1、Optimal Stopping For Stochastic FunctionalDifferential Equations Mou-Hsiung Chang Tao Pang Moustapha PemyDecember 9, 2005 (Revised February 12, 2007)AbstractWe consider a finite time horizon optimal stopping problem for asystem of stochastic functional differential equations with a boundedmemory. Un
2、dersomesufficientlysmoothconditions, aHamilton-Jacobi-Bellman (HJB) variational inequality for the value function is derivedvia dynamical programming principle. It is shown that the value func-tion is the unique viscosity solution of the HJB variational inequality.As an application of the results ob
3、tained, a pricing problem is con-sidered for American options in a financial market with one risklessbank account that grows according to a deterministic linear functionaldifferential equation and one stock whose price dynamics follows a non-linear stochastic functional differential equation. It is
4、shown that theoption pricing can be formulated into an optimal stopping problemconsidered in this paper and therefore all results obtained are applica-ble under very realistic assumptions.KEYWORDS: optimal stopping, stochastic control, stochastic functionaldifferential equations.AMS 2000 subject cla
5、ssifications: primary 60G40, 60G20; secondary60H30, 93E20The research of this paper is partially supported by a grant W911NF-04-D-0003 fromthe U. S. Army Research OfficeMathematics Division, U. S. Army Research Office, P. O. Box 12211, RTP, NC 27709,USA. Email: mouhsiung.changus.army.milDepartment o
6、f Mathematics, North Carolina State University, Raleigh, NC 27695,USA. Email: tpangunity.ncsu.edu (Corresponding Author)Department of Mathematics, Towson University, Towson, MD 21252-0001, USA.Email: mnpemytowson.edu11 IntroductionOptimal control and optimal stopping problems over a finite or an inf
7、initetime horizon for Itos diffusion processes arise in many areas of science,engineering, and finance (see e.g. Fleming and Soner 14, ksendal 35,Shiryaev 39, Karazas and Shreve 19 and references contained therein).The value function of these problems are normally expressed as a viscosityor a genera
8、lized solution of a Hamilton-Jacobi-Bellman equation (HJBE)or a Hamilton-Jacobi-Bellman variational inequality (HJBVI) that involvesa second order parabolic or elliptic partial differential equation in a finitedimensional Euclidean space (see e.g. Lions 30 and 32).In an attempt to achieve better acc
9、uracy and to account for the de-layed effect of the state variables in the modelling of real world stochasticcontrol problems, the stochastic delay equations and controlled stochasticdelay equations have been the subject of intensive studies in recent years byElsanousi 11, Larrssen 25, Elsanousi et
10、al 13, ksendal and Sulem 37,Larrssen and Risebro 27, and Bauer and Rieder 5.The controlled or uncontrolled stochastic delay equations considered bythe aforementioned researchers are described by the following special classesof equations that contain discrete and averaged delays:dX(s) = parenleftBigs
11、,X(s),X(sr),integraldisplay 0reX(s+)d,u(s)parenrightBigds (1)+ parenleftBigs,X(s),X(sr),integraldisplay 0reX(s+)d,u(s)parenrightBigdW(s), s t,T,ordX(s) = parenleftBigs,X(s),X(sr),integraldisplay 0reX(s+)dparenrightBigds (2)+ parenleftBigs,X(s),X(sr),integraldisplay 0reX(s+)dparenrightBigdW(s), s t,T
12、.In the above equations, W() = W(s),s 0 is anm-dimensional standardBrownian motion defined on a certain filtered probability space (,F,P,F),u() = u(s),s t,T is a control process taking values in the control setUin an Euclidean space, and are Rn and Rnm-valued functions definedon0,TRn Rn Rn U,2or0,TR
13、n Rn Rn,and 0 is a given constant.Based on the above two equations or their variants, Larssen 25 ob-tained an HJB equation for an optimal control problem, Elsanousi et al13 considered a singular control problem and obtained certain explicitlyavailable solutions, ksendal and Sulem 37 derived the maxi
14、mum prin-ciple for the optimal stochastic control. If the dynamics of the controlproblem with delay exhibit a special structure, Larssen and Risebro 27and Bauer and Rieder 5 showed that the value function actually lives ina finite-dimensional space and the original problem can be reduced to aclassic
15、al stochastic control problem without delay. Elsanousi and Larssen12 treated an optimal control problem and its applications to consumptionfor (1) under partial observation. We also mention that optimal stoppingproblems were studied in Elsanousis unpublished dissertation 11 for suchspecial type of e
16、quations.This paper extends the results obtained for finite dimensional diffusionprocesses and stochastic delay equations described in (2) and investigatesan optimal stopping problem over a finite time horizon for a general systemof stochastic functional differential equations (SFDE) described below
