An introduction to the (local) martingale problem

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1 An inroducion o he (local) maringale problem Chri Janjigian Ocober 14, 214 Abrac Thee are my preenaion noe for a alk in he Univeriy of Wiconin - Madion graduae probabiliy eminar. Thee noe are primarily baed on he exbook by Ehier and Kurz[2, Karaza and Shreve[3, Sroock and Varadhan[5, and he lecure noe by Bovier[1. The goal of hi alk i o preen an inroducion o he local maringale problem a a level underandable o omeone who i familiar wih coninuou ochaic calculu and who ha been inroduced o he abrac heory of Markov procee. I have ried o make implifying aumpion wherever poible o avoid any echnical iue, a hi i mean o be preened over wo hor alk. I expec ha I have overimplified he hypohee in place and ha here are iue wih he proof. Inroducion Moivaion We ar wih he heuriic ha he Markov propery i he naural random analogue of he key propery of a well-poed differenial equaion: given he preen ae, we can deermine he fuure of he yem. In he cae of a differenial equaion, hi propery ake an infinieimal form. If ẋ = g(x) and if f i a ufficienly nice funcion, hen hi propery i he fundamenal heorem of calculu f(x()) = f(x()) + f (x(u))g(x(u))du. One migh herefore hope ha for ome cla of Markov procee a imilar, infinieimal conrucion i poible. In order o ue he Markov propery, we are led 1

2 o conider condiional expecaion of e funcion f and o poi he exience of a deerminiic operaor L o ha E [ f(x + ) f(x ) F X = L f(x ) + o( ) In direc analogy o he eing of differenial equaion, one migh herefore hope ha we have he equaliy E [ [ f(x + ) f(x ) F X + = E L u f(x u )du F X E We may rewrie hi line uggeively a [ + ( f(x + ) f(x ) L u f(x u )du f(x ) f(x ) ) L u f(x u )du F X In oher word, we migh expec he Markov propery o imply ha for any e funcion f, he proce f(x ) f(x ) L u f(x u )du i a maringale. Auming ha hi i rue, i i naural a hi poin o queion wheher here i a ufficienly nice cla of operaor L for which he convere migh alo hold. Thi i he maringale problem of Sroock and Varadhan. = Dicree ime and finie pace: he eay cae To give ome inuiion for why one migh reaonably expec he heuriic above o be accurae in any generaliy, we fir how hey are correc when one remove all echnical obrucion. Lemma 1. Given a d d ochaic marix P, define L = P I. Then X n i a ime-homogeneou Markov chain wih raniion probabiliy marix P if and only if for all funcion f i a maringale. n 1 Mn f = f(x n ) f(x ) Lf(X k ) k= Proof. Boh par eenially follow from he fac ha we can rewrie he marix P n I a a elecoping um: n 1 P n I = P k (P I) }{{} k= L 2

3 Fix m, n N and uppoe ha X n i a Markov wih raniion probabiliy marix P. [ [ m 1 E M f n+m M n f Fn X = E f(x m+n ) f(x n ) + Lf(X n+k ) Fn X k= m 1 = P m f(x n ) f(x n ) P k Lf(X n ) = Converely, uppoe ha X n i a ochaic proce and M f n i a maringale wih repec o he filraion Fn X for all f. We will how he Markov propery by inducion; i.e. we how ha for all f, E [ f(x n+m ) Fn X = P m f(x n ), where P = I + L. The bae cae i immediae from he maringale propery. k= E [f(x n+1 ) F n = f(x n ) + Lf(X n ) and in general inducion give m 1 E [f(x n+m ) F n = f(x n ) + E [ Lf(X n+k ) Fn X k= m 1 = f(x n ) + P k Lf(X n ) = P m f(x n ) k= Ouline To avoid echnicaliie, we will place much ronger rericion on our ochaic procee han are aboluely neceary for he reul o hold. Indeed, we will reric aenion o one dimenional uniformly ellipic diffuion wih bounded coefficien. My goal i o convey he flavor of proof ha generalize, raher han o preen complee proof. We will herefore have o rely on ome reul which are imilar o wha we prove, bu no acually wha we prove. The goal of he alk will be o (eenially) prove hree heorem: 1. Soluion o he maringale problem for he generaor of a uniformly ellipic diffuion wih bounded coefficien exi if and only if a weak oluion o he correponding ochaic differenial equaion exi. 3

