Viscosity Solution of Optimal Stopping Problem for Stochastic Systems with Bounded Memory

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1 Viscosiy Soluion of Opimal Sopping Problem for Sochasic Sysems wih Bounded Memory Mou-Hsiung Chang Tao Pang Mousapha Pemy April 5, 202 Absrac We consider a finie ime horizon opimal sopping problem for a sysem of sochasic funcional differenial equaions wih a bounded memory. Under some sufficienly smooh condiions, a Hamilon-Jacobi-Bellman (HJB) variaional inequaliy for he value funcion is derived via dynamical programming principle. I is shown ha he value funcion is he unique viscosiy soluion of he HJB variaional inequaliy. KEYWORDS: opimal sopping, sochasic conrol, sochasic funcional differenial equaions. AMS 2000 subjec classificaions: primary 60G40, 60G20; secondary 60H30, 93E20 The research of his paper is parially suppored by a gran W9NF-04-D-0003 from he U. S. Army Research Office Mahemaical Sciences Division, U. S. Army Research Office, P. O. Box 22, RTP, NC 27709, USA. Deparmen of Mahemaics, Norh Carolina Sae Universiy, Raleigh, NC 27695, USA. (Corresponding Auhor) Deparmen of Mahemaics, Towson Universiy, Towson, MD , USA.

2 Inroducion Opimal conrol and opimal sopping problems over a finie or an infinie ime horizon for Iô s diffusion processes arise in many areas of science, engineering, and finance (see e.g. Fleming and Soner 4, Øksendal 35, Shiryaev 39, Karazas and Shreve 9 and references conained herein). The value funcion of hese problems are normally expressed as a viscosiy or a generalized soluion of a Hamilon-Jacobi-Bellman equaion (HJBE) or a Hamilon-Jacobi-Bellman variaional inequaliy (HJBVI) ha involves a second order parabolic or ellipic parial differenial equaion in a finie dimensional Euclidean space (see e.g. Lions 30 and 32). In an aemp o achieve beer accuracy and o accoun for he delayed effec of he sae variables in he modelling of real world sochasic conrol problems, he sochasic delay equaions and conrolled sochasic delay equaions have been he subjec of inensive sudies in recen years by many researchers such as Elsanousi, Larrssen 25, Elsanousi e al 3, Øksendal and Sulem 37, Larrssen and Risebro 27, and Bauer and Rieder 5. The conrolled or unconrolled sochasic delay equaions considered by he aforemenioned researchers are described by he following special classes of equaions ha conain discree and averaged delays: ( dx(s) = α s, X(s), X(s r), or ( + β s, X(s), X(s r), 0 r 0 r ( 0 dx(s) = α s, X(s), X(s r), ( + β s, X(s), X(s r), ) e λθ X(s + θ)dθ, u(s) ds () ) e λθ X(s + θ)dθ, u(s) dw (s), s, T, r 0 r ) e λθ X(s + θ)dθ ds (2) ) e λθ X(s + θ)dθ dw (s), s, T. In he above equaions, W ( ) = {W (s), s 0} is an m-dimensional sandard Brownian moion defined on a cerain filered probabiliy space (Ω, F, P, F), u( ) = {u(s), s, T } is a conrol process aking values in he conrol se U in an Euclidean space, α and β are R n and R n m -valued funcions defined on or 0, T R n R n R n U, 0, T R n R n R n, and λ > 0 is a given consan. Based on he above wo equaions or heir varians, Larssen 25 obained an HJB equaion for an opimal conrol problem, Elsanousi e al 3 considered a singular conrol problem and obained cerain explicily available soluions, Øksendal and Sulem 37 derived he maximum principle for he opimal 2

3 sochasic conrol. If he dynamics of he conrol problem wih delay exhibi a special srucure, Larssen and Risebro 27 and Bauer and Rieder 5 showed ha he value funcion acually lives in a finie-dimensional space and he original problem can be reduced o a classical sochasic conrol problem wihou delay. Elsanousi and Larssen 2 reaed an opimal conrol problem and is applicaions o consumpion for () under parial observaion. We also menion ha opimal sopping problems were sudied in Elsanousi s unpublished disseraion for such special ype of equaions. This paper exends he resuls obained for finie dimensional diffusion processes and sochasic delay equaions described in (2) and invesigaes an opimal sopping problem over a finie ime horizon for a general sysem of sochasic funcional differenial equaions (SFDE) described below: dx(s) = f(s, X s )d + g(s, X s )dw (s), s, T, (3) where T > 0 and 0, T, respecively, denoe he erminal ime and an iniial ime of he opimal sopping problem. Again, W ( ) = {W (s), s 0} is a sandard m-dimensional Brownian moion, and he drif f(s, X s ) and he diffusion coefficien g(s, X s ) (aking values in R n and R n m, respecively) depend explicily on he segmen X s of he sae process X( ) = {X(s), s r, T } over he ime inerval s r, s, where X s : r, 0 R n is defined by X s (θ) = X(s + θ), θ r, 0. The consideraion of such a sysem enables us o model many real world problems ha have afereffecs (see e.g. Kolmanovsky and Shaikhe 24 and references conained herein for applicaion examples). I is clear ha his equaion also includes (2) as a special case and many oher equaions ha can no be modelled in he form of (2). When r = 0, i is also clear ha he SFDE (3) reduces o he following Iô s diffusion process (wihou delay): dx(s) = f(s, X(s))ds + g(s, X(s))dW (s), s, T. This paper reas a finie ime horizon opimal sopping problem (see Secion 2 for he problem saemen). We derive an infinie dimensional HJB variaional inequaliy (HJBVI) for he value funcion via a dynamic programming principle (see e.g. Larssen 25). I is shown ha he value funcion is he unique viscosiy soluion of he HJBVI. The proof of uniqueness involves embedding he funcion space W,2 (( r, 0); R n ) ino he Banach space C( r, 0; R n ) and exending he concep of viscosiy soluion for conrolled Iô s diffusion process developed by Crandall e al 9, and Lions 30 and 32 o an infinie dimensional seing. Alhough infinie dimensional HJBVIs for opimal sopping problems and heir applicaions o pricing of American opion have been sudied very recenly by a few researchers, hey eiher considered sochasic delay equaion of special form (2)(see e.g. Gapeev and Reiss 6 and 7) or sochasic equaions in Hilber spaces (see e.g. Gaarek and Świech 5 and Barbu and Marinelli 4). This paper differs from he aforemenioned papers in he following significan ways: i) The segmened soluion process {X s, s, T } is a srong Markov process in he Banach space C( r, 0; R n ) whose norm is no differeniable and is herefore more difficul o handle han any Hilber space considered in 3

