On Doob s Maximal Inequality for Brownian Motion

Transcription

1 Stochastic Process. Al. Vol. 69, No., 997, (-5) Research Reort No. 337, 995, Det. Theoret. Statist. Aarhus On Doob s Maximal Inequality for Brownian Motion S. E. GRAVERSEN and G. PESKIR If B = (B t ) t is a standard Brownian motion started at x under P x for x, and is any stoing time for B with E x ( ) <, then for each > the following inequality is shown to be shar: E x max jb t j E x jb j x. The sharness is realized through the stoing times of the form: ;" for which it is comuted: whenever " > E ;" = " () max B s B t " st and < <. Hence, for the stoing time: ;" max jb s jjb t j " st which is shown to be a convolution of ;" with the first hitting time of " by jbj = (jb t j) t, we have: E ;" = " () for all " > and all < <. The method of roof relies uon the rincile of smooth fit and the maximality rincile for a Stehan roblem with moving (free) boundary, and Itˆo-Tanaka s formula (being alied two-dimensionally). The main emhasis is on the exlicit formulas throughout obtained.. Introduction The main urose of the aer is to derive and investigate a shar maximal inequality of Doob s tye for linear Brownian motion which may start at any oint. To describe this in more detail, let us assume we are given a standard Brownian motion B = (B t ) t which is defined on the robability sace (; F ; P ) and which starts at under P. Then the well-known Doob s maximal inequality states: (.) E max jb t j 4 EjB j where may be any stoing time for B with finite exectation (see [4].353 or [8].5). The constant 4 is known to be the best ossible in (.). For this one can consider the stoing times: AMS 98 subject classifications. Primary 6E5, 6G4, 6J65. Secondary 6G44, 6J6. Key words and hrases: Doob s maximal inequality, Brownian motion, otimal stoing (time), the rincile of smooth fit, submartingale, the maximality rincile, Stehan s roblem with moving boundary, Itˆo-Tanaka s formula, Burkholder-Gundy s inequality. (Second edition) goran@imf.au.dk

2 (.) ;" max jb s j jb t j " st where ; " >. It is well-known that E( ;" ) = < if and only if < =() whenever " > (see [9]). Alying Doob s maximal inequality with a general constant K > to the stoing time in (.) with some " > when < <, we get: (.3) E max jb t j = EjB ;" j + "EjB ;" j + " KEjB ;" j ;" Dividing through in (3.) by EjB ;" j and using that EjB ;" j = E( ;" )! together with EjB ;" j=ejb ;" j = E( ;" )! as " 4, we see that K 4. Motivated by these facts our main aim in this aer is to find an analogue of the inequality (.) when the Brownian motion B does not necessarily start from, but may start at any given oint x under P x for x. Thus P x (B = x) = for all x, and we identify P with P. Our main result (Theorem.) is the inequality: (.4) E x max jb t j 4E x jb j x which is valid for any stoing time for B with E x ( ) <, and which is shown to be shar as such. This is obtained as a consequence of the following inequality: x (.5) E x max jb t j c E x ( ) + c r 4 c which is valid for all c 4. When c > 4, the stoing time: (.6) c max jb s j st + 4=c jb t j is the one at which the equality in (.5) is attained, and moreover we have: 4=c (.7) E x ( c ) = x 4 4=c for all x and all c > 4. In articular, if we consider the stoing time: (.8) ;" max B s B t " st then (.7) can be rewritten to read as follows: (.9) E ;" = " () for all " > and all < <. Quite indeendently from this formula and its roof, below we resent a simle argument for E( ;" ) = which is based uon Tanaka s formula.

