2π(t s) (3) B(t, ω) has independent increments, i.e., for any 0 t 1 <t 2 < <t n, the random variables

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1 2 Brownin Motion 2.1 Definition of Brownin Motion Let Ω,F,P) be probbility pce. A tochtic proce i meurble function Xt, ω) defined on the product pce [, ) Ω. In prticulr, ) for ech t, Xt, ) i rndom vrible, b) for ech ω, X,ω) i meurble function clled mple pth). For convenience, the rndom vrible Xt, ) will be written Xt) orx t. Thu tochtic proce Xt, ω) cn lo be expreed Xt)ω) or imply Xt) orx t. Definition A tochtic proce Bt, ω) i clled Brownin motion if it tifie the following condition: 1) P {ω ; B,ω)=} =1. 2) For ny <t, the rndom vrible Bt) B) i normlly ditributed with men nd vrince t, i.e., for ny <b, P { Bt) B) b } 1 b = e x2 /2t ) dx. 2πt ) 3) Bt, ω) h independent increment, i.e., for ny t 1 <t 2 < <t n, the rndom vrible Bt 1 ),Bt 2 ) Bt 1 ),...,Bt n ) Bt n 1 ), re independent. 4) Almot ll mple pth of Bt, ω) re continuou function, i.e., P { ω ; B,ω) i continuou } =1.

2 8 2 Brownin Motion In Remrk we mentioned tht the limit Bt) = lim Y δ δ, δ t) i Brownin motion. However, thi fct come only conequence of n intuitive obervtion. In the next chpter we will give everl contruction of Brownin motion. But before thee contruction we hll give ome imple propertie of Brownin motion nd define the Wiener integrl. A Brownin motion i ometime defined tochtic proce Bt, ω) tifying condition 1), 2), 3) in Definition Such tochtic proce lwy h continuou reliztion, i.e., there exit Ω uch tht P Ω )=1 nd for ny ω Ω,Bt, ω) i continuou function of t. Thi fct cn be eily checked by pplying the Kolmogorov continuity theorem in Section 3.3. Thu condition 4) i utomticlly tified. The Brownin motion Bt) in the bove definition trt t. Sometime we will need Brownin motion trting t x. Such proce i given by x + Bt). If the trting point i not, we will explicitly mention the trting point x. 2.2 Simple Propertie of Brownin Motion Let Bt) be fixed Brownin motion. We give below ome imple propertie tht follow directly from the definition of Brownin motion. Propoition For ny t>, Bt) i normlly ditributed with men nd vrince t. For ny, t, we hve E[B)Bt)] = min{, t}. Remrk Regrding Definition 2.1.1, it cn be proved tht condition 2) nd E[B)Bt)] = min{, t} imply condition 3). Proof. By condition 1), we hve Bt) =Bt) B) nd o the firt ertion follow from condition 2). To how tht EB)Bt) = min{, t} we my ume tht <t. Then by condition 2) nd 3), E [ B)Bt) ] = E [ B) Bt) B) ) + B) 2] =+ =, which i equl to min{, t}. Propoition Trnltion invrince) For fixed t, the tochtic proce Bt) =Bt + t ) Bt ) i lo Brownin motion. Proof. The tochtic proce Bt) obviouly tifie condition 1) nd 4) of Brownin motion. For ny <t, Bt) B) =Bt + t ) B + t ) )

