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Brownian Motion on the Sierpinski Gasket and a Stochastic Maximum Principle XuanLiu ∗ 7 Abstract 1 0 Inthispaper,weestablishamartingalerepresentationtheoremfortheBrownianfiltration 2 ontheinfiniteSierpinskigasket,andtheexponentialintegrabilityofthequadraticprocessof the representing martingale. We prove that the martingale dimension of the infinite Sierp- n a inskigasketisone. Asa consequence,the existenceanduniquenessofsolutionsofa class J of backward stochastic differentialequations(BSDEs) on the infinite Sierpinskigasket are 5 obtained. UsingBSDEs,wefurtherestablishastochasticmaximumprincipleforstochastic 2 control problems on both the finite and the infinite Sierpinski gaskets, which has different behaviour from its counterpart on Euclidean spaces due to the singularity between certain ] R measures. P . h 1 Introduction t a m Recently, tostudy non-linear analysis onthefinite (orcompact) Sierpinski gasket, X.LiuandZ. [ Qian [8] developed a theory of backward stochastic differential equations (BSDEs)on the finite 2 Sierpinski gasket. BSDEsand related stochastic analysis on fractals, though initially considered v asefficient toolstotreat quasi-linear parabolic PDEsonfractals, alsohaveinterests ontheirown 3 from a mathematical finance point of view. Several interesting mathematical finance problems 6 5 are formulated as stochastic control problems on Euclidean spaces, which are based upon the 2 assumption that uncertainties in financial models are sourced from Brownian filtration on Euc- 0 . lidean spaces. However, ithadbeen widelyobserved from the realdata that manyfinancial time 1 series exhibit fractal behaviours (see, for example, M. Ausloos et al. [1], Y. Dong and J. Wang 0 7 [3], B. B. Mandelbrot [9], and etc.), which suggests the possibility that uncertainties in the mar- 1 kets might come from filtrations exhibiting fractal structures. Therefore, it is of significance to : v considerstochasticcontrolproblemsforcontrolledsystemswithnoisecomingfromfiltrationsde- i X terminedbythediffusions onfractals. SincethenaturalanalogueofEuclideanspacesareinfinite r (or s -compact) fractals (rather than the finite ones), it is interesting to establish a corresponding a theoryoninfinitefractals. The motivation of this paper is therefore to: (i) extend the results in X. Liu and Z. Qian [8] to the infinite Sierpinski gasket; (ii) establish a stochastic maximum principle for stochastic controlproblemsonthefiniteandinfiniteSierpinskigaskets,whereuncertaintiesinthecontrolled dynamic systems are sourced from the Brownian filtrations. It turns out that, in contrast to its counterpart on Euclidean spaces, the stochastic maximum principle on the Sierpinski gaskets consists of two necessity equations rather than a single one (see S. Peng [10], J. Yong and X. Zhou[12,Section3.2]). Asweshallsee,thisisduetothesingularity betweenmeasures. This paper is organized as follows. In Section 2, we introduce notations which will be enforced throughout this paper, and review some related results in literature. The main results of Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom. Email: ∗ [email protected]. 