3 Finally, since ;" defined by (.) is shown to be a convolution of ;" and H ", where f : jb t j = " g, from (.9) we obtain the formula: H " (.) E ;" = " () for all " > and all < <. The method of roof relies uon the rincile of smooth fit (see [5]) and the maximality rincile (see [6]) for a Stehan roblem with moving (free) boundary, and Itˆo-Tanaka s formula (being alied two-dimensionally). The result and method of Theorem. easily extend to the case > (Corollary.), while this further extend to all non-negative submartingales (Corollary.3) by using the maximal embedding theorem of Jacka [7]. The main emhasis is on the exlicit formulas throughout obtained.. The inequality and roof In this and next section we resent the main results of the aer. After this work was comleted we learned from D: Burkholder that the inequalities (.36) below follow as a by-roduct from his new roof of Doob s inequality for discrete non-negative submartingales (see [].4). While the roof given there in essence relies on the submartingale roerty, the roof given here is based on the (strong) Markov roerty. The advantage of the aroach taken here lies in its alicability to all diffusions (see [6]). Yet another advantage is that during the roof we exlicitly write down the otimal stoing times (those through which the equality is attained). Recently we also learned that Cox [3] derived the analogue of these inequalities for discrete martingales by a method which is based on results from the theory of moments. In his aer Cox also notes that the method does have the drawback of comutational comlexity, which sometimes makes it difficult or imossible to ush the calculations through. Theorem. Let B = (B t ) t be a standard Brownian motion started at x under P x for x, and let be any stoing time for B such that E x ( ) <. Then the following inequality is shar: (.) E x max jb t j 4E x jb j x. The constants 4 and are the best ossible. Proof. We shall begin by considering the following otimal stoing roblem: (.) V (x; s) = su E x;s S c where the suremum is taken over all stoing times for B satisfying E x;s ( ) <, while the maximum rocess S = (S t ) t is defined by: (.3) S t = max jb r j _ s rt 3

4 where s x are given and fixed. The exectation in (.) is taken with resect to the robability measure P x;s under which S starts at s, and the rocess X = (X t ) t defined by: (.4) X t = jb t j starts at x. The Brownian motion B from (.3) and (.4) may be realized as: (.5) B t = ~ B t + x where ~ B = ( ~ B t ) t is a standard Brownian motion started at under P. Thus the (strong) Markov rocess (X; S) starts at (x; s) under P, and P x;s may be identified with P. By Itˆo formula we find: (.6) dx t = dt + X t db t. Hence we see that the infinitesimal oerator of the (strong) Markov rocess X in ]; [ acts like: (.7) while the boundary oint is a oint of the instantaneous reflection. If we assume that the suremum in (.) is attained at the exit time from an oen set by the (strong) Markov rocess (X; S) which is degenerated in the second comonent, then by the general Markov rocesses theory (see [8].87-99) it is lausible to assume that the ayoff x 7! V (x; s) satisfies the following equation: (.8) LXV (x; s) = c for x ]g 3 (s); s[ with s > given and fixed, where s 7! g 3 (s) is an otimal stoing boundary to be found. The boundary conditions which may be fulfilled are the following: (.9) V (x; s)x=g3(s)+ = s (instantaneous stoing) (.) (x; (x; x=s = (smooth fit) = (normal reflection). The general solution to the equation (.8) is given by: (.) V (x; s) = A(s) x + B(s) + cx where A(s) and B(s) are unsecified constants. From (.9)+(.) we find that: (.3) A(s) = c g 3 (s) (.4) B(s) = s + cg 3 (s). Inserting this into (.) gives: 4

5 (.5) V (x; s) = c g 3 (s) x + s + cg3 (s) + cx. By (.) we find that s 7! g 3 (s) (.6) cg (s) is to satisfy the (nonlinear) differential equation:! r s g(s) + =. The general solution of the equation (.6) may be exressed in a closed form. Instead of going into this direction we shall rather note that this equation admits a linear solution of the form: (.7) g 3 (s) = s where the given > is to satisfy: (.8) + =c =. Motivated by the maximality rincile (see [6]) we shall chose the greater satisfying (.8) as our candidate:! (.9) = + 4=c. Inserting this into (.5) gives: (.) V 3 (x; s) = c xs + ( + c)s + cx, if s x s = s, if x s as a candidate for the ayoff V (x; s) defined in (.). The otimal stoing time is then to be: (.) 3 f j X t g 3 (S t ) g where s 7! g 3 (s) is defined by (.7)+(.9). To verify that the formulas (.) and (.) are indeed correct, we shall use Itˆo-Tanaka s formula being alied two-dimensionally (see [6] for a formal justification of its use in this context; note that (x; s) 7! V 3 (x; s) is C outside f (g 3 (s); s) j s > g while x 7! V 3 (x; s) is convex and C on ]; s[ but at g 3 (s) where it is only C whenever s > is given and fixed). In this way we obtain: (.) V 3 (X t ; S t ) = V 3 (X ; S ) + V (X r; S r ) dx r (X r; S r ) d X; X (X r; S r ) ds r where we set (@ V 3 =@x )(g 3 (s); s) =. Since the increment ds r equals zero outside the diagonal x = s, and V 3 (x; s) at the diagonal satisfies (.), we see that the second integral in (.) is identically zero. Thus by (.6)+(.7) and the fact that d X; X = 4X t dt, we see that t 5