3 2.3 Wiener Integrl 9 By condition 2) of Bt), we ee tht Bt) B) i normlly ditributed with men nd vrince t + t ) + t )=t. Thu Bt) tifie condition 2). To check condition 3) for Bt), we my ume tht t >. Then for ny t 1 <t 2 < <t n,wehve<t t 1 + t < <t n + t. Hence by condition 3) of Bt), Bt k + t ) Bt k 1 + t ),k=1, 2,...,n, re independent rndom vrible. Thu by Eqution 2.2.1), the rndom vrible Bt k ) Bt k 1 ),k=1, 2,...,n,re independent nd o Bt) tifie condition 3) of Brownin motion. The bove trnltion invrince property y tht Brownin motion trt freh t ny moment new Brownin motion. Propoition Scling invrince) For ny rel number λ >, the tochtic proce Bt) =Bλt)/ λ i lo Brownin motion. Proof. Condition 1), 3), nd 4) of Brownin motion cn be redily checked for the tochtic proce Bt). To check condition 2), note tht for ny <t, Bt) B) = 1 λ Bλt) Bλ) ), which how tht Bt) B) i normlly ditributed with men nd vrince 1 λ λt λ) =t. Hence Bt) tifie condition 2). It follow from the cling invrince property tht for ny λ>nd t 1 <t 2 < <t n the rndom vector Bλt1 ),Bλt 2 ),...,Bλt n ) ), λbt1 ), λbt 2 ),..., λbt n ) ) hve the me ditribution. 2.3 Wiener Integrl In Section 1.1 we ried the quetion of defining the integrl ft) dgt). We ee from Exmple tht in generl thi integrl cnnot be defined Riemnn Stieltje integrl. Now let u conider the following integrl: ft) dbt, ω), where f i determinitic function i.e., it doe not depend on ω) nd Bt, ω) i Brownin motion. Suppoe for ech ω Ω we wnt to ue Eqution 1.1.2) to define thi integrl in the Riemnn Stieltje ene by ] b RS) ft) dbt, ω) =ft)bt, ω) RS) Bt, ω) df t) )

4 1 2 Brownin Motion Then the cl of function ft) for which the integrl RS) ft) dbt, ω) i defined for ech ω Ω i rther limited, i.e., ft) need to be continuou function of bounded vrition. Hence for continuou function of unbounded vrition uch ft) =t in 1 t, <t 1, nd f) =, we cnnot ue Eqution 2.3.1) to define the integrl 1 ft) dbt, ω) for ech ω Ω. We need different ide in order to define the integrl ft) dbt, ω) for wider cl of function ft). Thi new integrl, clled the Wiener integrl of f, i defined for ll function f L 2 [, b]. Here L 2 [, b] denote the Hilbert pce of ll rel-vlued qure integrble function on [, b]. For exmple, 1 t in 1 t dbt) i Wiener integrl. Now we define the Wiener integrl in two tep: Step 1. Suppoe f i tep function given by f = n i=1 i 1 [ti 1,t i), where t = nd t n = b. In thi ce, define If) = n i Bti ) Bt i 1 ) ) ) i=1 Obviouly, If + bg) =If)+bIg) for ny, b R nd tep function f nd g. Moreover, we hve the following lemm. Lemm For tep function f, the rndom vrible If) i Guin with men nd vrince E If) 2) = ft) 2 dt ) Proof. It i well known tht liner combintion of independent Guin rndom vrible i lo Guin rndom vrible. Hence by condition 2) nd 3) of Brownin motion, the rndom vrible If) defined by Eqution 2.3.2) i Guin with men. To check Eqution 2.3.3), note tht E If) 2) = E n i j Bti ) Bt i 1 ) ) Bt j ) Bt j 1 ) ). i,j=1 By condition 2) nd 3) of Brownin motion, nd for i j, E Bt i ) Bt i 1 ) ) 2 = ti t i 1, E Bt i ) Bt i 1 ) ) Bt j ) Bt j 1 ) ) =. Therefore, E If) 2) n = 2 i t i t i 1 )= i=1 ft) 2 dt.