4 2. Soluion exi o uniformly ellipic ochaic differenial equaion wih bounded coefficien. 3. If a ufficienly nice (defined laer) maringale problem i well poed, hen he oluion ha he Markov propery. Sochaic differenial equaion Inroducion In 1951, Iô inroduced he modern framwork of ochaic differenial equaion, providing a rigorou generalizaion of he heory of ordinary differenial equaion o a eing appropriae for funcion which are of unbounded variaion. Alhough hi heory i mo generally applicable o càdlàg emi-maringale, we will reric our aenion o he ime-homogeneou coninuou eing, where a number of echnical and noaional problem can be avoided. In pecific, we conider ochaic differenial equaion of he form dx = b(x)d + σ(x)dw (1) wih given iniial condiion X and where W i andard Brownian moion. Thee differenial equaion hould be inerpreed a inegral equaion in he ene of Iô: X X = b(x )d + σ(x )dw (2) We will be concerned wih exience, uniquene, and (Markov) regulariy of equaion of hi ype. For eae of noaion, we will conider one dimenional procee hroughou mo of he alk. The reul and proof we will dicu do no change meaningfully in higher dimenion. Before proceding, le u record wha we mean by a oluion o a ochaic differenial equaion in hi conex. Definiion 2. A weak oluion o (1) i a ochaic bai (Ω, F, F, P ) aifying he uual condiion, an F Brownian moion W and a coninuou and F adaped proce X aifying (2). Sanding aumpion To avoid mo echnical problem, we will place exremely rong rericion on our diffuion coefficien. 4

5 a. b(x) and σ(x) are Borel meaurable. b. There exi a conan K o ha for all x b(x) + σ(x) K. c. There exi a conan C > o ha for all x, σ(x) > C. The maringale problem Moivaional example We begin wih an example of a local maringale problem which i well poed on C[, ) and moivae he main heorem of hi alk. Theorem 3. (Lévy) [3, Theorem 3.16 Define an operaor L by Lf = 1 2 for f(x) dx 2 {x, x 2 } = D(L). Then X i andard Brownian moion if and only if X i a coninuou ochaic proce wih X = and f(x ) f(x ) i a local maringale for all f D(L). Lf(X )d The local maringale condiion above i wrien uggeively, bu impler noaion would be o oberve ha i ay i X and X 2 are local maringale. We will need ignificanly more han wo e funcion in he domain of our generaor o ge uniquene in general. For example, hi maringale problem i no well-poed on he Skorokhod pace D[, ), a he Poion proce alo ha he propery ha X and X 2 are local maringale. d 2 Generaor of a diffuion There i an error here. I need weak oluion for all x. Fix ϕ C c (R). Suppoe ha for each x, we have weak a oluion X o (1) wih X() = x. Then Iô lemma give ha ϕ(x ) ϕ(x ) = + b(x )ϕ (X ) σ2 (X )ϕ (X )d (3) σ(x )ϕ (X )dw (4) For he momen, aume ha σ and b are bounded coninuou funcion. Then i i no hard o how ha we have he poinwie limi E x [ϕ(x ) ϕ(x) lim = b(x)ϕ (x) σ2 (x)ϕ (x) = Lϕ(x) 5

6 Pu anoher way (and informally) if we define a emi-group by P f(x) = E x [f(x ), hen d d P = P L. Converely, uppoe we know ha here i a coninuou weak oluion o (1) under aumpion (a.) and (b.). If we define L for ϕ Cc (R) (mapping ino B b (R) hi ime) hen we can rewrie he applicaion of Iô lemma a ϕ(x ) ϕ(x ) Lϕ(X )d = σ(x )ϕ (X )dw where he righ hand ide i a maringale and herefore o i he lef. To ee hi, oberve ha he ochaic inegral on he righ hand ide i well defined and σ(x ) 2 ϕ (X ) 2 d ϕ 2 K 2 <. Generaor In general, given a Polih pace E and an E valued Markov proce X, we can define a enible noion of a emi-group of operaor from a pace of e funcion like B b (E), C b (E) or, if he pace i locally compac on C (E) o ielf by P f(x) = E x [f(x ) The generaor of uch a emigroup i given by a pair: an operaor Lf(x) = lim 1 (P f(x) f(x)) and he domain for which he operaor i well defined D(L). I follow from he emigroup propery ha P + f(x) P f(x) P Lf(x) = P P f(x) f(x) o ha d d P = P L. A imilar argumen how ha P Lf(x) = LP f(x) for f DL. A a commen, I am inenionally leaving he meaning of hee limi ambiguou, bu for concreene hey can be aken in he upremum norm, which urn all of he above pace ino Banach pace. Theorem 4. Suppoe ha X i a Feller proce wih coninuou pah aking value in a locally compac Polih pace E and ha (L, D(L)) i i generaor on C (E), hen M = f(x ) f(x ) i a maringale. Lf(X )d 6