4 5 and 4; ii) he infinie-dimensional HJBVI uniquely involves he exensions DV (, ψ) and D 2 V (, ψ) of firs and second order Fréche derivaives DV (, ψ) and D 2 V (, ψ) from C and C o (C B) and (C B) (see Subsecion 3. for definiions of hese spaces), respecively; and iii) he infinie-dimensional HJBVI also involves he infiniesimal generaor SV (, ψ) of he semigroup of shif operaors value funcions ha does no appear in he special class of equaions (2) in he aforemenioned papers. This paper is organized as follows. The noaion and preliminary resuls ha are needed for formulaing he opimal sopping problem as well as he problem saemen are conained in Secion 2. In Secion 3, he HJBVI for he value funcion is heurisically derived using Bellman s ype dynamic programming principle. The verificaion heorem is also proved here. In Secion 4, he coninuiy of he value funcion is proved. Alhough coninuous, he value funcion is no known o be smooh enough o be a classical soluion of he HJBVI in general cases. I is shown in Secion 4, however, ha he value funcion is he unique viscosiy soluion of he HJBVI. 2 The Opimal Sopping Problem Le r > 0 be a fixed consan, and le J = r, 0 denoe he duraion of he bounded memory of he sochasic funcional differenial equaions considered in his paper. For he sake of simpliciy, denoe C(J; R n ), he space of coninuous funcions φ : J R n, by C. Noe ha C is a real separable Banach space under he sup-norm defined by φ = sup φ(θ), θ J φ C where is he Euclidean norm in R n. In addiion o he space C, we also consider L 2 (J; R n ), he Hilber space of Lebesque squared-inegrable funcions φ : J R n equipped wih he inner produc ( ) and he Hilberian norm 2 defined by (φ ψ) = 0 r φ(θ), ψ(θ) dθ and φ 2 = (φ φ) 2, φ, ψ L 2 (J, R n ), where, he inner produc in R n. Noe ha he space C can be coninuously embedded ino L 2 (J; R n ) (see e.g. Rudin 38). In fac, i is easy o verify ha φ 2 r φ, φ C. Convenion 2. Throughou he res of he paper, le T > 0 denoe he erminal ime and 0, T be an iniial ime of he opimal sopping problem. We shall use he following convenional noaion for funcional differenial equaions (see Hale 8): If ψ C( r, T ; R n ) and s, T, le ψ s C be defined by ψ s (θ) = ψ(s + θ), θ J. 4

5 In addiion, hroughou he res of he paper, we will use K and Λ o denoe generic consans and heir values may change from line o line. Le {W (s), s 0} be an m-dimensional sandard Brownian moion defined on a complee filered probabiliy space (Ω, F, P ; F), where F denoes he smalles σ-field ha conains all subses of Ω, and F = {F(s), s 0} is he P -augmenaion of he naural filraion {F W (s), s 0} generaed by he Brownian moion {W (s), s 0}, i.e., if s 0, and F W (s) = σ{w (), 0 s} F(s) = F W (s) {A Ω B F such ha A B and P (B) = 0 }, where he operaor denoes ha F(s) is he smalles σ-algebra such ha F W (s) F(s) and {A Ω B F such ha A B and P (B) = 0 } F(s). Consider he following sysem of sochasic funcional differenial equaions: dx(s) = f(s, X s )ds + g(s, X s )dw (s), s, T ; (4) wih he iniial funcion X = ψ, where ψ is a given C-valued random variable ha is F()-measurable. Here, f : 0, T C R n and g : 0, T C R n m are given deerminisic funcionals. Le L 2 (Ω, C) be he space of C-valued random variables Ξ : Ω C such ha ( ) Ξ L 2 = Ξ(ω) 2 2 dp (ω) <. Ω Le L 2 (Ω, C; F()) be he space of hose Ξ L 2 (Ω, C) which are F()-measurable. Definiion 2.2 A process {X(s;, ψ ), s r, T } is said o be a (srong) soluion of (4) on he inerval r, T and hrough he iniial daum (, ψ ) 0, T L 2 (Ω, C; F()) if i saisfies he following condiions:. X ( ;, ψ ) = ψ ; 2. X(s;, ψ ) is F(s)-measurable for each s, T ; 3. The process {X(s;, ψ ), s, T } is coninuous and i saisfies he following sochasic inegral equaion P -a.s. X(s) = ψ (0) + s f(λ, X λ )dλ + s g(λ, X λ )dw (λ), s, T. (5) 5