1 this paper is formulated and collected in Section 3. Section 4 is devoted to two results on the Brownian motion on the infinite Sierpinski gasket: a martingale representation theorem and the exponentialintegrability ofthequadraticprocessoftherepresenting martingale. InSection5,we establish the stochastic maximum principle for stochastic control problems on the finite and the infinite Sierpinski gasket. Though results of this paper are established for 2-dimensional Sierp- inski gasket, we however believe that our results also hold for higher-dimensional cases, where argumentinthispapershouldremainvalid. 2 Notations In this section, we introduce notations which will be enforced throughout this paper. We also reviewseveralresultsinliterature neededinthefollowingsections. 2.1 The Standard DirichletFormonthe FiniteSierpinski Gasket Let V = p ,p ,p R2 with p =(0,0), p =(1,0), p =(1,√3), and F :R2 R2, i= 0 { 1 2 3}⊆ 1 2 3 2 2 i → 1,2,3bethecontractionmappingsgivenbyF(x)= 1(x+p), x R2, i=1,2,3.DefineV , m N inductively by V = F (V ) F (V )i F (V2 ), mi N∈, and V = ¥ V . Tmhe (2∈- dimensional)finiteSime+rp1inski1gasmket∪isd2efinmed∪tob3ethmeclosu∈reS=V¯ of∗V inRm=2.0LemtW˜ = w = w w :w 1,2,3 , i N . Foreachw =w w W˜,let[w ]∗ =w ∗ Sw , m N {,and 1 2 i + 1 2 m 1 m + denote··W·˜[w ]0 =∈{W˜, W˜[w }]m =∈{w ′∈}W˜ :[w ′]m=[w ]m},an·d··S∈[w ]0=S, S[w ]m=Fw·m··◦···◦Fw∈1(S), m∈ N . In view of the contractivity of F, i=1,2,3, it is easily seen that, for each w W˜, the set + i ¥m=1S[w ]m contains a unique element p (w )∈S. It can be shown that the map p :W˜∈→S, w 7→ p (w )issurjective,andcard p 1(x) =1ifandonlyifx V (S V )(seeJ.Kigami[6,Lemma 1T.3.14]). Clearly,p W˜[w ]m (cid:0)=−S[w ]m(cid:1)foreachw ∈W˜, m∈∈N.0∪ \ ∗ Foragiven set V,wedenote by ℓ(V)the space of all real-valued functions on V. Define the (cid:0) (cid:1) relation onVˆ by: x yifand only if x y =2 m. Thestandard Dirichlet form (E,F(S)) m − ∼m ∼m | − | onthefiniteSierpinski gasketSisdefinedby E(u,v)= lim E(m)(u,v), u,v F(S), m ¥ ∈ →  F(S)= u ℓ(S): lim E(m)(u,u)<¥ ,  ∈ m ¥ → n o wheretheformsE(m), mNaregivenby ∈ E(m)(u,v)= 5 m(cid:229) u(x) u(y) v(x) v(y) , u,v ℓ(V ). m 3 − − ∈ x y (cid:16) (cid:17) ∼m (cid:0) (cid:1)(cid:0) (cid:1) Let n be the self-similar measure on S with weight (1,1,1), that is, n is the unique Borel 3 3 3 probability measure on S such that n (S[w ]m)=3−m for each w ∈W˜ and each m∈N. Then the formE isaregularDirichletformonL2(S;n ),andF(S)isthecorresponding Dirichletspace. AccordingtothegeneraltheoryofDirichletformsandMarkovprocesses(seeM.Fukushima et al. [5, Chapter 7]), associated to the form (E,F(S)) there exists a standard Hunt process M= W ,F, Xt t [0,¥ ], Px x S D with state space S, where D is the “cemetery” of S. The process X { }is∈called{Br}ow∈ni∪a{n}motion on S. The semigroup of X will be denoted by t t 0 t t 0 (cid:0) { }≥ (cid:1) { }≥ P . t t 0 { }L≥et P(S) be the family of all Borel probability measures