7 Proof. In he dicree eing, hi argumen follow eenially immediaely from wriing P I a a elecoping um. The analogue of ha argumen in he coninuou eing i he fundamenal heorem of calculu. We have [+ E [M + M F = E [f(x + ) f(x ) F E Lf(X u )du F = P f(x ) f(x ) = P f(x ) f(x ) = E [Lf(X +u ) F du P u Lf(X )du The la ep follow from he fac ha for any f C (E) and any x E we have Lf C b (E) (hi i he Feller aumpion) and E x f(x + ) E x f(x ). la aemen i rue becaue we have for any ω X + (ω) X (ω) a, o ha P Lf(X ) i a coninuou funcion of. The The maringale problem and weak oluion Definiion 5. Given an operaor (L, D(L)) a oluion o he (L, D(L)) maringale problem i an adaped ochaic proce X on a ochaic bai (Ω, F, F X, P ) aifying he uual condiion o ha for every f D(L), f(x ) f(x ) Lf(X )d i an F X maringale. We now procede o he fir genuine heorem we will prove in hi alk. Thi reul hold in far greaer generaliy han he reul given here. See for example [2, The main heuriic behind hi reul i ha if X i a oluion o (1), hen if we define M = X X + b(x )d hen formally dm = σ(x )dw and herefore we migh expec ha σ 1 (X )dm = W. Theorem 6. Suppoe ha aumpion a., b. and c. hold. Then a coninuou ochaic proce X on a ochaic bai (Ω, F, F X oluion o he maringale problem for, P ) aifying he uual condiion i a L = b(x) d dx + 1 d2 σ(x)2 2 dx 2 (5) 7

8 wih D(L) = Cc (R) and X = if and only if here exi a Brownian moion W ( on a ochaic bai Ω, F, F ), P aifying he uual condiion o ha (X, W ) i a oluion o he ochaic differenial equaion dx = b(x )d + σ(x )db (6) X = Proof. We have already een ha if (X, W ) i a weak oluion o (6) on (Ω, F, P, F ), hen X olve he maringale problem on he ame pace in equaion (4) of he previou ecion. Suppoe now ha X i a oluion of he maringale problem. Our goal here i going o be o build a Brownian moion ou of X. Define hiing ime by τ N = inf{ : X N} and ake any f C (R). Applying he hypohei o a funcion g C c (R) which i equal o f on [ 2N, 2N, we can ee ha f(x τn ) f() τn Lf(X )d i a maringale. Pah coninuiy give ha τ N, o i follow ha f(x ) f() Lf(X )d i a local maringale for every f C (R). In paricular, hi hold for f(x) = x and f(x) = x 2. Then he ochaic procee M = X N = X 2 b(x )d are coninuou local maringale. We claim ha B = 2X b(x ) + σ(x ) 2 d σ(x ) 1 dm i Brownian moion. Our aumpion ha σ(x ) 1 i bounded and previible implie ha B i a coninuou local maringale. To prove ha B i Brownian moion, i herefore uffice by Lévy characerizaion of Brownian moion (Theorem 3) o how ha B ha quadraic variaion. I i a andard reul ha [B = σ(x ) 2 d[m o i uffice o how ha [M = σ2 (X )d. Oberve ha ( 2 M 2 = N + 2 (X X )b(x )d + b(x )d) + 8 σ(x ) 2 d