6 In addiion, he (srong) soluion process {X(s;, ψ ), s r, T } of (4) is said o be (srongly) unique if { X(s;, ψ ), s r, T } is also a (srong) soluion of (4) on r, T and hrough he same iniial daum (, ψ ), hen P {X(s;, ψ ) = X(s;, ψ ), s, T } =. Throughou he res of he paper, we assume ha he funcionals f : 0, T C R n, and g : 0, T C R n m are coninuous funcionals and hey saisfy he following condiions (Assumpion 2.3 & 2.4) o ensure he exisence and uniqueness of a (srong) soluion {X(s;, ψ ), s r, T } for each (, ψ ) 0, T L 2 (Ω, C; F()). (See Mohammed 33, 34.) Assumpion 2.3 There exiss a consan K > 0 such ha f(, ϕ) f(s, φ) + g(, ϕ) g(s, φ) K( s + ϕ φ ) (, ϕ), (s, φ) 0, T C. Assumpion 2.4 There exiss a consan K > 0 such ha f(, φ) + g(, φ) K( + φ ) (, φ) 0, T C. Le {X(s;, ψ ), s, T } be he soluion of (4) hrough he iniial daum (, ψ ) 0, T C. We consider he corresponding C-valued process {X s (, ψ ), s, T } defined by X s (θ;, ψ ) X(s + θ;, ψ ), θ J. (6) For each 0, T, define G() s,t G(, s) where G(, s) is defined by G(, s) = σ(x(u;, ψ ), u s). Noe ha, i can be shown ha for each s, T, G(, s) = σ(x u (, ψ ), u s). This is due o he sample pah s coninuiy of he process {X(s;, ψ ), s, T }. One can hen esablish he following Markov propery (see Mohammed 33, 34). Theorem 2.5 Le Assumpions 2.3 and 2.4 hold. Then he corresponding C- valued soluion process of (4) describes a C-valued Markov process in he following sense: For any (, ψ ) 0, T L 2 (Ω, C), he Markovian propery P {X s (, ψ ) B G(, α)} = P {X s (, ψ ) B X α (, ψ )} p(α, X α (, ψ ), s, B) holds a.s. for α s and B B(C), where B(C) is he Borel σ-algebra of subses of C. 6

7 In he above, he funcion p :, T C, T B(C) 0, denoes he ransiion probabiliies of he C-valued Markov process {X s (, ψ ), s, T }. A random funcion τ : Ω 0, is said o be a G()-sopping ime if {τ s} G(, s), s. Le T be he collecion of all G()-sopping imes, and le T T be he collecion of all G()-sopping imes τ T such ha τ T a.s.. For each τ T T, le he sub-σ-algebra G(, τ) of F be defined by G(, τ) = {A F A { τ s} G(, s) s, T }. Wih a lile bi more effor, one can show ha he corresponding C-valued soluion process of (4) is also a srong Markov process in C. Tha is P {X s (, ψ ) B G(, τ)} = P {X s (, ψ ) B X τ (, ψ )} p(τ, X τ (, ψ ), s, B) holds a.s. for all τ T T, all deerminisic s τ, T, and B B(C). If he drif f and he diffusion coefficien g are ime-independen, i.e., f(s, φ) f(φ) and g(s, φ) g(φ), hen (4) reduces o he following auonomous sysem: dx(s) = f(x s )ds + g(x s )dw (s). (7) In his case, we usually assume he iniial daum (, ψ ) = (0, ψ) and denoe he soluion process of (7) hrough (0, ψ) and on he inerval r, T by {X(s; ψ), s r, T }. Then he corresponding C-valued soluion process {X s (ψ), s r, T } of (7) is a srong Markov process wih ime-homogeneous probabiliy ransiion funcion p(ψ, s, B) p(0, ψ, s, B) = p(, ψ, + s, B) for all s, 0, ψ C, and B B(C). Assume L and Ψ are wo 2 -Lipschiz coninuous real-valued funcionals on 0, T C wih a mos polynomial growh in L 2 (J; R n ). In oher words, here exis posiive consans K, Λ, and k such ha and L(, ψ) L(s, φ) + Ψ(, ψ) Ψ(s, φ) K( s + ψ φ 2 ) K( s + ψ φ ) (, ψ), (s, φ) 0, T C. (8) L(, φ) + Ψ(, φ) Λ( + φ 2 ) k, (, φ) 0, T C. (9) Given he iniial daum (, ψ) 0, T C, our objecive is o find an opimal sopping ime τ T T ha maximizes he following expeced performance index: τ J(τ;, ψ) E e ρ(s ) L(s, X s )ds + e ρ(τ ) Ψ(τ, X τ ), (0) where ρ > 0 denoes a discoun facor. In his case, he value funcion V : 0, T C R is defined o be V (, ψ) sup J(τ;, ψ). () τ T T 7

8 For he auonomous case, i.e., dx(s) = f(x s )d + σ(x s )dw (s), s, T, (2) he following opimal sopping problem is a special case of wha will be reaed in his paper: Find an opimal sopping ime τ T0 T ha maximizes he following expeced performance index: τ J(τ; ψ) E e ρs L(X s )ds + e ρτ Ψ(X τ ). (3) 0 In his case, he value funcion V : C R is defined o be 3 HJB Variaional Inequaliy 3. The Infiniesimal Generaor V (ψ) sup J(τ; ψ). (4) τ T0 T Le C and C be he space of bounded linear funcionals Φ : C R and bounded bilinear funcionals Φ : C C R, of he space C, respecively. They are equipped wih he operaor norms which will be, respecively, denoed by and. Le B = {v {0}, v R n }, where {0} : r, 0 R is defined by { 0 for θ r, 0), {0} (θ) = for θ = 0. We form he direc sum C B = {φ + v {0} φ C, v R n } and equip i wih he norm, also denoed by, defined by φ + v {0} = sup φ(θ) + v, φ C, v R n. θ r,0 Again, le (C B) and (C B) be spaces of bounded linear and bilinear funcionals of C B, respecively. The following wo resuls can be found in Lemma 3. and Lemma 3.2 on pp of Mohammed 33. If Γ C, hen Γ has a unique (coninuous) bilinear exension Γ : (C B) R saisfying he following weak coninuiy propery: (W) If {ξ (k) } k= is a bounded sequence in C such ha ξ(k) (θ) ξ(θ) as k for all θ J for some ξ C B, hen Γ(ξ (k) ) Γ(ξ) as k. The exension map C (C B), Γ Γ is a linear isomeric injecive map. 8