6 (.) can be equivalently written as follows: (.3) V3(X t ; S t ) = V3(x; s) + LXV3(X r ; S r ) dr + (X r; S r ) db r. Next note that LXV3(y; s) = c for g3(s) < y < s, and LXV3(y; s) = for y g3(s). Moreover, due to the normal reflection of X, the set of those r > for which X r = S r is of Lebesgue measure zero. This by (.3) shows that: (.4) V3(X ; S ) V3(x; s) + c + M for any stoing time for B, where M = (M t ) t is a continuous local martingale defined by: (.5) M t = Moreover, this also shows that: (X r; S r ) db r. (.6) V3(X ; S ) = V3(x; s) + c + M for any stoing time for B satisfying 3. Next we show that: (.7) E x;s (M ) = whenever is a stoing time for B with E x;s ( ) <. For (.7), by Burkholder-Gundy s inequality for continuous local martingales (see [8].53), it is sufficient to show that: Z (.8) E x;s Xr From (.) we (X r; S r ) fxrg3(sr)g dr := I <. c s (y; s) = + c y for s y s. Inserting this into (.8) we get: Z (.3) I = c E x;s X r = S r fxrsrg dr c ( ) E x;s Z c ( ) = c ( ) q E x;s (S ) Ex;s S r dr = c ( ) E x;s q max E x;s ( ) ~ Bt + x =q _ s S E x;s ( ) 6

7 E x;s c ( ) max j B ~ t j + x+s =q c ( ) 8E x;s ( ) + x + s =q E x;s ( ) < E x;s ( ) where we used Hölder s inequality, Doob s inequality (.), and the fact that E x;s j B ~ j = E x;s ( ) whenever E x;s ( ) <. Since V3(x; s) s, from (.4)+(.7) we find: (.3) V (x; s) = su + su E x;s S c su E x;s S V3(X ; S ) E x;s V3 (X ; S )c V3(x; s). Moreover, from (.6)+(.7) with = 3 we see that: (.3) E x;s S3 c3 = E x;s V3 (X 3 ; S 3 )c3 = V3(x; s) rovided that E x;s (3) <, which is known to be true if and only if c > 4 (see [9]). (Below we resent a different roof of this fact and moreover comute the value E x;s (3) exactly.) Matching (.3) and (.3) we see that the ayoff (.) is indeed given by the formula (.), and an otimal stoing time for (.) (stoing time at which the suremum is attained) is given by (.) with s 7! g3(s) from (.7) and ]; [ from (.9). In articular, note that from (.) with from (.9) we get: (.33) V3(x; x) = c r 4 c x. Alying the very definition of V (x; x) = V3(x; x) and letting c # 4, this yields: (.34) E x max jb t j 4E x ( ) + x. Finally, standard arguments show that: (.35) E x jb j = E x j ~ B + xj = E x j ~ B j + x E x ( ~ B ) + x = E x ( ) + x. Inserting this into (.34) we obtain (.). The sharness clearly follows from the definition of the ayoff in (.). This comletes the roof of the theorem. The revious result and method easily extend to the case we state this extension and sketch the roof. >. For reader s convenience Corollary. Let B = (B t ) t be a standard Brownian motion started at x under P x for x, let > be given and fixed, and let be any stoing time for B such that E x ( = ) <. Then the following inequality is shar: 7