9 If we can how ha 2 ( (X 2 X )b(x )d + )d) b(x i a local maringale, hen we are done, a hi will how ha M 2 σ2 (X )d i a local maringale, which idenifie [M = σ(x ) 2 d. We (ochaically) inegrae by par o find ha 2 X b(x )d = X b(x )d Subiuing hi yield ha ( 2 (X X )b(x )d + b(x )d) = 2 We claim ha 2 ( 2 b(x u )dudx + b(x )d) = 2 b(x u )dudx ( ) 2 b(x u )dudx + b(x )d b(x u )dudm which i a local maringale. One can ee hi by oberving ha boh funcion below are aboluely coninuou funcion which are a zero, wih he ame derivaive. ( 2 b(x )d) = 2 b(x ) b(x u )dud o i follow ha B i Brownian moion. To complee he proof, we oberve ha σ(x )db = dm = X X b(x )d Remark 7. I have choen o minimize he echnicaliie in he previou proof. To acually ue he la reul o prove exience, we will need o eiher work on he Skorokhod pace D R ([, )) or o allow b and σ o be ime inhomogeneou funcional of he pah of he proce. Exience One of he main advanage o hi approach i ha i lend ielf exremely naurally o proof of exience of Markov procee of ome deired form. The ypical ouline of an exience proof uing hi mehod i o prove ighne of an approximaing equence and hen o argue ha any ubequenial limi olve he maringale problem for he proce we are udying. The nex reul follow from very general exience reul baed on he rucure of he generaor and a compacificaion argumen [2, Theorem

10 In he argumen ha follow, we will hink of coninuou ochaic procee a meaure on C([, ), R). Thi argumen cloely follow he proof of [3, Theorem Propoiion 8. Suppoe ha b, σ, σ 1 C b (R). Then here exi a weak oluion o he ochaic differenial equaion dx = b(x )d + σ(x )dw X = Proof. (kech) Take a andard Brownian moion W on C([, ), R) le (n) j = j2 n. Se ϕ n () = j= (n) j 1 (n) [ j, (n) )() = max j{ j : j }. Then e X (n) = and for j+1 ( j, j+1 define X (n) recurively by X (n) = X (n) j + b(x (n) j )( (n) j Thi proce olve a ochaic inegral equaion X (n) = b (n) (, X (n) )d + ) + σ(x (n) j )(W W (n) ) j σ (n) (, X (n) )dw where he coefficien b (n) and σ (n) are ime inhomogeneou funcional on [, ) C[, ) b (n) (, y) = b(y(ϕ n ()) σ (n) (, y) = σ(y(ϕ n ())) We have defined he proce and he funcional in uch a way ha he map (ω, ) b (n) (, X (n) (ω)) (ω, ) σ (n) (, X (n) (ω)) are progreive, o he ochaic inegraion heory i well behaved. Tha aid, he echnical iue coming from working on funcion pace and he fac ha I have no dicued he ime-inhomogeneou maringale problem are he reaon why I am calling hi a kech. By eenially he ame argumen a above, X (n) problem for he ime-inhomogeneou family of operaor L (n) C c (R) mapping C (R) C(C([, ), R), R) defined by (L (n) f)(y) = b (n) (, y)f (y()) σ(n) (, y) 2 f (y()) olve he maringale wih domain D(L (n) ) = 1

11 Afer dealing wih ome echnical iue, we could have proven Theorem 6 in he ame eing, wih ime-inhomogeneou generaor (L f)(y) = b(y())f (y()) σ2 (y())f (y()) I am going o aume ha we have done o. Thi approach i dicued in deail in [3, Secion 5.4. We will how ighne by checking he hypohee of Kolmogorov condiion and appealing o he compac embedding of C α [, T ino C[, T coming from he Arzelà- Acoli heorem. ( E X (n) X (n) 4 8 E b (n) (u, X (n) u )du 4 + E σ (n) (u, X (n) u )dw u 4 ) Now, σ (n) (, X (n) ) i progreive and bounded, o σ(n) (u, X u (n) )dw u i a maringale. We can herefore apply he Burkholder-Davi-Gundy inequaliy [4, Theorem IV.4.1, o obain he upper bound E σ (n) (u, X (n) u )dw u 4 CE [ ( ) 2 σ (n) (u, X u (n) ) 2 du Combining hi wih a upremum bound on he fir erm, we find ha E X (n) X (n) 4 8( b C σ 4 ) 2 Rericing o a compac e [, T give he bound E X (n) X (n) 4 C T,b,σ 2. In paricular, he bound doe no depend on n. I follow [6, Corollary 1.2 ha for each γ < 1 4 here i a uniform conan c γ o ha E[ X (n) C γ [,T < c γ for all n. Arzelá-Acoli herefore give ighne of X (n) in C[, ). Fix f Cc (R) and y n C[, ) wih y n y uniformly on compac e. For any i follow from coninuiy of b and σ and uniform convergence on [, + 1 ha b (n) (, y n )f (y n ()) σ(n) (, y n ) 2 f (y n ()) b(y())f (y()) σ(y())2 f (y()) Fix a family of bounded coninuou funcion {h i } k i=1 and 1 < 2 < < k < k+1. For each n, we know ha [ ( E f(x (n) k+1 ) f(x (n) k ) k+1 k L (n) u f(x u (n) ) k )du i=1 h(x (n) i ) For any convergen ubequence, we may aume wihou lo of generaliy ha he convergence i almo ure ince C[, ) i Polih. We may ake limi in he above expecaion ince everyhing i uniformly bounded o find ha any ubequenial limi olve he maringale problem for L and i herefore a weak oluion o he SDE. 11 =