9 If Γ C, hen Γ has a unique (coninuous) linear exension Γ (C B) saisfying he following weak coninuiy propery: (W2) If {ξ (k) } k= and {ζ(k) } k= are bounded sequences in C such ha ξ (k) (θ) ξ(θ) and ζ (k) (θ) ζ(θ) as k for all θ J for some ξ, ζ C B, hen Γ(ξ (k), ζ (k) ) Γ(ξ, ζ) as k. For a sufficienly smooh funcional Φ : C R, we can define is Fréche derivaives wih respec o φ C. (For more deails abou Fréche derivaives, please refer o 2.) From he resuls saed above, we know ha is firs order Fréche derivaive, DΦ(φ) C, has a unique and (coninuous) linear exension DΦ(φ) (C B). Similarly, is second order Fréche derivaive, D 2 Φ(φ) C, has a unique and coninuous linear exension D 2 Φ(φ) (C B). For a Borel measurable funcion Φ : C R, we also define S(Φ)(φ) lim Φ( h 0+ h φ h ) Φ(φ) for all φ C, where φ : r, T R n is an exension of φ : r, 0 R n defined by φ() = { φ() if r, 0) φ(0) if 0, and again φ C is defined by φ (θ) = φ( + θ), θ r, 0. (5) Le ˆD(S), he domain of he operaor S, be he se of Φ : C R such ha he above limi exiss for each φ C. Define D(S) as he se of all funcionals Ψ : 0, T C R such ha Ψ(, ) ˆD(S), 0, T. Throughou he end, le C,2 lip (0, T C) be he space of funcions Φ : 0, T C R which saisfies he following condiions:. Φ : 0, T C R, DΦ : 0, T C C and D 2 Φ : 0, T C C exis and are coninuous. 2. Is second order Fréche derivaive D 2 Φ saisfies he following global Lipschiz condiion: D 2 Φ(, φ) D 2 Φ(, ϕ) K φ ϕ 0, T, φ, ϕ C. The following example shows ha ˆD(S) is no a subse of Clip 2 (C). Therefore, i is no redundan o require ha a funcion Φ C,2 lip (0, T C) D(S) for deriving he infinie-dimensional HJB variaional inequaliy. Example. Assume n =. Le Φ : C R be independen of he ime variable and be defined by ( Φ(φ) = φ r ), φ C. 2 9

10 Then i is clear ha Φ C 2 (C) bu Φ / ˆD(S), since ϕ C, ( DΦ(φ)(ϕ) = ϕ r ), 2 D 2 Φ(φ)(ϕ, ϕ) = 0, and D 2 Φ is globally Lipschiz. However, S(Φ)(φ) = lim Φ( 0 + φ ) Φ(φ) = lim 0 + φ ( ) ( ) r 2 φ r 2 which exiss only if φ has a righ derivaive a r 2. Le Φ : 0, T C R be a Borel measurable funcion and consider he following wo smoohness condiions: Smoohness Condiion (i). Φ C,2 lip (0, T C). Smoohness Condiion (ii). Φ D(S), i.e., Φ(, ) ˆD(S) for each 0, T. The following resul will be used laer on in his paper. Theorem 3. (Mohammed 33, 34) Suppose ha Φ C(0, T C) saisfies Smoohness condiions (i) and (ii). Le {X s, s, T } be he C-valued Markov soluion process of Equaion (4) wih he iniial daa (, ϕ ) 0, T C. Then EΦ( + ɛ, X +ɛ ) Φ(, ϕ ) lim ɛ 0 ɛ = Φ(, ϕ ) + S(Φ)(, ϕ ) + DΦ(, ϕ )(f(, ϕ ) {0} ) + 2 m D 2 Φ(, ϕ )(g(, ϕ )(e j ) {0}, g(, ϕ )(e j ) {0} ), (6) j= where e j, j =, 2,, m, is he jh vecor of he sandard basis in R m. 3.2 Heurisic Derivaion We consider he following HJBVI: { max Ψ V, V on 0, T C, where } + AV + L ρv = 0 (7) AV (, ψ) S(V )(, ψ) + DV (, ψ)(f(, ψ) {0} ) + m D 2 2 V (, ψ)(g(, ψ)e i {0}, g(, ψ)e i {0} ). (8) i= 0