5 2.3 Wiener Integrl 11 Step 2. We will ue L 2 Ω) to denote the Hilbert pce of qure integrble rel-vlued rndom vrible on Ω with inner product X, Y = EXY ). Let f L 2 [, b]. Chooe equence {f n } of tep function uch tht f n f in L 2 [, b]. By Lemm the equence {If n )} i Cuchy in L 2 Ω). Hence it converge in L 2 Ω). Define If) = lim n), n in L 2 Ω) ) Quetion I If) well-defined? In order for If) to be well-defined, we need to how tht the limit in Eqution 2.3.4) i independent of the choice of the equence {f n }. Suppoe {g m } i nother uch equence, i.e., the g m re tep function nd g m f in L 2 [, b]. Then by the linerity of the mpping I nd Eqution 2.3.3), E If n ) Ig m ) 2) = E If n g m ) 2) = fn t) g m t) ) 2 dt. Write f n t) g m t) = [ f n t) ft) ] [ g m t) ft) ] nd then ue the inequlity x y) 2 2x 2 + y 2 )toget fn t) g m t) ) 2 [fn dt 2 t) ft) ] 2 [ + gm t) ft) ] ) 2 dt, n, m. It follow tht lim n If n ) = lim m Ig m )inl 2 Ω). Thi how tht If) i well-defined. Definition Let f L 2 [, b]. The limit If) defined in Eqution 2.3.4) i clled the Wiener integrl of f. The Wiener integrl If) off will be denoted by ) If)ω) = ft) dbt) ω), ω Ω, lmot urely. For implicity, it will be denoted by ft) dbt) or ft) dbt, ω). Note tht the mpping I i liner on L 2 [, b]. Theorem For ech f L 2 [, b], the Wiener integrl ft) dbt) i Guin rndom vrible with men nd vrince f 2 = ft)2 dt. Proof. By Lemm 2.3.1, the ertion i true when f i tep function. For generl f L 2 [, b], the ertion follow from the following well-known fct: If X n i Guin with men μ n nd vrince σn 2 nd X n converge to X in L 2 Ω), then X i Guin with men μ = lim n μ n nd vrince σ 2 = lim n σn. 2

6 12 2 Brownin Motion Thu the Wiener integrl I : L 2 [, b] L 2 Ω) i n iometry. In fct, it preerve the inner product, hown by the next corollry. Corollry If f,g L 2 [, b], then E If) Ig) ) = ft)gt) dt ) In prticulr, if f nd g re orthogonl, then the Guin rndom vrible If) nd Ig) re independent. Proof. By the linerity of I nd Theorem we hve E [ If)+Ig)) 2] = E [ If + g)) 2] ) 2 = ft)+gt) dt = ft) 2 dt +2 ft)gt) dt + On the other hnd, we cn lo ue Theorem to obtin E [ If)+Ig)) 2] = E [ If) 2 +2If)Ig)+Ig) 2] = ft) 2 dt +2E [ If)Ig) ] + gt) 2 dt ) gt) 2 dt ) Obviouly, Eqution 2.3.5) follow from Eqution 2.3.6) nd 2.3.7). Exmple The Wiener integrl 1 db) i Guin rndom vrible with men nd vrince 1 2 d = 1 3. Theorem Let f be continuou function of bounded vrition. Then for lmot ll ω Ω, ) ft) dbt) ω) =RS) ft) dbt, ω), where the left-hnd ide i the Wiener integrl of f nd the right-hnd ide i the Riemnn Stieltje integrl of f defined by Eqution 2.3.1). Proof. For ech prtition Δ n = {t,t 1,...,t n 1,t n } of [, b], we define tep function f n by n f n = ft i 1 )1 [ti 1,t i). i=1 Note tht f n converge to f in L 2 [, b] n, i.e., Δ n. Hence by the definition of the Wiener integrl in Eqution 2.3.4),