12 Uniquene and he Markov propery Reurning now o our original moivaion, we would like o argue ha in ome generaliy he oluion o a maringale problem hould have he Markov propery. Clearly hi canno hold in general: for any differenial operaor, any coninuou ochaic proce wih he correc iniial condiion i a oluion o he maringale problem wih domain given by he conan funcion. So long a he maringale problem i well-poed, however, he Markov propery hold. Again, we ae and prove he reul in nohing approaching i full generaliy, ee [2, Theorem Indeed, we only how he Markov propery, bu under he well-poedne hypohei, boh diribuional uniquene and he rong Markov propery hold. Propoiion 9. Suppoe ha L i an operaor on C (R) and uniquene hold for he he (L, D(L)) maringale problem in ha for any B B(R) and any wo coninuou oluion X and Y of he maringale problem wih he ame iniial condiion, we have for any > Then X aifie he Markov propery. P (X B) = P (Y B) Proof. Suppoe ha X i a oluion of he maringale problem wih filraion F and le >. Fix F F wih P (F ) >. For A F we define meaure on (Ω, F) by P 1 (A) = E [1 F P (A F ) P (F ) P 2 (A) = E [1 F P (A X ) P (F ) and a ochaic proce Y ( ) = X( + ). Then for any B B(R) we have becaue Y = X r i G r meaurable. X r. P 1 (Y B) = P (X r B F ) = P 2 (Y B) We would like o how ha Y olve he maringale problem wih iniial condiion Fix 1 < 2 < < n < n+1, f D(L) and h k B b (R) and define a funcional by n+1 η(y ) = f(y n+1 ) f(y n ) n Lf(X u )du n h k (Y k ) I i no hard o ee ha ha Y olve he maringale problem if and only if E[η(Y ) = for admiible choice above. k=1 Bu ince X already olve he (L, D(L)) maringale 12

13 problem, we find ha E[η(X + ) F X = and herefore he ame hold for he condiional expecaion given X. Therefore E P1 [η(y ) = = E P2 [η(y ). Now ince Y i a oluion o he maringale problem under P 1 and P 2 wih he ame iniial condiion, by hypohei, E P1 [h(y ) = E P2 [h(y ) for any bounded Borel funcion h. F F X i arbirary ubjec o P (F ) >, o we find ha E [ 1 F E[h(X + ) F X = E [1 F E[h(X + ) X which a verion of he Markov propery. If we wan hi o imply The ame proof ouline can be ued o how finie dimenional uniquene and auming ha oluion o he maringale problem exi wih iniial condiion x for any x R ha he rong Markov propery hold [2, Theorem Reference [1 A. Bovier Markov Procee: Lecure Noe Summer 212 Unpublihed lecure noe 212 [2 S. Ehier and T. Kurz. Markov Procee: Characerizaion and Convergence. John Wiley & Son Inc., 25. [3 I. Karaza and S.E. Shreve Brownian Moion and Sochaic Calculu Springer 1998 [4 D. Revuz and M. Yor Coninuou Maringale and Brownian Moion Springer- Verlag 1991 [5 D.W. Sroock and S.R.S. Varadhan. Mulidimenional Diffuion Procee. Springer-Verlag [6 J.B. Walh An Inroducion o Sochaic Parial Differenial Equaion Springer- Verlag École d Ée de Probabilié de Sain-Flour XIV