11 The above inequaliy shall be inerpreed as follows. and V + AV + L ρv 0 and V Ψ (9) ( V ) + AV + L ρv (V Ψ) = 0 (20) on 0, T C, where A is defined by (8). We heurisically derive he above variaional inequaliy as follows. A rigorous derivaion will be provided as a byproduc when we prove ha he value funcion is a viscosiy soluion of he HJBVI in he nex secion. Firs, we prove ha V (, ψ) Ψ(, ψ) for all (, ψ) 0, T C. Le τ T T. If τ =, hen by (0), we can ge J( τ;, ψ) = Ψ(, ψ). Therefore, V (, ψ) = sup J(τ;, ψ) J( τ;, ψ) = Ψ(, ψ). (2) τ T T The following dynamic programming principle (see Larssen 25) will be used o derive our HJBVI: +δ V (, ψ) E e ρ(s ) L(s, X s )ds + e ρδ V ( + δ, X +δ ), δ 0. (22) From his principle and Theorem 3., we have Ee ρδ V ( + δ, X +δ ) V (, ψ) lim δ 0 δ Ee ρδ V ( + δ, X +δ ) V ( + δ, X +δ ) = lim δ 0 δ EV ( + δ, X +δ ) V (, ψ) + lim δ 0 δ = ρv (, ψ) + V (, ψ) + AV (, ψ) L(, ψ), for all (, ψ) 0, T C provided ha V C,2 lip (0, T C) D(S). Tha is, V From (2) and (23), i follows ha max{ψ V, V + AV + L ρv 0. (23) + AV + L ρv } 0 (24)

12 on 0, T C. The derivaion of he inequaliy max{ψ V, V can be found in he nex secion. We herefore have he following resul. + AV + L ρv } 0 (25) Theorem 3.2 Suppose V : 0, T C is he value funcion defined by () and saisfies Smoohness Condiions (i) and (ii). Then he value funcion V saisfies he following HJB variaional inequaliy: { max Ψ V, V } + AV + L ρv = 0 (26) on 0, T C, and V (T, ψ) = Ψ(ψ), ψ C. Noe ha i is no known ha he value funcion V saisfies he Smoohness Condiions (i) and (ii). Therefore, we need o consider viscosiy soluions insead of classical soluions for HJB variaional inequaliy (26). In fac, i will be shown ha he value funcion is a unique viscosiy soluion of he HJB variaional inequaliy (26). These resuls shall be given in he nex secion. 4 Viscosiy Soluion In his secion, we shall show ha he value funcion V defined by () is acually he unique viscosiy soluion of he HJB variaional inequaliy (26). Firs, le us define he viscosiy soluion of (26) as follows. Definiion 4. Le w C(0, T C). We say ha w is a viscosiy subsoluion of (26) if for every Γ : 0, T C R saisfying Smoohness Condiions (i)-(ii) on 0, T C, we have { min Γ(, ψ) Ψ(, ψ), ργ(, ψ) Γ } (, ψ) AΓ(, ψ) L(, ψ) 0 a every (, ψ) 0, T C which is a local maximum of w Γ, wih Γ(, ψ) = w(, ψ). We say ha w is a viscosiy supersoluion of (26) if, for every Γ : 0, T C R saisfying Smoohness Condiions (i)-(ii) on 0, T C, we have { min Γ(, ψ) Ψ(, ψ), ργ(, ψ) Γ } (, ψ) AΓ(, ψ) L(, ψ) 0. a every (, ψ) 0, T C which is a local minimum of w Γ, wih Γ(, ψ) = w(, ψ). We say ha w is a viscosiy soluion of (26) if i is a viscosiy supersoluion and a viscosiy subsoluion of (26). 2

13 As we can see in he definiion, a viscosiy soluion mus be coninuous. So firs we will show ha he value funcion V defined by () has his propery. Acually, we have he following resul: Lemma 4.2 The value funcion V : 0, T C R defined in () is coninuous and here exis consans K > 0 and k such ha, for every (, ψ) 0, T C, we have V (, ψ) K( + ψ 2 ) k. (27) Proof. I is clear ha V has a mos polynomial growh, since L and Φ have a mos polynomial growh wih he same k as in (9). Le Ξ(s) = X s (, ψ), s, T, be he C-valued soluion of (4) wih iniial daa (, ψ) 0, T C. In view of Remark (v) following he proof of Theorem I.2 and in he proof of Theorem III. pp. 33 of Mohammed 34, he rajecory map (, ψ) X s (, ψ) from 0, T C o L 2 (Ω, C) is globally Lipschiz in ψ uniformly wih respec o on compac ses, and coninuous in for fixed ψ. Therefore, given wo C-valued soluions Ξ (s) = X s (, ψ ) and Ξ 2 (s) = X s (, ψ 2 ), s, T of (4) wih iniial daa (, ψ ) and (, ψ 2 ), respecively, we have E Ξ (s) Ξ 2 (s) K ψ ψ 2 2, (28) where K is a posiive consan ha depends on he Lipschiz consan in Assumpions 2.3 and T. Using he Lipshiz coninuiy of L, Ψ : 0, T C R, here exiss ye anoher consan Λ > 0 such ha, J(τ;, ψ ) J(τ;, ψ 2 ) ΛE Ξ (τ) Ξ 2 (τ). (29) Therefore, using (29) and (28), we have V (, ψ ) V (, ψ 2 ) sup J(τ;, ψ ) J(τ;, ψ 2 ) τ T T Λ sup E Ξ (τ) Ξ 2 (τ) τ T T Λ ψ ψ 2 2. (30) This implies he (uniform) coninuiy of V (, ψ) wih respec o ψ. We nex show he coninuiy of V (, ψ) wih respec o. Le Ξ (s) = X s (, ψ), s, T and Ξ 2 (s) = X s ( 2, ψ), s 2, T, be wo C-valued soluions of (4) wih iniial daa (, ψ) and ( 2, ψ), respecively. Wihou loss of generaliy, we assume ha < 2. Then we can ge J(τ;, ψ) J(τ; 2, ψ) 3