7 ft) dbt) = lim n i=1 2.3 Wiener Integrl 13 n ft i 1 ) Bt i ) Bt i 1 ) ), in L 2 Ω) ) On the other hnd, by Eqution 2.3.1), the following limit hold for ech ω Ω for ome Ω with P Ω )=1, RS) ft) dbt, ω) = fb)bb, ω) f)b, ω) lim n i=1 = lim fb)bb, ω) f)b, ω) n n Bt i,ω) ft i ) ft i 1 ) ) n Bt i,ω) ft i ) ft i 1 ) )), which, fter regrouping the term, yield the following equlity for ech ω in Ω : RS) ft) dbt, ω) = lim n i=1 i=1 n ft i 1 ) Bt i ) Bt i 1 ) ) ) Since L 2 Ω)-convergence implie the exitence of ubequence converging lmot urely, we cn pick uch ubequence of {f n } to get the concluion of the theorem from Eqution 2.3.8) nd 2.3.9). Exmple Conider the Riemnn integrl 1 Bt, ω) dt defined for ech ω Ω for ome Ω with P Ω ) = 1. Let u find the ditribution of thi rndom vrible. Ue the integrtion by prt formul to get 1 ] 1 1 Bt, ω) dt = Bt, ω)t 1) t 1) dbt, ω) =RS) 1 1 t) dbt, ω). Hence by Theorem we ee tht for lmot ll ω Ω, 1 1 ) Bt, ω) dt = 1 t) dbt) ω), where the right-hnd ide i Wiener integrl. Thu 1 Bt) dt nd the Wiener integrl 1 1 t) dbt) hve the me ditribution, which i eily een to be Guin with men nd vrince E 1 t) dbt)) = 1 t) 2 dt = 1 3.

8 14 2 Brownin Motion 2.4 Conditionl Expecttion In thi ection we explin the concept of conditionl expecttion, which will be needed in the next ection nd other plce. Let Ω,F,P) be fixed probbility pce. For 1 p<, we will ue L p Ω) to denote the pce of ll rndom vrible X with E X p ) <. It i Bnch pce with norm X p = E X p)) 1/p. In prticulr, L 2 Ω) i the Hilbert pce ued in Section 2.3. In thi ection we ue the pce L 1 Ω) with norm given by X 1 = E X. Sometime we will write L 1 Ω,F) when we wnt to emphize the σ-field F. Suppoe we hve nother σ-field G F. Let X be rndom vrible with E X <, i.e., X L 1 Ω). Define rel-vlued function μ on G by μa) = Xω) dp ω), A G ) A Note tht μa) A X dp X dp = E X for ll A G. Moreover, Ω the function μ tifie the following condition: ) μ ) =; b) μ ) n 1 A n = n 1 μa n) for ny dijoint et A n G,, 2,...; c) If P A) =nda G, then μa) =. A function μ : G Rtifying condition ) nd b) i clled igned meure on Ω,G). A igned meure μ i id to be bolutely continuou with repect to P if it tifie condition c). Therefore, the function μ defined in Eqution 2.4.1) i igned meure on Ω,G) nd i bolutely continuou with repect to P. Apply the Rdon Nikodym theorem ee, e.g., the book by Royden [73]) to the igned meure μ defined in Eqution 2.4.1) to get G-meurble rndom vrible Y with E Y < uch tht μa) = Y ω) dp ω), A G ) A Suppoe Ỹ i nother uch rndom vrible, nmely, it i G-meurble with E Ỹ < nd tifie μa) = Ỹ ω) dp ω), A G ) A Then by Eqution 2.4.2) nd 2.4.3), we hve Y Ỹ ) dp = for ll A A G. Thi implie tht Y = Ỹ lmot urely. The bove dicuion how the exitence nd uniquene of the conditionl expecttion in the next definition.