on S. For each l P(S), the ∈ probability measure Pl on W is defined by Pl (A) = SPx(A)l (dx), A F. The expectation withrespecttoPl willbedenotedbyEl . LetFt0=s (RXr :r≤t), t ≥0,∈Ftl thePl -completion of F0 in F, and F the minimal completed admissible filtration of X , that is, F = t t t 0 t t 0 t l P(S)Ftl , t {0. }≥ { }≥ ∈ ≥ T 2 2.2 The Standard DirichletFormonthe Infinite Sierpinski Gasket LetS [0,1]2 bethe(unit)finiteSierpinskigasket. ForanyA R2,let AbethereflectionofA withr⊆especttothey-axis. ThentheinfiniteSierpinskigasketSˆ⊆isdefined−tobeSˆ = ¥ F n S ( S) . DenotebyVˆ them-verticessetofSˆ,thatis,Vˆ = ¥ F n V ( V ) ,n=m0 1−N. W∪e a−lsodenoteVˆ = ¥ m Vˆ . m n=0 1− m∪ − mS ∈ (cid:0) C(cid:1)learly,th∗ereSexmis=t0tramnslations ti, i∈NofR2 suchthatSt0=idR2(cid:0), (cid:1) ¥ Sˆ = t (S), t (S) t (S) Vˆ , i= j. i i j 0 ∩ ⊆ 6 i=0 [ Throughout thispaper, wewillfixachoiceofthetranslations t satisfying theproperties above. i Definetherelation onVˆ by: x yifandonly if x y =2 m. Foreach m Nandeach m − ∼m ∼m | − | ∈ u,v ℓ(Vˆ ),let m ∈ Eˆ(m)(u,v)= 5 m(cid:229) u(x) u(y) v(x) v(y) , 3 − − x y (cid:16) (cid:17) ∼m (cid:0) (cid:1)(cid:0) (cid:1) whenever the sum on the right hand side converges. It can be easily shown that Eˆ(m)(u,u) is non-decreasing in m for any u ℓ(Sˆ). The standard Dirichlet form (Eˆ,F(Sˆ)) on the infinite SierpinskigasketSˆ isthendefin∈edby Eˆ(u,v)= lim Eˆ(m)(u,v), u,v F(Sˆ), m ¥ ∈ →  F(Sˆ)= u ℓ(Sˆ): lim Eˆ(m)(u,u)<¥ .  ∈ m ¥ → n o Let nˆ be the self-similar measure on Sˆ with weight (1,1,1), that is, nˆ is the unique Radon 3 3 3 measure onSˆ such that n ti S[w ]m =3−m for eachw ∈W˜ and eachi,m∈N. Thenthe form Eˆ isaregularDirichletformonL2(Sˆ;nˆ),andF(Sˆ)isthecorresponding Dirichletspace. (cid:0) (cid:0) (cid:1)(cid:1) withWtehedfeonromte(bEˆy,FMˆ(=Sˆ)),Wâ,nFdˆ,b{yXˆt}Pˆt∈[0,¥ ],t{hPêx}sex∈mSî∪g{rDo}upthoef stXând.arTdhHeupnrotcpersoscesXˆs asissocciaaltleedd (cid:0) t t 0 (cid:1) t t the Brownian motion on Sˆ. It can{be}e≥asily shown that the {diff}usion Xˆ coinc{ide}s with the t { } BrownianmotionconstructedinM.T.BarlowandE.A.Perkins[2]asthelimitofsimplerandom walksongraphs. Inaccordance withM.Fukushima etal. [5],wedenote byF (Sˆ)thespace ofallfunctions loc which are locally in F(Sˆ). For any u,v F (Sˆ), the mutual energy measure of u and v is loc ∈ denoted bymˆ . u,v Let P(Sˆh) bie the family of all Borel probability measures on Sˆ. For each l P(Sˆ), the probabilitymeasurePˆl onWˆ isdefinedbyPˆl (A)= SˆPˆx(A)l (dx), A Fˆ. Theexp∈ectationwith respect to Pˆl will be denoted by Eˆl . Let Fˆt0 =s RXˆr :r≤t , t ≥0,∈Fˆtl the Pˆl -completion of Fˆ0 inFˆ,and Fˆ theminimalcompletedadmissible filtrationof Xˆ . t { t}t≥0 (cid:0) (cid:1) { t}t≥0 2.3 The Kusuoka Measureon theFiniteSierpinski Gasket Let 5 0 0 2 2 1 2 1 2 1 1 1 A = 2 2 1 , A = 0 5 0 , A = 1 2 2 , 1 2 3 5  5  5  2 1 2 1 2 2 0 0 5       2 1 1 1 − − P= 1 2 1 , 3 − −  1 1 2 − −   3 andletYi=PAiP, i=1,2,3. Letm˜ betheuniqueprobabilitymeasureonW˜ suchthatm˜(W˜[w ]m)= 53 mtrace Ytw 1···Ytw mYw m···Yw 1 , w ∈W˜. The Kusuoka measure m on S is then