8 (.36) E x max jb t j E x jb j x. The constants (=()) and =() are the best ossible. Proof. In arallel to (.) one is to consider the following otimal stoing roblem: (.37) V (x; s) = su E x;s S ci where the suremum is taken over all stoing times for B satisfying E x;s ( = ) <, and the underlying rocesses are given as follows: (.38) S t = (.39) I t = Z t max rt (.4) X t = jb t j X r (.4) B t = ~ B t + x = _ s Xr ()= dr where ~ B = ( ~ B t ) t is a standard Brownian motion started at under P = P x;s. This roblem can be solved in exactly the same way as the roblem (.) along the following lines. The infinitesimal oerator of X equals: (.4) LX = () The analogue of the equation (.8) is: x = (.43) LXV (x; s) = c The conditions (.9)-(.) are to be satisfied again. The analogue of the solution (.5) is: (.44) V (x; s) = c g= 3 (s) x = + s + c g 3(s) + c () x where s 7! g 3 (s) is to satisfy the equation: (.45) c g (s) s g(s)! =! + =. Again, like in (.6), this equation admits a linear solution of the form: (.46) g 3 (s) = s where < < is the maximal root (out of two ossible ones) of the equation: (.47) = + =c =. 8

9 By the standard argument one can verify that (.47) admits such a root if and only if + =( ) (). The otimal stoing time is then to be: c (.48) 3 f j X t g3(s t ) g where s 7! g3(s) is from (.46). To verify that the guessed formulas (.44) and (.48) are indeed correct we can use exactly the same rocedure as in the roof of Theorem.. For this, it should be recalled that E x;s (3 = ) < if and only if c > + =() () (see [9]). Note also that the analogue of (.35) is by Itˆo formula and the otional samling theorem given by: (.49) E x;s X = x + () E x;s (I ) whenever E x;s ( = ) <, which was the motivation for considering the roblem (.37) with (.39). The remaining details are easily comleted and will be left to the reader. Due to the extreme roerties of Brownian motion, the inequality (.36) extend to all nonnegative submartingales. This can be obtained by using the maximal embedding result of Jacka [7]. Corollary.3 Let X = (X t ) t be a non-negative cadlag (right continuous with left limits) uniformly integrable submartingale started at x under P. Let X denote the P -a.s. limit of X t for t!. Then the following inequality is satisfied and shar: (.5) E for all >. su X t t> E X Proof. Given such a submartingale X = (X t ) t satisfying E(X) <, and a Brownian motion B = (B t ) t started at X = x under P x, by the result of Jacka [7] we know that there exists a stoing time for B, such that jb j X and P f su t X t g P x f max jb j g for all >, with (B t^ ) t being uniformly integrable. The result then easily follows from Corollary. by using the integration by arts formula. Note that by the submartingale roerty of (jb t^ j) t we get su t E xjb t^ j = E x jb j for all >, so that E x ( = ) is finite if and only if E x jb j is so. x 3. The exected waiting time It is our main aim in this section to derive an exlicit formula for the exectation of the otimal stoing time 3 constructed in the roof of Theorem. (or Corollary.). This extend a result of Wang [9] who showed its finiteness. The stoing times of the form of 3 have been investigated by several eole. Instead of going into a historical exosition on this subject, we shall rather refer the reader to the aer of Azéma and Yor [] where more information in this direction can be 9

10 found. We d like to oint out however that as far as one is only concerned with the exectation of such a stoing time, the Lalace transform method develoed in some of these works may have the drawback of comutational comlexity in comarison with our method resented below which elegantly alies to all diffusions. Throughout we shall work within the setting and notation of Theorem. and its roof. By (.) with (.7) we have: (3.) 3 f j X t S t g where = (c) is a constant defined in (.9) for c > 4. Note that =4 < (c) " as c ". Our main task in this section is to comute exlicitly the function: (3.) m(x; s) = E x;s ( 3 ) for x s, where E x;s denotes the exectation with resect to P x;s under which X starts at x and S starts at s. Since clearly m(x; s) = for x s, we shall assume throughout that s < x s are given and fixed. Because 3 may be viewed as the exit time from an oen set by the (strong) Markov rocess (X; S) which is degenerated in the second comonent, by the general Markov rocesses theory (see [8].87-99) it is lausible to assume that x 7! m(x; s) satisfies the equation: (3.3) LX m(x; s) = for s < x < s with LX given by (.7). The following two boundary conditions seem aarent: (3.4) m(x; s) (x; x=s+ x=s The general solution to (3.3) is given by: = (instantaneous stoing) = (normal reflection). (3.6) m(x; s) = A(s) x + B(s) x where A(s) and B(s) are unsecified constants. By (3.4) and (3.5) we find: (3.7) A(s) = C s + (3.8) B(s) = C s += s s where C = C() is a constant to be determined, and where: (3.9) = ( ). In order to determine the constant C, we shall note by (3.6)-(3.9) that:

11 (3.) m(x; x) = C( ) x =( ) + ( ) x. Observe now that the ower =( ) >, due to the fact that = (c) > =4 when c > 4. However, the ayoff in (.33) is linear and given by: (3.) V 3 (x; x) := V 3 (x; c) = K(c) x where K(c) = (c=)( 4=c). This indicates that the constant C must be identically zero. Formally, this is verified as follows. Since c > 4 there is ]; [ such that c > 4. By definition of the ayoff we have: (3.) < V 3 (x; c) = E x;x S3 (c) c 3 (c) = E x;x S3 (c) c 3 (c) ()c Ex;x ( 3 (c)) V 3 (x; c) ()c E x;x ( 3 (c)) K(c) x ()c E x;x ( 3 (c)). This shows that x 7! m(x; x) is at most linear: (3.3) m(x; x) = E x;x ( 3 (c)) K(c) ()c x. Looking back to (3.) we conclude C. Thus by (3.6)-(3.8) with C we finish u with the following candidate for E x;s ( 3 ) : (3.4) m(x; s) = xs s x when s < x s. In order to verify that this formula is indeed correct we shall use Itˆo-Tanaka s formula in the roof below. Theorem 3. Let B = (B t ) t be a standard Brownian motion, and let X = (X t ) t and S = (S t ) t be associated with B by formulas (.3)+(.4). Then for the stoing time 3 defined in (3.): (3.5) E x;s (3) = where > =4. xs =, if x s. s x, if s x s Proof. Denote the function on the right-hand side of (3.5) by m(x; s). Note that x 7! m(x; s) is concave and non-negative on [s; s] for each fixed s >. By Itˆo-Tanaka s formula (see [6] for a justification of its use in this context) we get: (3.6) m(x t ; S t ) = m(x ; S ) + + (X r; S r ) ds r. LX m(x r ; S r ) dr + (X r; S r ) db r

12 Due to (3.5) the last integral in (3.6) is identically zero. In addition, let us consider the region G = f (x; s) j s < x < s + g. Given (x; s) G chose bounded oen sets G G... such that [ n= G n = G and (x; s) G. Denote the exit time of (X; S) from G n by n, then clearly n " 3 as n!. Denote further the second integral in (3.6) by M t. Then M = (M t ) t is a continuous local martingale, and we have: (3.7) E x;s (M n ) = for all n. For this note that: Z n (3.8) E (X r; S r ) dr! KE x;s ( n ) < with some K >, since (x; s) 7! x (@m=@x)(x; s) is bounded on the closure of G n. By (3.3) from (3.6)+(3.7) we find: (3.9) E x;s m(x n ; S n ) = m(x; s) E x;s ( n ). Since (x; s) 7! m(x; s) is non-negative, hence first of all we may deduce: (3.) E x;s (3) = lim n! E x;s( n ) m(x; s) <. This roves the finiteness of the exectation of 3 ( see [9] for another roof based uon random walk). Moreover, motivated by a uniform integrability argument we may note that: (3.) m(x n ; S n ) X n S n S 3 uniformly over all n. Moreover, by Doob s inequality (.) and (3.) we find: (3.) E x;s (S 3 ) 4E x;s (3) + x + s <. Thus the sequence (m(x n ; S n )) n is uniformly integrable, while it clearly converges to zero ointwise. For this reason we may conclude: (3.3) lim n! E x;sm(x n ; S n ) =. This shows that we have an equality in (3.), and the roof is comlete. Corollary 3. Let B = (B t ) t be a standard Brownian motion started at under P. Consider the stoing times: (3.4) ;" (3.5) ;" max st B s B t " max jb s j jb t j " st