9 2.4 Conditionl Expecttion 15 Definition Let X L 1 Ω,F). Suppoe G i σ-field nd G F. The conditionl expecttion of X given G i defined to be the unique rndom vrible Y up to P -meure 1) tifying the following condition: 1) Y i G-meurble; 2) A XdP = YdPfor ll A G. A We will freely ue E[X G], EX G), or E{X G} to denote the conditionl expecttion of X given G. Notice tht the G-meurbility in condition 1) i crucil requirement. Otherwie, we could tke Y = X to tify condition 2), nd the bove definition would not be o meningful. The conditionl expecttion E[X G] cn be interpreted the bet gue of the vlue of X bed on the informtion provided by G. Exmple Suppoe G = {,Ω}. Let X be rndom vrible in L 1 Ω) nd let Y = E[X G]. Since Y i G-meurble, it mut be contnt, y Y = c. Then ue condition 2) in Definition with A = Ω to get XdP = YdP= c. Ω Hence c = EX nd we hve E[X G] =EX. Thi concluion i intuitively obviou. Since the σ-field G = {,Ω} provide no informtion, the bet gue of the vlue of X i it expecttion. Exmple Suppoe Ω = n A n i dijoint union finite or countble) with P A n ) > for ech n. Let G = σ{a 1,A 2,...}, the σ-field generted by the A n. Let X L 1 Ω) nd Y = E[X G]. Since Y i G-meurble, it mut be contnt, y c n,ona n for ech n. Ue condition 2) in Definition with A = A n to how tht c n = P A n ) 1 A n XdP. Therefore, E[X G] i given by E[X G] = ) 1 XdP 1 An, P A n n ) A n where 1 An denote the chrcteritic function of A n. Exmple Let Z be dicrete rndom vrible tking vlue 1, 2,... finite or countble). Let σ{z} be the σ-field generted by Z. Then σ{z} = σ{a 1,A 2,...}, where A n = {Z = n }. Let X L 1 Ω). We cn ue Exmple to obtin E [ X σ{z} ] = ) 1 XdP 1 An, P A n n ) A n which cn be rewritten E [ X σ{z} ] = θz) with the function θ defined by 1 XdP, if x = n,n 1; θx) = P Z = n ) Z= n, if x/ { 1, 2,...}. Ω

10 16 2 Brownin Motion Note tht the conditionl expecttion E[X G] i rndom vrible, while the expecttion EX i rel number. Below we lit everl propertie of conditionl expecttion nd leve mot of the proof exercie t the end of thi chpter. Recll tht Ω,F,P) i fixed probbility pce. The rndom vrible X below i umed to be in L 1 Ω,F) nd G i ub-σ-field of F, nmely, G i σ-field nd G F. All equlitie nd inequlitie below hold lmot urely. 1. E E[X G] ) = EX. Remrk: Hence the conditionl expecttion E[X G] nd X hve the me expecttion. When written in the form EX = E E[X G] ), the equlity i often referred to computing expecttion by conditioning. To prove thi equlity, imply put A = Ω in condition 2) of Definition If X i G-meurble, then E[X G] =X. 3. If X nd G re independent, then E[X G] =EX. Remrk: Here X nd G being independent men tht {X U} nd A re independent event for ny Borel ubet U of R nd A G,or equivlently, the event {X x} nd A re independent for ny x R nd A G. 4. If Y i G-meurble nd E XY <, then E[XY G] =YE[X G]. 5. If H i ub-σ-field of G, then E[X H] =E [ E[X G] H ]. Remrk: Thi property i ueful when X i product of rndom vrible. In tht ce, in order to find E[X H], we cn ue ome fctor in X to chooe uitble σ-field G between H nd F nd then pply thi property. 6. If X, Y L 1 Ω) nd X Y, then E[X G] E[Y G]. 7. E[X G] E[ X G]. Remrk: For the proof, let X + = mx{x, } nd X = min{x, } be the poitive nd negtive prt of X, repectively. Then pply Property 6 to X + nd X. 8. E[X + by G] =E[X G]+bE[Y G],, b R nd X, Y L 1 Ω). Remrk: By Propertie 7 nd 8, the conditionl expecttion E[ G] i bounded liner opertor from L 1 Ω,F) intol 1 Ω,G) 9. Conditionl Ftou lemm) Let X n, X n L 1 Ω),, 2,..., nd ume tht lim inf n X n L 1 Ω). Then [ ] E lim inf X n G lim inf E[X n G]. n n 1. Conditionl monotone convergence theorem) Let X 1 X 2 X n nd ume tht X = lim n X n L 1 Ω). Then E[X G] = lim n E[X n G].