defined to be the push-forward m = m˜ p 1. In other words, m is the unique Borel probability measure on S − (cid:0)suc(cid:1)hthat m(cid:0) S[w ]m =m˜ W◦˜[w ]m , w (cid:1)∈W˜. We end this section with a review on the representing martingale on the finite Sierpinski (cid:0) (cid:1) (cid:0) (cid:1) gasket,ofwhichtheconstruction willbeneeded later. Let Zm(w )=trace(Ytw 1···Ytw mYw m···Yw 1)−1·Ytw 1···Ytw mYw m···Yw 1, w ∈W˜. Then Zm m 1 is a m˜-martingale, and Z=limm ¥ Zm exists for m˜-a.s. (see S.Kusuoka [7, Pro- positio{n (1}.7≥)]). Since card(p 1(x))=1, x S→V and m (V )=0, Z(x)=Z(p 1(x)) is well- − − defined for m -a.e.x S. The following resul∈t w\as fi∗rst shown∗in S. Kusuoka [7, Theorem (5.4)] ∈ (seealsoX.LiuandZ.Qian[8,Theorem2.1]). Theorem 2.1. (a) There exists an e = (e ,e ,e ) R3 such that e,e = 1 and Z(x)1/2e = 1 2 3 ∈ h i Z(x)1/2Pe=0for m -a.e.x S,where , istheEuclidean innerproduct onR3. 6 ∈ h· ·i (b)Foranye R3 satisfying thepropertyin(a),themartingale additivefunctional ∈ t W = e,Z(X )e 1/2dM[H0e], t 0, (2.1) t r − r 0 h i ≥ Z doesnotdepend ontheparticular choiceofe,andhas m asitsenergymeasure,whereH0eisthe harmonicfunctiononSwithboundaryvalue(H0e)(p)=e, i=1,2,3,andM[u]isthemartingale i i part in the Itô-Fukushima decomposition of u(X) for u F(S). Moreover, for each u F(S), t ∈ ∈ thereexistsauniqueg L2(S;m )suchthatM[u]= tg(X )dW , t 0. ∈ t 0 r r ≥ Definition2.2. Themartingale additive functionalRW givenby(2.1)iscalled theBrownianmar- tingaleonS. Definition2.3. Foranyu F(S),inviewofTheorem2.1,wedefinethegradient(cid:209) uofuasthe ∈ uniqueelementinL2(S;m )suchthatM[u]= t(cid:209) u(X )dW , t 0. t 0 r r ≥ The following result on the singularity bRetween the Lebesgue-Stieltjes measure induced by t W andtheLebesguemeasureon[0,¥ )wasprovedinX.LiuandZ.Qian[8,Lemma4.10]. t 7→h i Lemma 2.4. The Lebesgue-Stieltjes measure d W (w ) is singular to the Lebesgue measure dt t h i on[0,¥ )Pn -a.e.w W . ∈ 3 Formulation of Main Results Thefirstmainresultofthispaperisamartingalerepresentation theoremontheinfiniteSierpinski gasket. Westartwiththefollowingdefinition. Definition 3.1. The Kusuoka measure mˆ on Sˆ is unique Radon measure on Sˆ such that mˆ|ti(S) = m t 1, i N,wheret :R2 R2, i Narethetranslations introduced inSection4.1. ◦ i− ∈ i → ∈ Remark. Itiswell-knownthattheKusuokameasurem onSisnon-atomicandsingulartotheself- similarmeasuren onS(seeS.Kusuoka[7,Corollary(2.15)andExample1,p. 678]). Therefore, theKusuokameasure mˆ onSˆ isalsonon-atomic andsingular totheself-similar measurenˆ onSˆ. Withtheabovedefinition, wecannowformulatethemartingalerepresentation theorem. Theorem3.2. ThemartingaledimensionoftheBrownianfiltrationontheinfiniteSierpinskigas- ket Sˆ is one. Moreover, there exists a martingale additive functional Wˆ such that Wˆ has the representing propertyandhas mˆ asitsenergymeasure. 