In Chapter 1, the results on the pure BSDEs, especially the fundamen- tal well-posedness result, are based on the method introduced in the seminal paper of Pardoux-Peng [1].. The well-po
Trang 1250 Chapter 9 Numerical Methods of FBSDEs Add the two inequalities above and apply Gronwall's lemma; we see t h a t
sup(H~kI[ + IIckil) = V(h + At)
k Applying the arguments similar to those in Theorem 2.3 we can derive the following theorem
T h e o r e m 2.4 Suppose that (A1)-(A3) hold Then,
-u(t, z)] + IV(~)(t,x) -u~(t,z)[} = O(1)
sup { ]V(n) (t, x)
Moreover, for each fixed x E IR, U (n) (., x) and V (~) (., x) are left-continuous; for fixed t C [0, T], U (~) (t, ) and V ('~) (t, -) are uniformly Lipschitz, with the same Lipschitz constant that is independent of n
w N u m e r i c a l A p p r o x i m a t i o n o f t h e Forward S D E
Having derive the numerical solution of the PDE (1.5), we are now ready
to complete the final step: approximating the Forward SDE (1.4) Recall
t h a t the FSDE to be approximated has the following form:
/0 (3.1) X t = x + s, X s ) d s + ~(s, X s ) d W s ,
where
b(t, x) = b(t, x, O(t, x), - a ( t , x, O(t, x)O~ (t, x)) = bo (t, x, O(t, x), O~ (t, x));
~(t, x) = ~(t, x, o(t, ~))
for (t, x) e [0, T] • IR
To define the approximate SDEs, we need some notations For each
n E IN, set At~ = T / n , t ~,k = kAtn, k = 0 , 1 , 2 , , n , and
n - - 1 (3.2) ~/n(t) = Etn'kl[tn.~,t~.k+x)(t), t 9 [0, T);
k = 0 gn(T) = T
Next, for each n, let (U (n), V (n)) be the approximate solution to the PDE (1.5), defined by (2.35) (in the special case we may consider only u (n)
defined by (2.24)) Set
(3.3) o ~ ( t , x ) = U ( ~ ) ( T - t , x ) , o n ( t , ~ ) = V ( n ) ( T - t , x ) ,
and
bn(t,x) = bo(t,x, On(t,x),O'~(t,x)); ~ ( t , x ) = a ( t , x , On(t,x))
By Theorem 2.4 we know that 0 = is right continuous in t and uniformly Lipschitz in x, with the Lipschitz constant being independent of t and n;
Trang 2w Numerical approximation of the FSDE 251 thus, so also are the functions ~n and yn We henceforth assume that there exists a constant K such that, for all t and n,
(3.4) Ibn(t,x) - b n ( t , x ' ) l + I~n(t,x) - ~ n ( t , x ' ) l _ K I x - x'l, x,x' e IR
Also, from Theorem 3.4,
(3.5) s u p ( b ~ ( t , x ) - b ( t , x ) l ' + s u p l ~ n ( t , x ) - ~ ( t , x ) l = O ( 1 )
We now introduce two SDEs: the first one is a discretized SDE given
by
(3.6) 2 / ~ = x + g"(.,2?Lo(s)as + ~(.,2~),~(s)aw~,
where ~n is defined by (3.2) The other is an intermediate approximate SDE given by
It is clear from the properties of ~n and ~n mentioned above that both SDEs (3.6) and (3.7) above possess unique strong solutions
We shall estimate the differences )(~ - X~ * and X n - X , separately
L e I n m a 3.1 Assume (A1) (A3) Then,
E { sup I X T : - X : I 2} = 0 ( 1 )
O < t < T
Proof T o simplify notation, w e shall suppress the sign " - " for the coefficients in the sequel We first rewrite (3.6) as follows:
where
u~ = L k', 9 )n~(s)-
Applying Doob's inequality, Jensen's inequality, and using the Lipschitz property of the coefficients (3.4) we have
E { sup IX: - ~7:12 }
s < t
(~) _<~{ s~p i~nl ~ } + ~ ]/~{I~ n - ~s~,~}~s
s<t
+ 12K 2 - - fE{lX$ -
J0
Trang 3252 Chapter 9 Numerical Methods of FBSDEs Now, set as(t) = E { sups_< t [X~ - Xsnl2} Then, from (3.8),
/o'
an(t) < 3E{ sup [un] 2 } + 3 K 2 ( T + 4 ) an(s)ds,
s<_t
and Gronwall's inequality leads to
(3.9) E ~ s u p [ X n - Xsni 2 } < 3e3K~(T+4)E~ sup ]Uy]2~
s < t ~ s < t J
We now estimate E{sups_< t ]uyl2} Note t h a t if s E [tn'k,tn'k+l), for some 1 < k < n, then ??n(s) = kAtn (whence T - ~n(s) = (n - k ) A t n , as
