DSpace at VNU: Some remarks on the finite-time behavior of Wiener paths

DSpace at VNU: Some remarks on the finite-time behavior of Wiener paths tài liệu, giáo án, bài giảng , luận văn, luận án...

VNU JOURNAL OF SCIENCE, M athem atics - Physics T X V III, N()3 - 2002 SO M E R EM A R K S ON B E H A V IO R THE F IN IT E -T IM E O F W IE N E R PA TH S D ang P h u o c H uy D e p a rtm e n t o f M a th e m a tic s, U n iv ersity o f I)a L a t A b s tr a c t W e establish, so m e pro p erties o f the fin ite -tim e behavior o f W iener paths S o m e applications o f these results are also given Keywords: W iener’s measure, stopping-tim e, W iener scaling invariance I n tr o d u c tio n T hroughout this note, by 93(R;V) we shall denote the Polish space of all continuous paths $ : [0 ,oo) — > R*v , and let M i(© (R iV)) be the space of Borel probability m easures on 23(IR'V)( see, for example, (1, Section 1]) Define, for each X £ R ‘v , the transform ation Tx :® ( R * ) — >) = r y *W(rf«&) \ l ) /(« (O jw ïïi# ) [ /(ybi(A,)( y - x - z ) d y , this completes the proof □ We need the following well known notions Define, for each Í € [0,oo), the coordinate projections 7Tt : S8 (R /V) — >• R'v by 7rt $ = * (t), (2.8) S o m e r e m a r k s ori the f i n i t e - t i m e b e h a vio r of 21 and let, = < *{*3 • s € [0 , i]), t € [0 ,oo) (2.9) be the ơ-algebra over Í8 (R'V) generated by all m aps 7TS, s € Ị(M] Given a {© fr : £ £ [0 , oo)}-stopping tim e T, we will use the notation = {.4 ầ đ (R 'V) : A n { r < t } [0, +oo] is a 33^ m easurable fu n c tio n T hen, f o r each f € J3(IRjV; /{), X € R A? a n d h € C (R ^ ;R ^ ) , we have (H ere we use i '( r ) in place o f ÿ ( r ( i ' ) ) ) Proof Define, from the above assum ptions, the function I I : Q3(R/V) X *B(RiV) — » R by / / ( $ , # ) = Z|0.o o)(t(*)) -IỊ0.OO) (*?(*)) • F (< J> )/(*(r,()) + /1 ( < % ( $ ) ) ) ) , where I A denotes the characteristic function of a set A T hen H is 93^ X 93(d¥) Ĩ B Ị $ ( i ( í ) ) j w ỉ [ i )w j w < w)( '» ) Corollary 2.4 is thus completely proved □ C o ro lla ry 2.5 L et R b e a n y o rth o g o n a l m a tr ix o f order /V, d e fin e th e tra n sfo rm a tio n h : R'v — »by /i(y) — R (y — R y ) U nder th e a ssu m p tio n s o f T h e o re m 2.3, we have F { ) J [ # ( t + 7/) + R t Ơ ( t ) - đ (r ) ) W X { N ) {( M) [ J ( tị< oo ) V / = / 7(r, It is easy to see (see (2.8)) th a t 7rt~ l (fí/v(a; r)) = { í G QÍ(Rn ) : I # ( - a |< r} e by Identify 'ĩ' € 53(K'V) ■( * ! ,• • • , N ) e ( » ( « ) ) w *(«) = ( # 1(s ), - ,< M « ) ) r , s G [0,oo), (see, for e x a m p le , [l, p 16], [2, P-179Ị) T h e n , for e a c h X = (x*i, • • • , T/v)7 £ s a tis f y in g condition rii= i ^ 7^ 0> putting r = \ / n for any £ > 0, by the independence of the coordinates under W x%' (see [2, Exercise 3.3.28]) we have, for any fixed t € [0,oo), W jW f { * € ® (R'V) : I 4»(0 |< y /V e } ^ > W ^ N) ^ { * « ( R * ) : I ỵ ' j ( i ) | < e; f o r I < j =n < N}^J ({*€*(/*): I 0(0 |< t}) Next, taking a , = x ” 2, < i < N , use (2.4) and (2.5) to see th at f j W Xj ( V € © (* ) : I 4>(t) |< c} j = n w ssni< ( { ậ € ® (/ỉ) : I ự.