14 2 = E + τ 2 e ρ(ξ ) L(ξ, Ξ (ξ))dξ e ρ(ξ 2) L(ξ, Ξ (ξ)) L(ξ, Ξ 2 (ξ))dξ + e ρ(τ ) Ψ(τ, Ξ (τ)) e ρ(τ 2) Ψ(τ, Ξ 2 (τ)). (3) Therefore, here exiss a consan Λ > 0 such ha J(τ;, ψ) J(τ; 2, ψ) ( ) Λ 2 E Ξ (τ) + E Ξ (τ) Ξ 2 (τ). (32) Le ε > 0 be any give small consan. Using he compacness of 0, T and he uniform coninuiy of he rajecory map in, we know ha here exiss a consan η > 0 such ha if 2 < η hen E Ξ (s) Ξ 2 (s) ε 2Λ. In addiion, here exiss a consan K > 0 such ha E Then for 2 < min { η, sup Ξ (s) s,t ε 2ΛK }, we have K 0, T. J(τ;, ψ) J(τ; 2, ψ) ε 2 + ε 2 = ε. Consequenly, This complees he proof. V (, ψ) V ( 2, ψ) ε. Before we show ha he value funcion is a viscosiy soluion of he HJB variaional inequaliy (26), we need o prove some resuls relaed o he dynamic programming principle (22) which can also be found in 25. The resuls are given nex in Lemma 4.3 and Lemma 4.5. Lemma 4.3 Le α, β T T be G()-sopping imes and > 0 such ha α β a.s.. Then we have E e ρ(α ) V (α, X α ) β E e ρ(ξ ) L(ξ, X ξ )dξ α + E e ρ(β ) V (β, X β ). (33) Proof. By virue of (0) and (), we can ge β E e ρ(β ) V (β, X β ) + e ρ(ξ ) L(ξ, X ξ )dξ α 4

15 τ = sup E e ρ(ξ β) e ρ(β ) L(ξ, X ξ )dξ + e ρ(τ β) e ρ(β ) Ψ(τ, X τ ) τ Tβ T β β + E e ρ(ξ ) L(ξ, X ξ )dξ α τ = sup E e ρ(ξ ) L(ξ, X ξ )dξ + e ρ(τ ) Ψ(τ, X τ ) τ Tβ T β β + E e ρ(ξ ) L(ξ, X ξ )dξ α τ = sup τ T T β E sup E τ Tα T α τ = Ee ρ(α ) V (α, X α ). α e ρ(ξ ) L(ξ, X ξ )dξ + e ρ(τ ) Ψ(τ, X τ ) e ρ(α ) e ρ(ξ α) L(ξ, X ξ )dξ + e ρ(α ) e ρ(τ α) Ψ(τ, X τ ) This complees he proof. Now le us give he definiion of ɛ-opimal sopping ime which will be used in he nex lemma. Definiion 4.4 For each ɛ > 0, a G()-sopping ime τ ɛ T T is said o be ɛ-opimal if τɛ 0 V (, ψ) E e ρ(ξ ) L(ξ, X ξ )dξ + e ρ(τɛ ) V (τ ɛ, X τɛ ) ɛ. (34) Lemma 4.5 Le θ be a sopping ime such ha θ τ ɛ a.s., for any ɛ > 0, where τ ɛ T T is ɛ-opimal. Then, θ V (, ψ) = E e ρ(ξ ) L(ξ, X ξ )dξ + e ρ(θ ) V (θ, X θ ). (35) Proof. Le θ be a sopping ime such ha θ τ ɛ a.s., for any ɛ-opimal τ ɛ T T. Using Lemma 4.3, we have τɛ Ee ρ(θ ) V (θ, X θ ) E e ρ(ξ ) L(ξ, X ξ )dξ θ + Ee ρ(τɛ ) V (τ ɛ, X τɛ ). (36) This implies ha θ Ee ρ(θ ) V (θ, X θ ) + E e ρ(ξ ) L(ξ, X ξ )dξ τɛ E e ρ(ξ ) L(ξ, X ξ )dξ 5 + Ee ρ(τɛ ) V (τ ɛ, X τɛ ). (37)

16 Noe ha τ ɛ is he ɛ-opimal sopping ime, hen we can ge τɛ 0 V (, ψ) E e ρ(ξ ) L(ξ, X ξ )dξ + e ρ(τɛ ) V (τ ɛ, X τɛ ) ɛ. (38) On he oher hand, by virue of (37), we can ge Thus, we can ge V (, ψ) E e ρ(θ ) V (θ, X θ ) + V (, ψ) E e ρ(τɛ ) V (τ ɛ, X τɛ ) + θ e ρ(ξ ) L(ξ, X ξ )dξ e ρ(ξ ) L(ξ, X ξ )dξ. (39) θ 0 V (, ψ) E e ρ(ξ ) L(ξ, X ξ )dξ + e ρ(θ ) V (θ, X θ ) ɛ. (40) τɛ Now we le ɛ 0 in he above inequaliy and we can ge θ V (, ψ) = E e ρ(ξ ) L(ξ, X ξ )dξ + Ee ρ(θ ) V (θ, X θ ). This complees he proof. Now we are ready o show ha he value funcion V defined by () is a viscosiy soluion of he HJBVI (26). Theorem 4.6 The value funcion V is a viscosiy soluion of he HJB variaional inequaliy (26). Proof. We need o prove ha V is boh a viscosiy supersoluion and a viscosiy subsoluion of (26). Firs we prove ha V is a viscosiy supersoluion. Le (, ψ) 0, T C and Γ C,2 lip (0, T C) D(S) saisfying Γ V on he neighborhood N(, ψ) of (, ψ) wih Γ(, ψ) = V (, ψ), we wan o prove he he viscosiy supersoluion inequaliy, i.e., { min Γ(, ψ) Ψ(, ψ), ργ(, ψ) Γ } (, ψ) AΓ(, ψ) L(, ψ) 0. (4) We know ha V Ψ on N(, ψ) and Γ(, ψ) = V (, ψ), so we have Therefore, we jus need o prove ha Γ(, ψ) Ψ(, ψ) 0. ργ(, ψ) Γ (, ψ) AΓ(, ψ) L(, ψ) 0. 6