4 LetT (0,¥ )andl P(Sˆ). ByvirtueofTheorem3.2,usinganargumentsimilartothatof ∈ ∈ X.LiuandZ.Qian[8,Theorem3.3],wehavetheexistenceanduniqueness oftheBSDE dYt =g(t,Yt)dt+ f(t,Yt,Zt)d Wˆ t+ZtdWˆt, t [0,T), Pˆl -a.s., h i ∈ (3.1) ( YT =x , where x is an Fˆl -measurable random variable, and f : [0,¥ ) R W R, g :[0,¥ ) R T × × → × × R W R are measurable functions such that the processes t g(t,y) and t f(t,y,z) are F×ˆl -a→dapted foreachy,z R. 7→ 7→ t { } ∈ Theorem3.3. Letb R2. Suppose thatx , g,and f satisfythefollowing: ∈ Eˆl x 2e2b1T+2b2hWîT <¥ , (3.2) (cid:16) (cid:17) and,forPl -a.e.w W , ∈ g(t,y,w ) g(t,y¯,w ) K y y¯, (3.3) | − |≤ | − | f(t,y,z,w ) f(t,y¯,z¯,w ) K (y y¯ + z z¯), (3.4) 0 | − |≤ | − | | − | forallt [0,T)andally,y¯,z,z¯ R,whereK,K >0aresomeconstants. 0 (a)T∈hereexistsatmostone∈pairof Fˆl -adapted processes satisfying (3.1)inthefollowing t sense: If(Y,Z)and(Y¯,Z¯)aretwopairs{of F}ˆl -adapted processes satisfying (3.1),then t { } T Yt =Y¯t, t [0,T], and (Zr Z¯r)2d Wˆ r =0, Pˆl -a.s. ∈ 0 − h i Z (b)If(3.2)-(3.4)andthefollowing T T Eˆl g(r,0)2e2b1r+2b2hWîrdr <¥ , Eˆl f(r,0,0)2e2b1r+2b2hWîrd Wˆ r <¥ , (3.5) 0 0 h i (cid:18)Z (cid:19) (cid:18)Z (cid:19) holdforsufficiently largeb ,b >0,thenthe(3.1)admitsauniquesolution (Y,Z). Moreover, 1 2 T k(Y,Z)k2Vlb [0,T]≤CEˆl (cid:18)x 2e2b1T+2b2hWîT +Z0 g(r,0)2e2b1r+2b2hWîrdr (3.6) T + f(r)2e2b1r+2b2 Wˆ rd Wˆ , h i r 0 h i Z (cid:19) whereC>0isconstant depending onlyonK,K andb ,and 0 t k(y,z)k2Vlb [0,T],Eˆl (cid:20)0sutpT(cid:18)yt2e2b1t+2b2hWît+Zt y2re2b1r+2b2hWîrdr ≤≤ t + y2+z2 e2b1r+2b2 Wˆ rd Wˆ <¥ . r r h i r t h i Z (cid:19)(cid:21) (cid:0) (cid:1) Thesecondmainresultofthispaperconcernsthestochasticmaximumprincipleforstochastic controlproblemsonthefiniteandinfiniteSierpinskigaskets,whereuncertaintiesinthecontrolled dynamic systems are sourced from the Brownian filtrations. For simplicity, we only give the formulateforthefiniteSierpinskigasket. ThecaseforinfiniteSierpinskigasketcanbeformulated similarly. Letl P(S)satisfyl n . Let(U,r )beadecisionspace,thatis,(U,r )isagivenseparable ∈ ≪ metricspace. Leth:R R, f :[0,T] R U R, f :[0,T] R U RbeBorelmeasurable 1 2 → × × → × × → functions. ForanyU-valued F -adapted processu(t),weintroduce thecostfunctional t { } T T J(u),El h(x(T))+ f1(t,x(t),u(t))dt+ f2(t,x(t),u(t))d W t , (3.7) 0 0 h i (cid:18) Z Z (cid:19) 5 wherethecontrolled processx(t)isgivenbythefollowingSDE1 on W ,F, Ftl t 0,Pl : { }≥ (cid:0) (cid:1) dx(t)=b (t,x(t),u(t))t+b (t,x(t),u(t))d W 1 2 t h i  +s (t,x(t),u(t))dWt, t (0,T], Pl -a.s., (3.8) ∈  x(0)=x , 0 wherej :[0,T] R U R, j =b ,b ,s areBorelmeasurable functions, andx Fl . × × → 1 2 0∈ 0 Definition3.4. DenotebyA[0,T]thefamilyofallU-valuedprocesses u(t)suchthat T T El h(x(T)) + f1(t,x(t),u(t))dt+ f2(t,x(t),u(t))dt <¥ , (3.9) | | 0 | | 0 | | (cid:18) Z Z (cid:19) where x(t) is the controlled process given by (3.8). Any u A[0,T] is called an admissible ∈ control, andthepair(x(),u()) iscalledanadmissiblepair. · · Weconsiderthefollowingoptimization problem minimize J(u). (P) u A[0,1] ∈ Theorem3.5. Letl P(S)beabsolutely continuous withrespectton . Assumethat: ∈ (A.1) j (t,x,u) j (t,xˆ,uˆ) M x xˆ +r (u,uˆ), t [0,T], x,xˆ R, u,uˆ U, | − |≤ | − | ∈ ∈ ∈ ( j (t,0,u) M, t [0,T], u U, | |≤ ∈ ∈ forj =b ,b ,s ,f ,f ,h,and 1 2 1 2 (A.2) ¶ j (t,x,u) ¶ j (t,xˆ,uˆ) + ¶ 2j (t,x,u) ¶ 2j (t,xˆ,uˆ) M x xˆ +r (u,uˆ), x x x x | − | | − |≤ | − | t [0,T], x,xˆ R, u,uˆ U, ∈ ∈ ∈ forj =b ,b ,s ,f ,f ,h,whereM>0isaconstant. 