17、:dX(s) = f(s,Xs)dt+g(s,Xs)dW(s), s t,T, (3)whereT 0 andt 0,T, respectively, denote the terminal time and an ini-tial time of the optimal stopping problem. Again, W() = W(s),s 0 isa standard m-dimensional Brownian motion, and the drift f(s,Xs) and thediffusion coefficient g(s,Xs) (taking values in Rn
18、 and Rnm, respectively)depend explicitly on the segment Xs of the state process X() = X(s),str,T over the time interval sr,s, where Xs : r,0 Rn is definedby Xs() = X(s+), r,0. The consideration of such a system enableus to model many real world problems that have aftereffects (see e.g. Kol-manovsky
19、and Shaikhet 24 and references contained therein for applicationexamples).When r = 0, it is clear that the SFDE (3) reduces to the following Itosdiffusion process (without delay):dX(s) = f(s,X(s)ds+g(s,X(s)dW(s), s t,T.3It is clear that this equation also includes (2) as a special case and manyother
20、 equations that can not be modelled in the form of (2).This paper treats a finite time horizon optimal stopping problem (seeSection 2 for the problem statement) and derives an infinite dimensionalHJB variational inequality (HJBVI) for the value function via a dynamicprogramming principle (see e.g. L
21、arssen 25). It is shown that the valuefunction is the unique viscosity solution of the HJBVI. The proof of unique-ness involves embedding the function space C = C(r,0;Rn) into theHilbert space L2(r,0;Rn) and extending the concept of viscosity solu-tion for controlled Itos diffusion process developed
22、 by Crandall et al 9,and Lions 30 and 32 to an infinite dimensional setting. As an applicationof the results obtained, a pricing problem is considered for American optionsin a financial market with one riskless bank account that grows accordingto a deterministic linear functional differential equati
23、on (see (52) and onestock whose price dynamics follows a nonlinear stochastic functional dif-ferential equation (see (51). It is shown that the option pricing can beformulated into an optimal stopping problem considered in this paper andtherefore all results obtained are applicable under very realis
24、tic assumptions.Although infinite dimensional HJBVIs for optimal stopping problemsand their applications to pricing of American option have been studied veryrecently by a few researchers, they either considered stochastic delay equa-tion of special form (2)(see e.g. Gapeev and Reiss 16 and 17) or st
25、ochasticequations in Hilbert spaces (see e.g. Gatarek and Swiech 15 and Barbuand Marinelli 4). This paper differs from the aforementioned papers in thefollowing significant ways: i) The segmented solution process Xs,s t,Tis a strong Markov process in the Banach space C whose norm is not dif-ferentia
26、ble and is therefore more difficult to handle than any Hilbert spaceconsidered in 15 and 4; ii) the infinite-dimensional HJBVI uniquely in-volves the extensions DV(t,) and D2V(t,) of first and second orderFrechet derivativesDV(t,) andD2V(t,) from C and C to (CB) and(C B) (see Subsection 3.1 for defi
27、nitions of these spaces), respectively;and iii) the infinite-dimensional HJBVI also involves the infinitesimal gener-ator SV(t,) of the semigroup of shift operators value functions that doesnot appear in the special class of equations (2) in the aforementioned papers.This paper is organized as follo
28、ws. The notation and preliminary resultsthat are needed for formulating the optimal stopping problem as well asthe problem statement are contained in Section 2. In Section 3, the HJBVI4for the value function is heuristically derived using Bellmans type dynamicprogramming principle. The verification
29、theorem is also proved there. InSection 4, the continuity of the value function is proved. Although continu-ous, the value function is not known to be smooth enough to be a classicalsolution of the HJBVI in general cases. It is shown in Section 4, however,that the value function is a viscosity solut
30、ion of the HJBVI. Section 5 consid-ers the pricing of American options as an application of the results obtainedin the previous sections. The proof for the comparison principle (Theorem4.7) as well as the necessary lemmas are given in the appendix section afterthe list of references for the readabil
31、ity of the paper. The uniqueness resultfor the value function being the unique viscosity solution of the HJBVI fol-lows immediately from the comparison principle.Due to the excessive length of the current paper, the computational is-sues of the viscosity solution of the infinite dimensional HJBVI de
32、rived hereshall be the subject of our future study.2 The Optimal Stopping ProblemLet r 0 be a fixed constant, and let J = r,0 denote the durationof the bounded memory of the stochastic functional differential equationsconsidered in this paper. For the sake of simplicity, denote C(J;Rn), thespace of