(2.9) and (3.2), for every x E IR
0~(nn(~), ~) = ~ ( ~ ) ( T - n n ( s ) , x) = ~(~)((~ - k ) ~ t ~ , x)
: ur - s,x) : O'~(s,x)
More generally, for all (s, x) E [0, T] x IR,
b'~(s,x) = b(s,x, On(s,x)) = b(s,x, On(~n(s),x))
Using this fact, it is easily seen t h a t
fo ~ b(v (~),X,o(~),O (v (~), n - n n n x,~(~))) - b(~,X n~ , O~(~,X:)) - n ds
/o'
b ~ n - n _
+ (~,x~,o (~,x,o(~))) b(~,x:,on(~,x:)) }d~
=11 + I2
Using the boundedness of the functions bt, b~ and by, we see t h a t
/o'
Thus,
where h" depends only on K, ]]btllc~, IIb~]l~ and Ilbvll~ Since
i n n ( s ) - sJds = (s - t k ) d s < -~ k=o
Trang 4w Numerical approximation of the FSDE 253
E{sup~<~ fo~b ~'~, 2 ~ ),~(s)- b~(8, X:)ds 2 }
(3.10)
Using the same reasoning for a with Doob's inequality, we can see that
u < t
(3.11) _< 8/~2 { EIy;n(~)-XFI2ds+ (s-~n(s))2ds}
< 8~2{ fo ~
Combining (3.10) and (3.11), we get
E{sup I,,~12} < ~ ( 4 T + 16) E[2~(~) - X~[2ds + R~T(T + 3 ~2"
s < t
Thus, by (3.9),
.{ su8 , ~ : - x : l ~ }
+ K 2 T ( T + ~ ) n ~ }- Finally, noting that ] ~-(s) - X~I < I ~(~) - -~21 + 122 - X21 and that
we see as before that
fo EI2~(~) - 2212ds <_ 2 IlbllL(s - v~(s)) 2 + [l~llLIs - v~(s)l ds
- 3 n 2 +II~II~T n
Therefore, (3.12) becomes
(3.13) E { s u p l X n - - x : I 2} < C , + C 2 z ~ + C 3 f E#suPlX~-X~l }as,
s < t - - I t J o t r < s
where C1, C2 and Cz are constants depending only on the coefficients b,
a and K and can be calculated explicitly from (3.12) Now, we conclude from (3.13) and Gronwall's inequality that
~n(t) <_ /3ne CT, Vt 6 [0, T],
Trang 5254 Chapter 9 Numerical Methods of FBSDEs
where /9~ = Cln -1 + C~n -2 and CT = C3T In particular, by slightly
changing the constants, we have
an(T):E~ sup I X : - X n l 2} < C, + 02 =0(i),
- 0<~<~ - ~ -
The main result of this chapter is the following theorem
T h e o r e m 3.2 Suppose that the standing assumptions (A1) (A3) hold Then, the adapted solution (X, Y, Z) to the FBSDE (1.1) can be approxi- mated by a sequence of adapted processes (X "n, Y~, Zn), where f(~ is the solution to the discretized SDE (3.6) and, for t 6 [0, T],
~ n : : 8~(t,2tn); Z? := - a ( t , 2 ~ , s n ( t , f ~ ) ) O ~ ( t , f ( ? ) ,
with O n and 0 n being defined by (3.3) and U (n) and V (~) by (2.34) Fur- thermore,
(3.14) E { 0<t<TSUp ]f(: XtI+O<t<TSUp ]~n Ytl+0<t<Tsup I ' ~ - Z t l } = O ( ~ n )
Moreover, if f is C 2 and uniformly Lipschitz, then for n large enough,
n
for a constant K
Proo] Recall that at the beginning of the proof of Lemma 3.1, we have suppressed the sign "-" for b and ~ to simplify notation Set
~n(t) = { sup Ibm(t, x) - b(t, x)l 2 + sup lan(t, x) a ( t , ~)l ~ },
where b, b n, a and a n are defined by (3.1) and (3.3) Then, from (3.5) we
know that sup t Izn(t)l = O(~A~) Now, applying Lemma 3.1, we have
~{ : ~ I~: - ~J~} _< ~{ ~u~ i ~ : - ~:l ~ } ~ { ~u~ i~: - ~sl ~ }
Trang 6w Numerical approximation of the FSDE 255 Further, observe that
<_4T fot Elbn(s, X2) - bn(s, X~)[2ds
+ 16 Elan(s, X2) - a n (s, Xs)12ds + 4(T + 4) r
~4(T + 4)K ~ E{ sup IX~ - X~I ~}es + 4(T + 4) ~n(s)e~
r<_s
Applying Gronwall's inequality, we get
(3.16) E sup [X• - Xs[ 2 < 4(T + 4) Sn(s)ds" e 4(T+4)K2 < n- ~,
s<t
where C is a constant depending only on K and T Now, note t h a t the functions 0 and On are both uniformly Lipschitz in x So, if we denote their Lipschitz constants by the same L, then
0<t<T
< 2E~ sup lO(t, X t ) - , , t :~ I
0<t<T
+ 2E{ sup 10~(t,22)-0(t,~)l 2}
0<t<T
by Theorem 3.4 and (3.16) The estimate (3.14) then follows from an easy application of Cauchy-Schwartz inequality To prove (3.15), note that Theorem 2.3 implies that, for n large enough, snp(t,x)10n(t,x) - 0 ( t , x)l