13 for " > and < <. Then ;" is a convolution of ;" and H ", where H " f : jb t j = " g, and the formulas are valid: (3.6) E ;" = " () (3.7) E ;" = " () for all " > and all < <. Proof. Consider the definition rule for ;" in (3.5). Clearly ;" > H " and after hitting ", the reflected Brownian motion jbj = (jb t j) t does not hit zero before ;". Thus its absolute value sign may be droed out during the time interval between H " and ;", and the claim about the convolution identity follows by the reflection roerty and the strong Markov roerty of Brownian motion. (3.6): Consider the stoing time 3 defined in (3.) for s = x. By the very definition it can be rewritten to read as follows: (3.8) 3 jb t j max jb s j st max jb s j jb t j st max j B ~ s + xj j B ~ t + xj st max ~Bs + x ~Bt + x st max st ~B s x B ~ t. Setting = = and " = (= ) x, by (3.5) hence we find: (3.9) E ;" = Ex;x (3) = ( ) x = " (3.7): Since E(H " ) = ", by (3.6) we get: (3.3) E ;" = E(;" ) + E(H " ) = " (). The roof is comlete. (). Remark 3.3 Let B = (B t ) t be a standard Brownian motion started at under P. Consider the stoing time: (3.3) ;" max B s B t " st 3

14 for ". It follows from (3.6) in Corollary 3. that: (3.3) E( ;" ) = + if " >. Here we resent yet another argument based uon Tanaka s formula which imlies (3.3). For this consider the rocess: (3.33) t = sign(b s ) db s where sign(x) = for x and sign(x) = for x >. Then = ( t ) t is a standard Brownian motion, and Tanaka s formula states (see [8].4-5): (3.34) jb t j = t + L t where L = (L t ) t is the local time rocess of B at given by: (3.35) L t = max ( s ). st Thus ;" is equally distributed as: (3.36) n n o jbt j " t max ( s ) ( t ) " st Note that is an (F t )-stoing time, and since F t = F jbj t F B t, we see that is an (F B t )-stoing time too. Assuming now that E( ;") which equals E() is finite, by the standard Wald s tye identity for Brownian motion we obtain: (3.37) E() = EjB j = E(" ) = " "E( ) + Ej j = " + E(). Hence we see that " must be zero. This comletes the roof of (3.3).. o REFERENCES [] AZEMA, J: and YOR, M: (979): Une solution simle au robleme de Skorokhod: Lecture Notes in Math: 7, Sringer (9-5). [] BURKHOLDER, D: L: (99): Exlorations in martingale theory and its alications: Ecole d Eté de Probabilités de Saint-Flour XIX-989, Lecture Notes in Math: 464, Sringer- Verlag (-66). [3] COX, D: C: (984): Some shar martingale inequalities related to Doob s inequality: IMS Lecture Notes Monograh Ser: 5 (78-83). [4] DOOB, J: L: (95): Stochastic Processes: Wiley, New York. 4

15 [5] DUBINS, L: E: SHEPP, L: A: and SHIRYAEV, A: N: (993): Otimal stoing rules and maximal inequalities for Bessel rocesses: Theory Probab: Al: 38 (6-6). [6] GRAVERSEN, S. E. and PESKIR, G. (995). Otimal stoing and maximal inequalities for geometric Brownian motion. Research Reort No. 33, Det. Theoret. Statist. Aarhus, (8 ). J. Al. Probab. 35, 998 (856-87). [7] JACKA, S: D: (988): Doob s inequalities revisited: A maximal H -embedding: Stochastic Process: Al: 9 (8-9). [8] REVUZ, D: and YOR, M: (99): Continuous Martingales and Brownian Motion: Sringer- Verlag Berlin Heidelberg 994. [9] WANG, G: (99): Shar maximal inequalities for conditionally symmetric martingales and Brownian motion: Proc: Amer: Math: Soc: ( ). Svend Erik Graversen Deartment of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus matseg@imf.au.dk Goran Peskir Deartment of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus home.imf.au.dk/goran goran@imf.au.dk 5