1 2 1 2 Suppose that (x¯(),u¯()) is a solution of (P). Let(p(),q()) and (P(),Q()) be the solutions · · · · · · oftheadjoint equations dp(t)= [¶ b (t)p(t) ¶ f (t)]dt x 1 x 1 − − [¶ b (t)p(t)+¶ s (t)q(t) ¶ f (t)]d W  − x 2 x − x 2 h it (3.10)  +q(t)dWt, t ∈[0,T], Pl -a.s., p(T)= ¶ h(x¯(T)), x −    and  dP(t)= [2¶ b (t)P(t)+¶ 2b (t)p(t) ¶ 2f (t)]dt x 1 x 1 x 1 − −  2¶ xb2(t)+¶ xs (t)2 P(t)+¶ xs (t)Q(t)+¶ x2b2(t)p(t)+¶ x2s (t)q(t) ¶ x2f2(t) d W t − − h i  +(cid:2)Q(cid:0)(t)dWt, t ∈[0,T(cid:1)], Pl -a.s., (cid:3) P(T)= ¶ 2h(x¯(T)), x  − (3.11)    1TheexistenceanduniquenessofsolutionsofthisSDEcanbeeasilyshownbyanaprioriestimatesimilarto(3.6). 6 andletH (t,x,u), H (t,x,u)betheHamiltonians definedby 1 2 H (t,x,u),b (t,x,u)p(t) f (t,x,u), 1 1 1 − 1 H (t,x,u),b (t,x,u)p(t)+s (t,x,u)q(t) f (t,x,u)+ [s (t,x,u) s (t,x,u¯(t))]2P(t). 2 2 2 − 2 − Then H (t,x¯(t),u¯(t))=maxH (t,x¯(t),u), M -a.e., 1 1 1 u U ∈ (3.12) H (t,x¯(t),u¯(t))=maxH (t,x¯(t),u), M -a.e., 2 2 2  u U ∈ where  M1=dt Pl , (3.13) × andM istheuniquemeasureontheoptional s -field2 on[0,¥ ) W suchthat 2 × M2 Js 1,s 2M =El hWis 2−hWis 1 , (3.14) forany F -stoppingtimess ,s(cid:0) withs (cid:1) s ,w(cid:0)hereJs ,s M=(cid:1) (t,w ) [0,¥ ) W :s (w ) t 1 2 1 2 1 2 1 { } ≤ { ∈ × ≤ t <s (w ) . 2 } Remark. (a)Byl n andLemma2.4,themeasuresM andM aremutuallysingular. 1 2 ≪ (b)LetM=M +M . InviewofthesimilaritybetweenthenecessityequationforH (t,x¯(t),u¯(t)) 1 2 1 andPontryagin’smaximumprinciple,wecallu =u dM1 andu =u dM2 thequasi-deterministic 1 · dM 2 · dM andpurely-stochasticpartsofthecontrolu()respectively. Notethatsuchseparationphenomenon · doesnothappen forthestochastic control problemsonEuclidean spaces. 4 Brownian Martingale on the Infinite Sierpinski Gasket Inthissection,weproveamartingalerepresentation theorem(Theorem3.2)forsquare-integrable Fˆ -martingales and theexponential integrability ofquadratic process oftherepresenting mar- t { } tingale. 4.1 MartingaleRepresentations onthe Infinite Sierpinski Gasket Lemma4.1. Foranyu,v F(Sˆ), ∈ ¥ Eˆ(u,v)= (cid:229) E u t ,v t . (4.1) i i ◦ ◦ i=0 (cid:0) (cid:1) Moreover, foranyu,v F (Sˆ), loc ∈ ¥ mˆhu,vi = (cid:229) 1ti(S)· m hu◦ti,v◦tii◦ti−1 . (4.2) i=0 (cid:0) (cid:1) Proof. Itfollowsimmediatelyfromthedefinitionthat ¥ Eˆ(m)(u,v)= (cid:229) E(m)(u t ,v t ). i i ◦ ◦ i=0 ThisimpliesthatEˆ(u,v)=(cid:229) ¥ E u t ,v t . i=0 ◦ i ◦ i 2Thatis,thes -fieldon[0,¥ ) W gene(cid:0)ratedbythefa(cid:1)milyofallrightcontinuousleftlimitprocesses. × 7 Moreover, foranyu F(Sˆ)andany f C (Sˆ) F(Sˆ), 0 ∈ ∈ ∩ ¥ f dmˆ =2Eˆ(fu,u) Eˆ(f,u2)=(cid:229) 2E (f t )(u t ),u t E f t ,u2 t u i i i i i Sˆ h i − ◦ ◦ ◦ − ◦ ◦ Z i=0 ¥ ¥ (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3) =i(cid:229)=0ZS f ◦tidm hu◦tii=i(cid:229)=0Zti(S) f d m hu◦tii◦ti−1 . (cid:0) (cid:1) Therefore, mˆhui =(cid:229) ¥i=01ti(S)· m hu◦tii◦ti−1 foreachu∈F(Sˆ). Thisimpliestheotherstatement