11 2.5 Mrtingle Conditionl Lebegue dominted convergence theorem) Aume tht X n Y, Y L 1 Ω), nd X = lim n X n exit lmot urely. Then E[X G] = lim n E[X n G]. 12. Conditionl Jenen inequlity) Let X L 1 Ω). Suppoe φ i convex function on R nd φx) L 1 Ω). Then φ E[X G] ) E[φX) G]. 2.5 Mrtingle Let f L 2 [, b] nd conider the tochtic proce defined by M t = t f) db), t b ) We will how tht M t i mrtingle. But firt we review the concept of the mrtingle. Let T be either n intervl in R or the et of poitive integer. Definition A filtrtion on T i n increing fmily {F t t T } of σ-field. A tochtic proce X t,t T, i id to be dpted to {F t t T } if for ech t, the rndom vrible X t i F t -meurble. Remrk A σ-field F i clled complete if A F nd P A) = imply tht B F for ny ubet B of A. We will lwy ume tht ll σ-field F t re complete. Definition Let X t be tochtic proce dpted to filtrtion {F t } nd E X t < for ll t T. Then X t i clled mrtingle with repect to {F t } if for ny t in T, E{X t F } = X,.. lmot urely) ) In ce the filtrtion i not explicitly pecified, then the filtrtion {F t } i undertood to be the one given by F t = σ{x ; t}. The concept of the mrtingle i generliztion of the equence of prtil um riing from equence {X n } of independent nd identiclly ditributed rndom vrible with men. Let S n = X X n. Then the equence {S n } i mrtingle. Submrtingle nd upermrtingle re defined by replcing the equlity in Eqution 2.5.2) with nd, repectively, i.e., for ny t in T, E{X t F } X,.. ubmrtingle), E{X t F } X,.. upermrtingle).

12 18 2 Brownin Motion Let {X n } be equence of independent nd identiclly ditributed rndom vrible with finite expecttion nd let S n = X X n. Then {S n } i ubmrtingle if EX 1 nd upermrtingle if EX 1. A Brownin motion Bt) i mrtingle. To ee thi fct, let Then for ny t, F t = σ{b); t}. E{Bt) F } = E{Bt) B) F } + E{B) F }. Since Bt) B) i independent of F, we hve E{Bt) B) F } = E{Bt) B)}. But EBt) = for ny t. Hence E{Bt) B) F } =.On the other hnd, E{B) F } = B) becue B) if -meurble. Thu E{Bt) F } = B) for ny t nd thi how tht Bt) i mrtingle. In fct, it i the mot bic mrtingle tochtic proce with time prmeter in n intervl. Now we return to the tochtic proce M t defined in Eqution 2.5.1) nd how tht it i mrtingle in the next theorem. Theorem Let f L 2 [, b]. Then the tochtic proce M t = t f) db), t b, i mrtingle with repect to F t = σ{b); t}. Proof. Firt we need to how tht E M t < for ll t [, b] in order to tke the conditionl expecttion of M t. Apply Theorem to get E M t 2) = t f) 2 d f) 2 d. Hence E M t { E M t 2)} 1/2 <. Next we need to prove tht E{Mt F } = M.. for ny t. But M t = M + t fu) dbu) nd M i F -meurble. Hence { t E{M t F } = M + E fu) dbu) F }. Thu it uffice to how tht for ny t, { t } E fu) dbu) F = )

