X (0)u ) X (1)u V (0)u (A.44)at each point of time u, which is summed and discounted back to the current time. Bysumming the above two terms, one can easily conrm its equivalence to the result of the previous section. Applying Malliavin derivative automatically tells the consistency of volatility terms. Using the same arguments, one can check the consistency between thecurrent method and that of Sec. 6. Although we have solved V (i) separately, the sum

ik=0

k V (k ) can be shown equivalent to V [i] up to the error terms o(i ).

References

[1] Bismut, J.M. 1973, Conjugate Convex Functions in Optimal Stochastic Control, J.Political Econ., 3, 637-654.

[2] Duffie, D. and Epstein, L., 1992, Stochastic Differential Utility, Econometrica 60353-394.

[3] Duffie, D., Huang, M., 1996, Swap Rates and Credit Quality, Journal of Finance,Vol. 51, No. 3, 921.

[4] El Karoui, N., Peng, S.G., and Quenez, M.C., 1997, Backward stochastic differentialequations in nance, Math. Finance 7 1-71.

[5] Fujii, M., Takahashi, A., 2010, Derivative pricing under Asymmetric and Imper-fect Collateralization and CVA, CARF Working paper series F-240, available athttp://ssrn.com/abstract=1731763.

[6] Gobet, E., Lemor, J., Warin, X., 2005, A Regression-based Monte Carlo Method tosolve Backward Stochastic Differential Equations, The Annals of Applied Probabil-ity, 15, No.3, 2172-2202.

[7] Kunitomo, N. and Takahashi, A. 2003, On Validity of the Asymptotic ExpansionApproach in Contingent Claim Analysis, Annals of Applied Probability, 13, No.3,914-952.

5 See, for examples, [3] for similar calculation.

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[8] Shiraya, K., Takahashi, A., and Toda, M, 2009, Pricing Barrier and Average Optionsunder Stochastic Volatility Environment, CARF Working Paper F-242, available athttp://www.carf.e.u-tokyo.ac.jp/workingpaper/ , forthcoming in Journal of Compu-tational Finance.

[9] Takahashi, A. 1999, An Asymptotic Expansion Approach to Pricing ContingentClaims, Asia-Pacic Financial Markets, 6, 115-151.

[10] Takahashi, A., Takehara, K., and Toda, M., 2011, A General Computation Schemefor a High-Order Asymptotic Expansion Method, CARF Working Paper F-242,available at http://www.carf.e.u-tokyo.ac.jp/workingpaper/ .

[11] Takahashi, A. and Yoshida, N. 2004, An Asymptotic Expansion Scheme for OptimalInvestment Problems, Statistical Inference for Stochastic Processes, 7, No.2, 153-188.

[12] Ma, J., and Yong, J., 2000, Forward-Backward Stochastic Differential Equations andtheir Applications, Springer.

[13] Pardoux, E., and S. Peng, 1990, Adapted Solution of a Backward Stochastic Differ-ential Equation, Systems Control Lett., 14, 55-61.
http://www.carf.e.u-tokyo.ac.jp/workingpaper/http://www.carf.e.u-tokyo.ac.jp/workingpaper/http://www.carf.e.u-tokyo.ac.jp/workingpaper/http://www.carf.e.u-tokyo.ac.jp/workingpaper/

Non LinearFBSDEs Perturbation

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