ofthelemma. (cid:0) (cid:1) DefineafunctionZˆ :Sˆ R3 3byZˆ(t x)=Z(x), x S, i N. Lete R3satisfytheproperty × i (a)inTheorem2.1andbefi→xed,anddefineHˆ0e F (∈Sˆ)by∈setting Hˆ∈0e t =H0e, i N. loc i ∈ ◦ ∈ Wehavethefollowingrepresentations formartingaleadditivefunctionals. (cid:0) (cid:1) Proposition 4.2. Themartingaleadditive functional t Wˆ = e,Zˆ(Xˆ )e 1/2dMˆ[Hˆ0e], t 0, (4.3) t r − r 0 h i ≥ Z has mˆ as its energy measure, where, for any u F (Sˆ), Mˆ[u] is the martingale part in the loc Itô-Fukushima decomposition of u(Xˆ ). Moreove∈r, for each u F (Sˆ), there exists a unique t loc ∈ g L2 (Sˆ;mˆ)suchthatMˆ[u]= tg(Xˆ )dWˆ , t 0. ∈ loc t 0 r r ≥ Proof. Forany f C (Sˆ),wehRave 0 ∈ 1 t lim Eˆnˆ f(Xˆr)d Wˆ r t 0 2t 0 h i ↓ Z =lim(cid:0) 1Eˆnˆ t f(Xˆr)(cid:1) e,Zˆ(Xˆr)e −1d Mˆr[Hˆ0e] r t 0 2t 0 h i h i ↓ (cid:18)Z (cid:19) = f(x) e,Zˆ(x)e 1 mˆ (dx) ZSˆ h i− hHˆ0ei ¥ = (cid:229) f(t x) e,Zˆ(t x)e 1 m (dx) i=0ZS i h i i− hHˆ0e◦tii ¥ = (cid:229) f(t x) e,Z(x)e 1 m (dx) i − H0e i=0ZS h i h i = f(x)mˆ(dx). Sˆ Z Therefore,Wˆ has mˆ asitsenergymeasure. t We now turn to the second statement. It suffices to prove the result for u F(Sˆ). Clearly, ∈ (u ti)S F(S), i N. Letgi betheuniqueelementofL2(S;m )suchthat ◦ | ∈ ∈ t Mt[u◦ti]= gi(Xr)dWr, t 0, i N. 0 ≥ ∈ Z Defineg L2 (Sˆ;mˆ)byg t =g, i N. Then ∈ loc ◦ i i ∈ ¥ ¥ g(x)2mˆ(dx)= (cid:229) g(x)2m (dx)= (cid:229) E(u t ,u t )=Eˆ(u,u)<¥ . i i i Sˆ S ◦ ◦ Z i=0Z i=0 Therefore, g∈L2(Sˆ;mˆ),and 0·g(Xˆr)dWˆr isamartingaleadditivefunctional offiniteenergy. R 8 Furthermore, by(4.3)andLemma4.1, Eˆnˆ Mˆ[u] ·g(Xˆr)dWˆr − 0 1 Z hD E i =Eˆnˆ Mˆ[u] 1 2Eˆnˆ Mˆ[u], ·g(Xˆr)dWˆr +Eˆnˆ ·g(Xˆr)dWˆr h i − 0 1 0 1 Z Z hD E i hD E i =mˆ (cid:2)(Sˆ) 2(cid:3) g(x) e,Zˆ(x)e 1/2mˆ (dx)+ g(x)2mˆ(dx) hui − ZSˆ h i− hu,Hˆ0ei ZSˆ ¥ ¥ ¥ =i(cid:229)=0m hu◦tii(S)−2i(cid:229)=0ZSgi(x)he,Z(x)ei−1/2m hu◦ti,H0ei(dx)+i(cid:229)=0ZSgi(x)2m (dx) ¥ ¥ =2(cid:229) En M[u◦ti] 1 2(cid:229) En M[u◦ti], ·gi(Xr) e,Z(Xr)e −1/2dMr[H0e] i=0 h i − i=0 Z0 h i 1 hD E i ¥ (cid:2) (cid:3) ¥ =2(cid:229) En M[u◦ti] 1 2(cid:229) En M[u◦ti], ·gi(Xr)dWr i=0 h i − i=0 Z0 1 hD E i (cid:2) (cid:3) =0. Therefore, Mˆ[u]= tg(Xˆ )dWˆ , t 0. Thiscompletestheproof. t 0 r r ≥ We can now giRve a proof of the martingale representation theorem on the infinite Sierpinski gasket. ProofofTheorem3.2. Itfollows immediately from Proposition 4.2andZ.QianandJ. Ying[11, Theorem3](seealsoX.LiuandZ.Qian[8,Theorem4.8]andtheremarkthereafter). Definition4.3. Themartingale additive functionalWˆ givenby(4.3)iscalled theBrownianmar- tingaleonSˆ. Definition 4.4. For any u F (Sˆ), its gradient is defined to be the unique element (cid:209) u loc ∈ ∈ L2 (Sˆ;mˆ)suchthatMˆ[u]= t(cid:209) u(Xˆ )dWˆ , t 0. loc t 0 r r ≥ Remark. (a)Clearly, R Eˆ(u,v)= (cid:209) u(x)(cid:209) v(x)mˆ(dx), u,v F(Sˆ). Sˆ ∈ Z (b)ItiseasilyseenfromtheproofofProposition 4.2that ((cid:209) u◦ti−1)|S =(cid:209) (u◦ti|S), u∈Floc(Sˆ), where(cid:209) (u ti S)isthegradient ofu ti S asanelementinF(S). ◦ | ◦ | 4.2 Exponential Integrability oftheQuadraticVariationProcess In this subsection, we prove the exponential