(3.6) by fixing n and approximating the solution X ~ of (3.7) by a standard Euler scheme indexed by k:
2 ~ 'k = x + b ( , 2 ~ , k ) n , c ( s ) d s + a ( , 2 ? ' k ) , , , ( s ) d W s
It is then standard (see, for example, Kloeden-Platen [1, p.460]) t h a t
(3.17) E{:(X~)} - E{f(2~'k)} <_ K C1
Trang 7256 Chapter 9 Numerical Methods of FBSDEs
On the other hand, we have
O ~ t ~ T ) ?2
for Lipschitzian f, by (3.16) Therefore, noting that X~ as defined by (3.6)
is just _~n,n t , the triangle inequality, (3.17) and (3.18) lead to (3.15) []
Trang 8C o m m e n t s a n d R e m a r k s
The main body of this b o o k is built on the works of the authors, with various collaboration with other researchers, on this subject since 1993 Some significant results of other researchers are also included to enhance the book However, due to the limitation of our information, we inevitably might have overlooked some new development in this field while writing this book, for which we deeply regret
In Chapter 1, the results on the pure BSDEs, especially the fundamen- tal well-posedness result, are based on the method introduced in the seminal paper of Pardoux-Peng [1] The results on nonsolvability of FBSDEs are inspired by the example of Antonelli [1] The well-posedness results of FB- SDEs over small duration is also based in the spirit of the work of Antonelli [1] The whole Chapter 2 is based on the paper of Yong [4]
In Chapter 3 we begin to consider a general form of the FBSDE (1) with an arbitrarily given T > 0 The main references for this chapter are based on the works of Ma-Yong [1], virtually the first result regarding solvability of FBSDE in this generality; and Ma-Yong [4], in which the notion of approximate solvability is introduced A direct consequence of the method of optimal control is the Four Step Scheme presented in Chapter 4 The finite horizon case is initiated by Ma-Protter-Yong [1]; and the infinite horizon case is the theoretical part of the work on "Black's Consol Rate Conjecture" presented later in Chapter 8, by Duffie-Ma-Yong [1]
Chapter 5 can be viewed either as a tool needed to extend the Four Step Scheme to the situation when the coefficients are allowed to be random, or
as an independent subject in stochastic partial differential equations The main results come from the papers of Ma-Yong [2] and [3]; and the appli- cations in finance (e.g, the stochastic Black-Scholes formula) are collected
in Chapter 8
The method of continuation of Chapter 6 is based on the paper of Hu- Peng [2], and its generalization by Yong [1] The method adopted a widely used idea in the theory of partial differential equations Compared to the Four Step Scheme, this method allows the randomness of the coefficients and the degeneracy of the forward diffusion, but requires some analysis which readers might find difficult in a different way
Chapter 7 is based on the work of Cvitanic-Ma [2] The idea for the forward SDER using the solution mapping of Skorohod problem is due
to Anderson-Orey [1], while the Lipschitz property of such solution map- ping is adopted from Dupuis-Ishii [1] The proof of the backward SDER
is a modification of the arguments of Pardoux-Rascanu [1], [2], as well as some arguments from BuckdahmHu [1] The proof of the existence and uniqueness of FBSDER adopted the idea of Pardoux-Tang [1], a general- ized method of contraction mapping theorem, which can be viewed as an independent method for solving FBSDE as well