(x,-2í) = n Wi(V € |< 11 r }) : I xô^*r20 1^ el) ã Hence, we obtain the following inequality v \4 " > ( { * e © ( R w ) : I * ( | < n/7 v e } ^ > f ] w , ({ G * ( / ỉ ) : I X, ự>(x,-2 í) |< t } ) , for any c > 0, ỉ G [0,oo), and X is given in the above R e m a rk If p u ttin g Ai = { ộ € 33(/ỉ) : I Xj ộ ( x ~ 2t) |< e}, < i < Ny then (see (1.3)) ííw c * > -< v (íụ ) t=l \= = < } ,1)T( { * € ®(R/V) : Ix**/°r ^ i ^ A'})Thus, we have W r } , e © (R * ) (3.2) T hen is a {®;v : t € [0, oo)}-stopping time In this example, we will show th a t the W iener p ath s satisfy the following properties (a ) F or each X € B \ ( o ; r ) (see the n o ta tio n (3 )) a n d T > 0, then - ị ) + to r) Jio) O O B ^ Ĩ - ị ) + far) (3.3) and j 0) cos^ "*■ = e- ^ I*\irther, ( b) f o r a n y c > a n d X € #/v(0; c os ^ ■~ + > (3-4) /o r e v e ry f c ỗ Z w * v :} > T ) then > e" ^ ^ cos" - ) ) ( -5 ) and (4 ^ > T) > e -^ ^ c o s w^ i ệ l ) (3.6) In order to prove these assertions,we proceed in several steps Step L Using T h e same techniques of the proof of Theorem 7.2.4 in [2] with respect to the function / ( i , x ) = e ^T 'r* cos^7T ^- ” ) + w2 t f TT X resp ỡ(£,:r) = c * cosi e [0’°°) x Æ \ n )' ^ [0>°°) x ft we see th a t, th e assertion (3.3) [resp (3.4)1 is obtained from the fact th a t /( iA r > r\ 7T resp #(£ A T>r\ 7rtA e - £ ^ Ỵ ị c o s ^ — S te p Next, for any fixed X (E #/v (0; p u ttin g y = ••• I Then, there exists a R otation Ĩ Ỉ (relative to R ) on 23 (R ^ ) in which the orthogonal m atrix R satisfied D ang Phuoc H u y 32 the condition R y = X T hus, by (2.5), we conclude from (3.11) th a t = n * W yV) ( j i v W i N) = W ịN) > : T ^ i n v ) > T } ' JS = w W { * rrg0^ ) > r} i > e- i ' ^ c o s ^ l i i - ì ) ) Finally, by the sam e argum ent as above, we also obtain (3.6) and this exam ple is completely established R e m a r k We use E F [ X , A ] to denote the expected value under p of X over the set A Taking X = I in (3.3) and thereby obtain L „ T) “ » ( * ( ^ - i ) + fe rjW j(i® ) = ( - l ) V Í i Thus, E w * COS ^7T + kTr'j , r ị lJ > T = ( —l ) fc C0S^7T^~ ~ ) + COS ^7T ^ ~ ) + ^c7r) ’ > for any r > 0, 7’ > 0, k £ z and X € # i( ;r ) Similarly, by (3.4) (w ith Ew COS X f I •— + = 0), we have ẮC7T j , r l lJ > T = ( - l ) fc° ° s f ■“ + fc7r) £'VV co's ( for any r > T > 0, k £ z an d X ~ + fc7r) ’ T- r‘ > T € íỉi( ;r ) R e m a r k One could define th e sub sets of Ổ ạt(0; ^7= ) under w hich we will obtain the b e tte r estim ates than the inequalities of (3.5) and (3.6) Namely, letting X Ç ft/v (0; for each e > , then S o m e r e m a r k s o n th e f i n i t e - t i m e b e h a v io r o f 33 This implies th at, *(*(& x g M 2) ) - - l > ^ r ;^ ) X e ỉ f ij v ( ; §), (3.12) I N = 1, 2, where ổtì/v(0; I ) is the boundary of /í/V (0; I ) and Co s ( f M ) = l, , , « ISO (3.13) Then, it shows from (3.5)(3.12) th a t W V) / g o > > e~ * ị for TV = 1,2,3; € > 0, T > and X € Ó/Ỉ/V (0; I ) Similarly, by (3.6)(3.13), we o b tain the certain result of T heorem 7.2.4 in [2], yy(N) ^ r (W) > > e - - JS£ L t t > , T > E x a m p le 3.3 Let us consider two {23^ : t € [0,0 )} -stopping tim es as follows ơ(>ỉ) = inf { s > : I 'P(.s) |< ^ } (3.14) £t and r ( ^ ) = iiif1 > : I #(.v) |> r} , for each r > and every í* E Q3(R;V) 77 : ( r < + 00 ) — > [0, + 00 ] by Let t € (0 , 00 ) (3.15) be fixed an d X G Byv(0;r) Define 7/($) = ( t - r ( i ) ) v (3.16) Then 77 is a © ^-m easure function Furtherm ore, taking ;4 = ( r < £), it is obvious th a t A c (r] < 00 ) T he following n otations will be used: R tì = { R y : y G Ỡ } and /3 + z = { y -f z : y G / ĩ } , z € R 'v , B € ©RAT In this exam ple, we shall prove th a t the W iener p a th s satisfy th e following properties (a) F or a n y orthogonal m a tr ix R be g iv e n , a n d every B G w w € ® ( R W) : r(vt') < t, f(T)) + $ ( r ) |) , J / (3.17) D ang Phuoc H u y 34 € « ( R * ) : r { < b ) < t , I { t ) |< < w* )^r - ^ (d * ) Som e rem arks on the fin ite -tim e b e h a v io r o f 35 Using Lemma 2.2, the above relation becomes W