Differential Equations and Their Applications Part 14 docx

250 Chapter 9. Numerical Methods of FBSDEs Add the two inequalities above and apply Gronwall's lemma; we see that sup(H~kI[ + IIckil) = V(h + At). k Applying the arguments similar to those in Theorem 2.3 we can derive the following theorem. Theorem 2.4. Suppose that (A1)-(A3) hold. Then, -u(t, z)] + IV(~)(t,x) -u~(t,z)[} = O(1). sup { ]V(n) (t, x) (t,~) n 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 Numerical Approximation of the Forward SDE 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 that the FSDE to be approximated has the following form: /0 (3.1) Xt = x + s, Xs)ds + ~(s, Xs)dWs, 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), on(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; w 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) -bn(t,x')l + I~n(t,x) -~n(t,x')l _ KIx- x'l, x,x' e IR. Also, from Theorem 3.4, (3.5) sup(b~(t,x)-b(t,x)l'+supl~n(t,x)-~(t,x)l = O(1). t,~ t,x 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 // // (3.7) X~ = x + b~(s, X2)ds + Yn(s, X2)dW~. 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. LeInma 3.1. Assume (A1) (A3). Then, E{ sup IXT:-X:I 2} =0(1). O<t<T Proof. To simplify notation, we shall suppress the sign "-" for the coefficients in the sequel. We first rewrite (3.6) as follows: /o // 2~ = No + u~ + bn(s, Xn)ds + an(s, f(n)dWs, where L /o trbn, f(n\ bn(s,22)]ds+ [an(.,2.~), (~)_an(s,22)]dW~. 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 252 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 } + 3K2(T +4) an(s)ds, s<_t and Gronwall's inequality leads to (3.9) E~ sup[X n - Xsni 2 } < 3e3K~(T+4)E~ sup ]Uy]2~. s<t ~ s<t J We now estimate E{sups_< t ]uyl2}. Note that if s E [tn'k,tn'k+l), for some 1 < k < n, then ??n(s) = kAtn (whence T - ~n(s) = (n - k)Atn, as T = nat,) and T - s E ((n - k - 1)Atn, (n - k)At~]. Thus, by definitions (2.9) and (3.2), for every x E IR 0~(nn(~), ~) = ~(~)(T - nn(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 that fo ~ b(v (~),X,o(~),O (v (~), n -n n n -n x,~(~))) - b(~,X n~ , O~(~,X:)) ds /o' ~ { b(~n(s), f(vn~(~), On(s, 2~n~ (~))) - b(s, X2, 9~(s, 2v~(~)) ) 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 that { ~ ~/o' { lt~ll~l~~ § H~JJ~I~,~-~:1}~, /o' I2 < K]lbyH~ . If(vS(s) - X2]ds. Thus, - + x jlds, where h" depends only on K, ]]btllc~, IIb~]l~ and Ilbvll~. Since inn(s) - sJds = (s - tk)ds < -~ k=o w Numerical approximation of the FSDE 253 E{sup~<~ fo~b ~'~, 2 ~. ),~(s)- b~(8, X:)ds 2 } (3.10) 0 t T4 Using the same reasoning for a with Doob's inequality, we can see that ~{sup [ ~n(. X.~)~(~) _ ~~ X:)eW~ 2} u<t (3.11) _< 8/~2 { EIy;n(~)-XFI2ds+ (s-~n(s))2ds} < 8~2{ fo ~ Combining (3.10) and (3.11), we get fo t 16)1 E{sup I,,~12} < ~(4T + 16) E[2~(~) - X~[2ds + R~T(T + 3 ~2" s<t Thus, by (3.9), .{ su8 ,~: - x:l ~ } (3.12) <_ 3e 3K2(T+4){/~2(4T + 16) EI2,~.(~) - X2[2ds +K2T(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 < 2]lbll~T 1 1 - 3 n 2 +II~II~T n Therefore, (3.12) becomes i t -n n 2 (3.13) E{suplXn x:I 2} <C, +C2z~+C3 f E#suPlX~-X~l }as, s<t It Jo 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], 254 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 IX:-Xnl 2} < C, + 02 =0(i), - 0<~<~ -~- proving the lemma. [] The main result of this chapter is the following theorem. Theorem 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~,sn(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'~-Ztl}=O(~n). Moreover, if f is C 2 and uniformly Lipschitz, then for n large enough, (3.15) E{f(2~, 2~)} - EU(XT, Z~)}[ _< K 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 ~ }, x x 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 ~ } w 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 { } Jo (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 that 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 _ on(t 2n~121 < 2E~ sup lO(t, Xt)- , , t :~ I 0<t<T + 2E{ sup 10~(t,22)-0(t,~)l 2} 0<t<T 0<t<T (t,x) 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 Cn -1, for some (generic) constant C > 0. We modify )(~ as defined by (3.6) by fixing n and approximating the solution X ~ of (3.7) by a standard Euler scheme indexed by k: f0 f0 2~ 'k = x + b(.,2.~,k)n,c(s)ds + a(.,2?'k),,,(s)dWs. It is then standard (see, for example, Kloeden-Platen [1, p.460]) that (3.17) C1 E{:(X~)} - E{f(2~'k)} <_ K 256 Chapter 9. Numerical Methods of FBSDEs On the other hand, we have Ig{f(XT)} - E{f(X~)}[ <_ KE{IXT - X~[ } (3.18) C2 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). [] Comments and Remarks The main body of this book 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. 