Trang 9258 Comments and Remarks Chapter 8 collects some successful applications of the FBSDEs devel- oped so far The integral representation theorem is due to Ma-Protter-Yong [1]; the Nonlinear Feynman-Kac formula is in the spirit of Peng [4], but the argument of the proof follows more closely those of Cvitanic-Ma [2] The Black's consol rate conjecture is due to Duffie-Ma-Yong [1]; while hedging contingent claims for large investors comes from Cvitanic-Ma [1] for uncon- straint case, and from Buckdahn-Hu [1] for constraint case The section on stochastic Black-Scholes formula is based on the results of Ma-Yong [2] and [3], and the American game option is from Cvitanic-Ma [2]
Finally, the numerical method presented in Chapter 9 is essentially the paper of Douglas-Ma-Protter [1], with slight modifications We should point out that, to our best knowledge, the scheme presented here is the only numerical m e t h o d for (strongly coupled) FBSDEs discovered so far, and even when reduced to the pure BSDE case, it is still one of the very few existing numerical methods that can be found in the literature
In summary, F B S D E is a new type of Stochastic differential equations
t h a t has its own mathematical flavor and many applications Like a usual two-point b o u n d a r y value problem, there is no generic theory for its solv- ability, and many interesting insights of the equations has yet to be dis- covered In the meantime, although the theory exists only for such a short period of time (recall that the first paper on F B S D E was published in
19930, many topics in theoretical and applied mathematics have already been found closely related to it, and its applicability is quite impressive
It is our hope t h a t by presenting a lecture notes in the series of LNM, more attention would be drawn from the mathematics community, and the
b e a u t y of the problem would be further exposed
Trang 10R e f e r e n c e s
Ahn, H., Muni, A., and Swindle, G.,
[1] Misspecified asset pricing models and robust hedging strategies, preprint
Anderson, R F and Orey, S.,
[1] Small random perturbation of dynamical systems with reflecting
boundary, Nagoya Math J., 60 (1976), 189-216
Antonelli, F.,
[1] Backward-forward stochastic differential equations, Ann Appl Prob.,
3 (1993), 777-793
Bailey, P B., Shampine, L F., and Waltman, P E.,
[1] Nonlinear Two Point Boundary Value Problems, Academic Press, New
York, 1968
Barbu, V.,
[1] Nonlinear Semigroups and Differential Equations in Banach Spaces,
Noordhoff Internation Publishing, 1976
Barles, G., Buckdahn, R., and Pardoux, E.,
[1] Backward stochastic differential equations and integral-partial differen-
tial equations, Stochastics and Stochastics Reports, 60 (1997), 57-83
Bellman, R., and Wing, G M.,
[1] An Introduction to Invariant Imbedding, John Wiley & Sons, New
York, 1975
Bensoussan, A.,
[1] Stochastic maximum principle for distributed parameter systems, J
F~anMin Inst., 315 (1983), 387-406
[2] Maximum principle and dynamic programming approaches of the op-
timal control of partially observed diffusions, Stochastics, 9 (1983),
169-222
[3] On the theory of option pricing, Acta Appl Math., 2 (1984), 139-158 [4] Perturbation Methods in OptimM Control, John-Wiley & Sons, New
York, 1988
[5] Stochastic Control of Partially Observable Systems, Cambridge Uni-
versity Press, 1992
Bergman, Y.Z., Grundy, B.D., and Wiener, Z.,
[1] General Properties of Option Pricing, Preprint, 1996
Bismut, J M.,
[1] Th~orie Probabiliste du Contr61e des Diffusions, Mere Amer Math Soc 176, Providence, Rhode Island, 1973