integrability of the quadratic variation process Wˆ of the Brownian martingale on Sˆ. We shall need the following heat kernel estimate, whichhwais firstprovedinM.T.BarlowandE.A.Perkins[2,Theorem1.5]. Lemma4.5. Brownianmotion Xˆ onSˆ hastransitionkernels pˆ (x,y), t>0, x,y Sˆ thatare t t 0 t absolutely continuous withresp{ect}to≥nˆ. Moreover, thereexists universal constantsC∈ ,C >0 ,1 ,2 ∗ ∗ suchthat pˆt(x,y)≤C∗,1t−ds/2exp −C∗,2 |xt1−/dwy| dw/(dw−1) , t ≥0, x,y∈Sˆ, h (cid:16) (cid:17) i whered =2log3/log5, d =log5/log2arethespectral dimension andthewalkdimension of s w Xˆ respectively. t t 0 { }≥ 9 Definition4.6. ForeachRadonmeasurel onSˆ,wedefine Pˆl (x)= pˆ (x,y)l (dy), x Sˆ, t >0, t t Sˆ ∈ Z whenevertheaboveintegralexists. Lemma4.7. Foreacht >0, Pˆmˆ(x) C max 1,t ds/2 , x Sˆ, t − ≤ ∗ { } ∈ whereC >0isauniversal constant. ∗ Proof. For each x Sˆ, denote I = i N : dist(x,t (S)) 2n , n N. Clearly, card(I ) n i n ∈ { ∈ ≤ } ∈ ≤ 2 3n, n N. Ift 1,then,byLemma4.5, · ∈ ≤ ¥ Pˆmˆ(x) C t ds/2 (cid:229) mˆ(t (S))+ (cid:229) (cid:229) exp C 2ndw/(dw 1) mˆ(t (S)) t − i − i ≤ ∗ − ∗ (cid:26)i∈I0 ¥ n=0i∈In+1\In (cid:2) (cid:3) (cid:27) C t ds/2 1+ (cid:229) 3nexp C 2ndw/(dw 1) , − ,2 − ≤ ∗ − ∗ (cid:26) n=0 (cid:27) (cid:2) (cid:3) whereC >0denotesauniversalconstantwhichmightbedifferentatvariousappearances. Note tthhaatt(cid:229)kP¥nˆt=mˆ∗0k3Ln¥ e≤xpC(cid:2)∗−mCa∗x,2{21n,dtw−/d(ds/w2−}1,)(cid:3)t<>¥0.. Therefore,kPˆtmˆkL¥ ≤C∗t−ds/2, t∈(0,1]. Thisimplies The following can be easily shown by the same argument as that of X. Liu and Z. Qian [8, Lemma4.17andCorollary4.19]. Lemma4.8. LetAˆ(i), i=1,...,n be positive continuous additive functionals withrespect to the HuntprocessMˆ ,andhavemˆ , i=1,...,nastheirRevuzmeasures. Then,foreacht>0andeach i f B ([0,¥ ) Sˆ), i=1,...n, i + ∈ × Eˆ f(Xˆ ) f (t ,Xˆ ) f (t ,Xˆ )dAˆ(1) dAˆ(n) x(cid:18) t Z0<t1<···<tn<t 1 1 t1 ··· n n tn t1 ··· tn (cid:19) = Pˆ mˆ f (t , )Pˆ (4.4) Z0<t1<···<tn<t t1 1 1 1 · t2−t1 mˆ f (cid:0)(t , )Pˆ (cid:0)mˆ f (t , )Pˆ f (x)dt dt , x Sˆ. ··· n−1 n−1 n−1 · tn−tn−1 n n n · t−tn ··· 1··· n ∈ Proposition 4.9. Foreach f L1(Sˆ;nˆ)a(cid:0)ndeachb ,t >0,(cid:1) (cid:1)(cid:1) ∈ + sxupSˆ Eˆx f(Xˆt)eb hWît ≤max{1,t−ds/2}kfkL1(nˆ)Egs,gs C∗G (gs)(t+1)b , ∈ (cid:0) (cid:1) (cid:2) (cid:3) whereC >0 is a universal constant, g =1 d /2, G () is the Gamma function, and, for each s s ∗ − · a,b>0,E ()istheMittag-Leffler function a,b · E (z)= (cid:229)¥ zp , z C. a,b G (ap+b) ∈ p=0 5 A Stochastic Maximum Principle In this section, we prove Theorem 3.5 for the optimization problem (P) on the finite Sierpinski gasket. TheproofforthefiniteSierpinskigasketshouldbeeasilyadaptedtotheinfiniteSierpinski gasket by virtue of the results in Section 4. Ourargument is essentially amodification of that of S. Peng [10] while overcoming some difficulties concerning the Brownian martingales on the Sierpinskigaskets. Anexampleisalsoprovided attheendofthissection. Asweshallsee,acrucial ingredient ofourargument isanordercomparison lemma(Lemma 5.2),whichisneededwhenperforming stochastic Taylorexpansions. 10