13 2.5 Mrtingle 19 Firt uppoe f i tep function f = n i=1 i 1 [ti 1,t i), where t = nd t n = t. In thi ce, we hve t fu) dbu) = n i Bti ) Bt i 1 ) ). i=1 But Bt i ) Bt i 1 ),i=1,...,n, re ll independent of the σ-field F. Hence E{Bt i ) Bt i 1 ) F } = for ll i nd o Eqution 2.5.3) hold. Next uppoe f L 2 [, b]. Chooe equence {f n } of tep function converging to f in L 2 [, b]. Then by the conditionl Jenen inequlity with φx) =x 2 in Section 2.4 we hve the inequlity which implie tht { t E E{X F} 2 E{X 2 F}, fn u) fu) ) dbu) F } 2 { t E fn u) fu) ) ) 2 } dbu) F. Next we ue the property E E{X F} ) = EX of conditionl expecttion nd then pply Theorem to get { t E E fn u) fu) ) } 2 t dbu) F fn u) fu) ) 2 du, fn u) fu) ) 2 du n. Hence the equence E{ t f nu) dbu) F } of rndom vrible converge to E{ t fu) dbu) F } in L 2 Ω). Note tht the convergence of equence in L 2 Ω) implie convergence in probbility, which implie the exitence of ubequence converging lmot urely. Hence by chooing ubequence if necery, we cn conclude tht with probbility 1, { t } { t lim E f n u) dbu) F = E fu) dbu) F } ) n Now E { t f nu) dbu) } F = ince we hve lredy hown tht Eqution 2.5.3) hold for tep function. Hence by Eqution 2.5.4), { t } E fu) dbu) F =, nd o Eqution 2.5.3) hold for ny f L 2 [, b].

14 2 2 Brownin Motion 2.6 Serie Expnion of Wiener Integrl Let {φ n } be n orthonorml bi for the Hilbert pce L 2 [, b]. Ech f L 2 [, b] h the following expnion: f = f,φ n φ n, 2.6.1) where, i the inner product on L 2 [, b] given by f,g = ft)gt) dt. Moreover, we hve the Prevl identity f 2 = f,φ n ) Tke the Wiener integrl in both ide of Eqution 2.6.1) nd informlly interchnge the order of integrtion nd ummtion to get ft) dbt) = f,φ n φ n t) dbt) ) Quetion Doe the rndom erie in the right-hnd ide converge to the left-hnd ide nd in wht ene? Firt oberve tht by Theorem nd the remrk following Eqution 2.3.5), the rndom vrible φ nt) dbt),n 1, re independent nd hve the Guin ditribution with men nd vrince 1. Thu the right-hnd ide of Eqution 2.6.3) i rndom erie of independent nd identiclly ditributed rndom vrible. By the Lévy equivlence theorem [1] [37] thi rndom erie converge lmot urely if nd only if it converge in probbility nd, in turn, if nd only if it converge in ditribution. On the other hnd, we cn eily check the L 2 Ω) convergence of thi rndom erie follow. Apply Eqution 2.3.5) nd 2.6.2) to how tht E = = ft) dbt), N f,φ n ) 2 φ n t) dbt) N N ft) 2 dt 2 f,φ n 2 + f,φ n 2 ft) 2 dt N f,φ n 2 N. Hence the rndom erie in Eqution 2.6.3) converge in L 2 Ω)to the rndom vrible in the left-hnd ide of Eqution 2.6.3). But the L 2 Ω) convergence implie convergence in probbility. Therefore we hve proved the next theorem for the erie expnion of the Wiener integrl.