17 Since Γ C,2 lip (0, T C) D(S), by virue of Theorem 3. pp. 78 of Mohammed 33, for s T, we have Ee ρ(s ) Γ(s, X s ) Γ(, ψ) s ( ) Γ(ξ, = E e ρ(ξ ) Xξ ) + AΓ(ξ, X ξ ) ργ(ξ, X ξ ) dξ. (42) ξ For any s, T such ha (s, X s ) N(, ψ), from Lemma 4.3, we can ge s V (, ψ) E e ρ(ξ ) L(ξ, X ξ )dξ + E e ρ(s ) V (s, X s ). (43) By virue of (42), Γ V and V (, ψ) = Γ(, ψ), we can ge s 0 E e ρ(ξ ) L(ξ, X ξ )dξ + E e ρ(s ) V (s, X s ) V (, ψ) s E e ρ(ξ ) L(ξ, X ξ )dξ + E e ρ(s ) Γ(s, X s ) Γ(, ψ) s ( E e ρ(ξ ) L(ξ, X ξ ) + Γ(ξ, X ξ) + AΓ(ξ, X ξ ) ξ ) ργ(ξ, X ξ ) dξ. (44) Dividing boh sides of he above inequaliy by (s ), we have s ( 0 E e ρ(ξ ) L(ξ, X ξ ) + Γ(ξ, X ξ) + AΓ(ξ, X ξ ) s ξ ) ργ(ξ, X ξ ) dξ. (45) Now le s in (45), and we obain Γ (, ψ) + AΓ(, ψ) + L(, ψ) ργ(, ψ) 0. (46) which proves he inequaliy (4). Nex we wan o prove ha V is also a viscosiy subsoluion of (26). Le (, ψ) 0, T C and Γ C,2 lip (0, T C) D(S) saisfying Γ V on he neighborhood N(, ψ) of (, ψ) wih Γ(, ψ) = V (, ψ), we wan o prove ha { min Γ(, ψ) Ψ(, ψ), ργ(, ψ) Γ } (, ψ) AΓ(, ψ) L(, ψ) 0. (47) I is easy o ge ha Γ(, ψ) = V (, ψ) Ψ(, ψ). 7

18 Therefore, we need o show ha ργ(, ψ) Γ (, ψ) AΓ(, ψ) L(, ψ) 0. (48) Le θ T T be a sopping ime such ha θ τ ɛ for every τ ɛ, ɛ-opimal sopping ime. Using Lemma 4.5, we can ge θ V (, ψ) = E e ρ(ξ ) L(ξ, X ξ )dξ + E e ρ(θ ) V (θ, X θ ). (49) Using he Dynkin s formula (see Mohammed 33), we have E e ρ(θ ) Γ(θ, X θ ) Γ(, ψ) θ ( ) Γ(ξ, = E e ρ(ξ ) Xξ ) + AΓ(ξ, X ξ ) ργ(ξ, X ξ ) dξ. ξ Since Γ V on N(, ψ) and Γ(, ψ) = V (, ψ), for all θ such ha (θ, X θ ) N(, ψ), we can ge E e ρ(θ ) V (θ, X θ ) V (, ψ) E e ρ(θ ) Γ(θ, X θ ) Γ(, ψ) θ ( ) Γ(ξ, = E e ρ(ξ ) Xξ ) + AΓ(ξ, X ξ ) ργ(ξ, X ξ ) dξ ξ Combining his wih (49), he above inequaliy implies θ ( 0 E e ρ(ξ ) L(ξ, X ξ ) + Γ(ξ, X ) ξ) + AΓ(ξ, X ξ ) ργ(ξ, X ξ ) dξ. (50) ξ Dividing (50) by E(θ ) and sending Eθ, we deduce Γ (, ψ) + AΓ(, ψ) + L(, ψ) ργ(, ψ) 0, (5) which proves (48). Therefore, V is also a viscosiy subsoluion. This complees he proof of he heorem. In order o prove he uniqueness we need following resuls. We will use he nex resul proven in Ekeland and Lebourg 0 and also in a general form in Segall 4 and Bourgain 6. The reader is also referred o Crandall e al 9 and Lions 3 for an applicaion example of his resul in a seing similar o ours. Lemma 4.7 Le Φ be a bounded above and upper semiconinuous real-valued funcion on a closed ball B of a Banach space X which has he Radon-Nikodym propery. Then for any ɛ > 0 here exiss an elemen u X wih norm a mos ɛ, where X is he dual of X, such ha Φ + u aains is maximum on B. 8