33、continuous functions : J Rn, by C. Note that C is a realseparable Banach space under the sup-norm defined bybardblbardbl = supJ|()|, Cwhere | is the Euclidean norm in Rn.In addition to the space C, we also consider L2(J;Rn), the Hilbert spaceof Lebesque squared-integrable functions : J Rn equipped w
34、ith theinner product ( | ) and the Hilbertian norm bardblbardbl2 defined by(|) =integraldisplay 0r(),()d and bardblbardbl2 = (|)12, , L2(J,Rn),where , the inner product in Rn.5Note that the space C can be continuously embedded intoL2(J;Rn) (seee.g. Rudin 38). In factbardblbardbl2 rbardblbardbl, C.Co
35、nvention 2.1 Throughout the end, let T 0 denote the terminal timeand t 0,T be an initial time of the optimal stopping problem. We shalluse the following conventional notation for functional differential equations(see Hale 18):If C(tr,T;Rn) and s t,T, let s C be defined bys() = (s+), J.In addition, t
36、he values of the generic constants K and used throughoutthe end of the paper may change from line to line.Let W(s),s 0 be an m-dimensional standard Brownian motion definedon a complete filtered probability space (,F,P;F), where F = F(s),s0 is the P-augmentation of the natural filtration FW(s),s 0 ge
37、neratedby the Brownian motion W(s),s 0, i.e., if s 0,FW(s) = W(t),0 tsandF(s) = FW(s)A |B F such that AB and P(B) = 0 ,where the operator denotes that F(s) is the smallest -algebra such thatFW(s) F(s) andA |B F such that AB and P(B) = 0 F(s).Consider the following system of stochastic functional dif
38、ferential equa-tions:dX(s) = f(s,Xs)ds+g(s,Xs)dW(s), s t,T; (4)with the initial function Xt = t, where t is a given C-valued random vari-able that is F(t)-measurable. Here, f : 0,TC Rn and g : 0,TC Rnm are given deterministic functions.6Let L2(,C) be the space of C-valued random variables : C suchth
39、atbardblbardblL2 =parenleftbiggintegraldisplaybardbl()bardbl2dP()parenrightbigg12 0 such that|f(t,)f(s,)|+|g(t,)g(s,)| K(|ts|+bardblbardbl) (t,),(s,) 0,TC.7Assumption 2.4 There exists a constant K 0 such that|f(t,)|+|g(t,)| K(1+bardblbardbl) (t,) 0,TC.Let X(s;t,t),s t,T be the solution of (4) throug
40、h the initialdatum (t,t) 0,TC. We consider the corresponding C-valued processXs(t,t),s t,T defined byXs(;t,t) X(s+;t,t), J. (6)For eacht 0,T, let G(t) = G(t,s),s t,T be the filtration definedbyG(t,s) = (X(u;t,t),tus).Note that, it can be shown that for each s t,T,G(t,s) = (Xu(t,t),tus).This is due t
41、o the sample paths continuity of the process X(s;t,t),s t,T.One can then establish the following Markov property (see Mohammed33, 34).Theorem 2.5 Let Assumptions 2.3 and 2.4 hold. Then the correspondingC-valued solution process of (4) describes a C-valued Markov process in thefollowing sense:For any
42、 (t,t) 0,TL2(,C), the Markovian propertyPXs(t,t) B|G(t,) = PXs(t,t) B|X(t,t) p(,X(t,t),s,B)holds a.s. for t s and B B(C), where B(C) is the Borel -algebraof subsets of C.In the above, the function p : t,T C 0,T B(C) 0,1 denotesthe transition probabilities of the C-valued Markov process Xs(t,t),s t,T
43、.A random function : 0, is said to be a G(t)-stopping time if s G(t,s), st.8Let T be the collection of all G(t)-stopping times, and let TTt be the col-lection of all G(t)-stopping times T such that t T a.s For each TTt , let the sub-algebra G(t,) of F be defined byG(t,) = A F |At s G(t,s) s t,T.With
44、 a little bit more effort, one can also show that the correspondingC-valued solution process of (4) is also a strong Markov process in C. ThatisPXs(t,t) B|G(t,) = PXs(t,t) B|X(t,t) p(,X(t,t),s,B)holds a.s. for all TTt , all deterministic s ,T, and B B(C).If the drift f and the diffusion coefficient
45、g are time-independent, i.e.,f(t,) f() and g(t,) g(), then (4) reduces to the following au-tonomous system:dX(t) = f(Xt)dt+g(Xt)dW(t). (7)In this case, we usually assume the initial datum (t,t) = (0,) and de-note the solution process of (7) through (0,) and on the interval r,Tby X(s;),s r,T. Then th
46、e corresponding C-valued solution pro-cess Xs(),s r,T of (7) is a strong Markov process with time-homogeneous probability transition function p(,s,B) p(0,s,B) =p(t,t+s,B) for all s,t 0, C, and B B(C).AssumeLand are two bardblbardbl2-Lipschitz continuous real-valued functionson 0,TC with at most poly
47、nomial growth in L2(J;Rn). In other words,there exist a positive constant K, , and k 1 such that|L(t,)L(s,)|+|(t,)(s,)| K(|ts|+bardblbardbl2) K(|ts|+bardblbardbl) (t,),(s,) 0,TC. (8)and|L(t,)|+|(t,)| (1+bardblbardbl2)k, (t,) 0,TC. (9)Given the initial datum (t,) 0,T C, our objective is to find anoptimal stopping time TTt that maximizes the following expected per-formance index:J(;t,) Ebracketleftbiggintegraldisplay te(st)L(s,Xs)ds+e(t)(,X)bracketrightbigg, (10)9where 0 denotes a discount fac