258 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 method 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, FBSDE is a new type of Stochastic differential equations that has its own mathematical flavor and many applications. Like a usual two-point boundary 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 FBSDE 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 that by presenting a lecture notes in the series of LNM, more attention would be drawn from the mathematics community, and the beauty of the problem would be further exposed. References 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. [...]... stochastic differential equations and applications, Proc ICM, 1994 Pardoux, E., and Peng, S., [1] Adapted solution of a backward stochastic differential equation, Systems 35 Control Left., 14 (1990), 55-61 [2] Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS 176, 200-217, Springer 1992 [3] Backward doubly stochastic differential equations. .. principle and related backward stochastic differential equations, Proc 31st CDC, Tucson 1992 [8] Backward stochastic differential equation and its application in optimal control, Appl Math ~: Optim., 27 (1993), 125 -144 [9] Backward SDE and related g-expectation, preprint [10] Adapted solution of backward stochastic differential equations and related partial differential equations, preprint Peng, S., and. .. 1964 [2] Stochastic Differential Equations and Applications, Vol 1, Academic Press, 1975 Geoffard, P Y., [1] Discounting and optimizing, preprint Gilbarg, D., and Trudinger, N S., [1] Elliptic Partial Differential Equations of Second Order, SpringerVerlag, Berlin, 1977 Hamadene, S [1] Backward-forward stochastic differential equations and stochastic games, preprint Hamadene, S., and Lepeltier, J P.,... and Parabolic Equations of the Second Order, Reidel, Dordrecht, Holland, 1987 Krylov, N N., and Rozovskii, B L., [1] Stochastic partial differential equations and diffusion processes, Uspekhi Mat Nauk, 37 (1982), No.6, 75-95 Kunita, H., [1] Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990 Ladyzenskaja, O A., Solonnikov, V A., and Ural'ceva, N N., [1] Linear and. .. Equations and Diffusion Processes, North Hol- land, Amsterdam, 1981 John, F., [1] Partial Differential Equations, Springer-Verlag, 1975 Karatzas, I., and Shreve, S E., [1] Brownian Motion and Stochastic Calculus, Springer-Verlag, Berlin, 1988 Kloeden, P.E and Platen, E., [1] Numerical Solution of Stochastic Differential Equations, SpringerVerlag, 1992 Koopmans, T [1] Stationary ordinary utility and impatience,... backward SPDE, with applications, Stochastic Processes and their Applications, 70 (1997), 59-84 [3] On linear, degenerate backward stochastic partial differential equations, Probab Theory Re/at Fields, to appear [4] Approximate solvability of forward-backward stochastic differential equations, preprint Meyer, G H., [1] Initial Value Methods for Boundary Value Problems - - Theory and Application of Invariant... first order differential equation, J Diff Eqs., 5 (1969), 515-530 [2] Generalized Solutions in Optimal Stochastic Control, Differential games and control theory II, Lect Notes in Pure and Appl Math., 30, Dekker, 1977 Fleming, W F., and Soner, M., [1] Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993 References 263 Friedman, A., [1] Partial Differential Equations of... Buckdahn, R., and Hu, Y., [1] Hedging contingent claims for a large investor in an incomplete market, Advances in Applied Probability, 30 (1998), 239-255 Buckdahn, R., and Pardoux, E., [1] BSDE's with jumps and associated integral-stochastic differential equations, preprint Caffarelli, L A., and Friedman, A., [1] Partial regularity of the zero-set of semilinear and superlinear elliptic equations, J... partial differential equations under Neumann boundary conditions, Stoch Proc Appl., 48 (1993), 107-121 Hu, Y., and Peng, S., [1] Adapted solution of backward stochastic evolution equation, Stoch Anal ~: App1., 9 (1991), 445-459 [2] Solution of forward-backward stochastic differential equations, Probab Theory ~ Re1 Fields, 103 (1995), 273-283 Ikeda, N., and Watanabe, S., [1]Stochastic Differential Equations. .. equations and systems of quasilinear parabolic SPDEs, Probab Theory 35 Related Fields, 98 (1994), 209-227 [4] Some Backward stochastic differential equations with non-Lipschitz coefficients, preprint Pardoux, E., and Protter, P., [1] A two-sided stochastic integral and its calculus, Probab Fields, 76 (1987), 15-49 Th Rel Pardoux, E., and Rascanu, A., [1] Backward stochastic differential equations with subdifferential . differential equations and quasilinear parabolic partial differential equations, Lecture Notes in CIS 176, 200-217, Springer 1992. [3] Backward doubly stochastic differential equations and. Nonlinear Elliptic and Parabolic Equations of the Second Order, Reidel, Dordrecht, Holland, 1987. Krylov, N. N., and Rozovskii, B. L., [1] Stochastic partial differential equations and diffusion. 27 (1993), 125 -144 . [9] Backward SDE and related g-expectation, preprint. [10] Adapted solution of backward stochastic differential equations and re- lated partial differential equations, preprint.