19 Noe ha every separable Hilber space (X, X ) saisfies he Radon- Nikodym propery (see e.g. Ekeland and Lebourg 0). In order o apply Lemma 4.7, we shall herefore resric ourself o a subspace X of C = C( r, 0; R n ) which is a separable Hilber space and dense in C. One of he good candidaes is he Sobolev space W,2 (( r, 0); R n ), where W,2 (( r, 0); R n ) = {φ L 2 ( r, 0; R n ); φ,2 < }, where φ 2,2 = φ φ 2 2, wih φ being he firs derivaive in he sense of disribuion of φ. From he Sobolev embedding heorems (see e.g. Adams ), i is known ha W,2 (( r, 0); R n ) C and ha W,2 (( r, 0); R n ) is dense in C. For more abou Sobolev spaces and corresponding resuls, one can refer o Adams. Theorem 4.8 (Comparison Principle) Assume ha V (, ψ) and V 2 (, ψ) are boh coninuous wih respec o he argumen (, ψ) 0, T C and are respecively viscosiy subsoluion and supersoluion of (26) wih a mos a polynomial growh, i.e., here exis consans Λ > 0 and k such ha, V i (, ψ) Λ( + ψ 2 ) k, for (, ψ) 0, T C, i =, 2. Then, on every closed ball B of W,2 (( r, 0); R n ), we have V (, ψ) V 2 (, ψ) for all (, ψ) 0, T B. (52) Before we give he proof of he above heorem, we firs need o do some preparaion works. Le V and V 2 be, respecively, a viscosiy subsoluion and supersoluion of (26). For any 0 < δ, γ < and for all ψ, φ W,2 (( r, 0); R n ) and, s 0, T, define Θ δγ (, s, ψ, φ) ψ φ 22 + ψ 0 φ s 2 δ and + γ exp( + ψ ψ 0 2 2) + exp( + φ φ 0 2 2), (53) Φ δγ (, s, ψ, φ) V (, ψ) V 2 (s, φ) Θ δγ (, s, ψ, φ), (54) where ψ 0, φ 0 W,2 (( r, 0); R n ) wih ψ 0 (θ) = θ r ψ( r θ), φ0 (θ) = θ φ( r θ), θ r, 0. r Moreover, using he polynomial growh condiion for V and V 2, we have lim Φ δγ(, s, ψ, φ) =. (55) ψ 2+ φ 2 9

20 The funcion Φ δγ is a real-valued funcion ha is bounded above and coninuous on 0, T 0, T W,2 (( r, 0); R n ) W,2 (( r, 0); R n ), since he he embedding of W,2 (( r, 0); R n ) in C is coninuous. Therefore, from Lemma 4.7 (which is applicable by virue of (55)), for any > ɛ > 0, here exis a coninuous linear funcional T ɛ in he opological dual of W,2 (( r, 0); R n ) W,2 (( r, 0); R n ), wih norm a mos ɛ, such ha he funcion Φ δγ + T ɛ aains i maximum in 0, T 0, T B B, for any closed ball B of W,2 (( r, 0); R n ). (see Lemma 4.7). Le B be a closed ball of W,2 (( r, 0); R n ) cenered a 0. Denoe by ( δγɛ, s δγɛ, ψ δγɛ, φ δγɛ ) he maximum of Φ δγ + T ɛ on 0, T 0, T B B. Wihou loss of generaliy, we assume ha for any given δ, γ and ɛ, here exiss a consan M δγɛ such ha he maximum value Φ δγ + T ɛ + M δγɛ is zero. In oher words, we have Φ δγ ( δγɛ, s δγɛ, ψ δγɛ, φ δγɛ ) + T ɛ (ψ δγɛ, φ δγɛ ) + M δγɛ = 0. (56) We have he following lemmas. Lemma 4.9 Le B be a closed ball of W,2 (( r, 0); R n ) cenered a 0, and le ( δγɛ, s δγɛ, ψ δγɛ, φ δγɛ ) be he maximum of Φ δγ +T ɛ +M δγɛ on 0, T 0, T B B for some δ, γ, ɛ (0, ). Then, we have lim ɛ 0,δ 0 ( ψ δγɛ φ δγɛ ψ 0 δγɛ φ 0 δγɛ δγɛ s δγɛ 2) = 0, (57) Proof. have Le r B be he radius of he ball B. Then, for all δ, γ, ɛ (0, ), we ψ δγɛ 2 2 ψ δγɛ 2,2 < rb, 2 and φ δγɛ 2 2 φ δγɛ 2 2,2 < rb. 2 Noing ha ( δγɛ, s δγɛ, ψ δγɛ, φ δγɛ ) is he maximum of Φ δγ + T ɛ + M δγɛ, we ge Φ δγ ( δγɛ, δγɛ, ψ δγɛ, ψ δγɛ ) + T ɛ (ψ δγɛ, ψ δγɛ ) + Φ δγ (s δγɛ, s δγɛ, φ δγɛ, φ δγɛ ) + T ɛ (φ δγɛ, φ δγɛ ) 2Φ δγ ( δγɛ, s δγɛ, ψ δγɛ, φ δγɛ ) + 2T ɛ (ψ δγɛ, φ δγɛ ). (58) I implies V ( δγɛ, ψ δγɛ ) V 2 ( δγɛ, ψ δγɛ ) 2γ(exp( + ψ δγɛ ψ 0 δγɛ 2 2)) + T ɛ (ψ δγɛ, ψ δγɛ ) + V (s δγɛ, φ δγɛ ) V 2 (s δγɛ, φ δγɛ ) 2γ(exp( + φ δγɛ φ 0 δγɛ 2 2)) + T ɛ (φ δγɛ, φ δγɛ ) 2V ( δγɛ, ψ δγɛ ) 2V 2 (s δγɛ, φ δγɛ ) 2 ψ δγɛ φ δγɛ ψδγɛ 0 φ 0 δ δγɛ 2 2 ( + δγɛ s δγɛ 2 2γ exp( + ψ δγɛ ψδγɛ 0 2 2) ) + exp( + φ δγɛ φ 0 δγɛ 2 2) + 2T ɛ (ψ δγɛ, φ δγɛ ). (59) 20

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