15 Exercie 21 Theorem Let {φ n } be n orthonorml bi for L 2 [, b]. Then for ech f L 2 [, b], the Wiener integrl of f h the erie expnion ft) dbt) = f,φ n φ n t) dbt), with probbility 1, where the rndom erie converge lmot urely. In prticulr, pply the theorem to =,b= 1, nd f =1 [,t), t 1. Then 1 f) db) =Bt) nd we hve the rndom erie expnion, t ) 1 ) Bt, ω) = φ n ) d φ n ) db, ω). Note tht the vrible t nd ω re eprted in the right-hnd ide. In view of thi expnion, we expect tht Bt) cn be repreented by Bt, ω) = ξ n ω) t φ n ) d, where {ξ n } i equence of independent rndom vrible hving the me Guin ditribution with men nd vrince 1. Thi method of defining Brownin motion h been tudied in [29] [41] [67]. Exercie 1. Let Bt) be Brownin motion. Show tht E B) Bt) 4 =3 t Show tht the mrginl ditribution of Brownin motion Bt) t time <t 1 <t 2 < t n i given by P {Bt 1 ) 1,Bt 2 ) 2,...,Bt n ) n } = 1 2π)n t 1 t 2 t 1 ) t n t n 1 ) n 1 [ exp 1 x x 2 x 1 ) x n x n 1 ) 2 )] dx 1 dx 2 dx n. 2 t 1 t 2 t 1 t n t n 1 3. Let Bt) be Brownin motion. For fixed t nd, find the ditribution function of the rndom vrible X = Bt)+ B). 4. Let Bt) be Brownin motion nd let < t u v. Show tht the rndom vrible 1 t Bt) 1 B) nd Bu)+bBv) re independent for ny, b R.

16 22 2 Brownin Motion 5. Let Bt) be Brownin motion nd let < t u v. Show tht the rndom vrible B)+bBt) nd 1 v Bv) 1 ubu) re independent for ny, b R tifying the condition + bt =. 6. Let Bt) be Brownin motion. Show tht lim t + tb1/t) = lmot urely. Define W ) = nd W t) =tb1/t) for t>. Prove tht W t) i Brownin motion. 7. Let Bt) be Brownin motion. Find ll contnt nd b uch tht Xt) = t ) + b u t dbu) i lo Brownin motion. 8. Let Bt) be Brownin motion. Find ll contnt, b, nd c uch tht + b u t + c u2 t 2 ) dbu) i lo Brownin motion. Xt) = t 9. Let Bt) be Brownin motion. Show tht for ny integer n 1, there exit nonzero contnt, 1,..., n uch tht Xt) = t 2 u 2 t n u n t n ) dbu) i lo Brownin motion. + 1 u t + 1. Let Bt) be Brownin motion. Show tht both Xt) = t 2t u) dbu) nd Y t) = t 3t 4u) dbu) re Guin procee with men function nd the me covrince function 3 2 t for t. 11. Let Bt) =B 1 t),...,b n t)) be n R n -vlued Brownin motion. Find the denity function of Rt) = Bt) nd St) = Bt) For ech n 1, let X n be Guin rndom vrible with men μ n nd vrince σn. 2 Suppoe the equence X n converge to X in L 2 Ω). Show tht the limit μ = lim n μ n nd σ 2 = lim n σn 2 exit nd tht X i Guin rndom vrible with men μ nd vrince σ Let fx, y) be the joint denity function of rndom vrible X nd Y. The mrginl denity function of Y i given by f Y y) = fx, y) dx. The conditionl denity function of X given Y = y i defined by f X Y x y) = fx, y)/f Y y). The conditionl expecttion of X given Y = y i defined by E[X Y = y] = xf X Y x y) dx. Let σy )betheσ-field generted by Y. Prove tht E[X σy )] = θy ), where θ i the function θy) =E[X Y = y]. 14. Prove the propertie of conditionl expecttion lited in Section Let Bt) be Brownin motion. Find the ditribution of t et db). Check whether X t = t et db) i mrtingle. 16. Let Bt) be Brownin motion. Find the ditribution of t B) d. Check whether Y t = t B) d i mrtingle. 17. Let Bt) be Brownin motion. Find the ditribution of the integrl t B) cot ) d. 18. Let Bt) be Brownin motion. Show tht X t = 1 3 Bt)3 t mrtingle. B) d i

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