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Optimal Stopping Inequalities for the Integral of Brownian Paths
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
222, 244]254 Ž1998.
AY985937
Optimal Stopping Inequalities for the Integral of Brownian Paths Goran Peskir* ˇ Uni¨ ersity of Aarhus and Uni¨ ersity of Zagreb Submitted by Sandy L. Zabell Received November 18, 1996
Let B s Ž Bt . t G 0 be a Brownian motion started at x g R. Given a stopping time t for B and a real valued map F, we show how one can optimally bound: E
žH
t
0
F Ž < Bt < . dt
/
in terms of EŽt .. The method of proof relies upon solving the optimal stopping problem where one minimizes or maximizes E
žH
t
0
F Ž < Bt < . dt y ct
/
over all stopping times t for B with c ) 0. By Ito’s ˆ formula and the optional sampling theorem, this problem simplifies to the form where explicit computations are possible. The method is quantitatively demonstrated through the example of F Ž x . s x r , with r ) y1. This yields some new sharp inequalities. Q 1998 Academic Press
Key Words: Brownian motion; integral of Brownian path; optimal stopping; Burkholder]Davis’ inequality; Ito]Tanaka formula; occupation times formula; ˆ Doob’s optional sampling theorem; hitting time; square root boundary.
1. DESCRIPTION OF THE PROBLEM AND METHOD OF PROOF Let Ž Bt . t G 0 be standard Brownian motion. For simplicity in this section, we assume that B starts at zero, but most of the considerations can be extended to the case in which B starts at any x g R. Our starting point is * E-mail: [email protected]. http:rrwww.mi.aau.dkr; goran 244 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
OPTIMAL STOPPING FOR BROWNIAN PATHS
245
Theorem 2.1, where we prove the following inequalities: t
A p E Ž t 1qp r2 . F E
žH
0
< Bt < p dt F Bp E Ž t 1qp r2 .
/
Ž 1.1.
for all stopping times t for B, and all p ) 0, where A p and Bp are numerical constants. Although the best values for the constants A p and Bp in Ž1.1. are also identified below, in most of the cases it is much easier to evaluate EŽt . rather than EŽt 1q p r2 .. In this paper we shall answer the question on how the inequality Ž1.1. can be optimally modified if the quantity EŽt 1q p r2 . is replaced with a function of EŽt .. It turns out that the left-hand inequality in Ž1.1. admits such a modification. In Theorem 2.2 we prove that E
ž
t
H0 < B <
p
t
2
dt G
/
Ž2 q p. Ž1 q p.
Ž Et .
1q pr2
Ž 1.2.
for all stopping times t for B and all p ) 0, and, moreover, we show that 2rŽ2 q p .Ž1 q p . is the best possible constant. The analogue of such an inequality does not extend to the right-hand side in Ž1.1.. In Example 2.4 we show that the hitting times of the square root boundaries violate its validity. For this reason we are forced to change the functional of the Brownian path in Ž1.1. from t ¬ < Bt < to t ¬ < Bt
ž
t
H0
1 < Bt <
q
dt F
/
2
Ž2 y q. Ž1 y q.
Ž Et .
1y qr2
Ž 1.3.
for all stopping times t for B and all 0 F q - 1, and, moreover, we again show that 2rŽ2 y q .Ž1 y q . is the best possible constant. The optimality of the constant in Ž1.2. and Ž1.3. is achieved through the hitting time of any positive number by the reflected Brownian motion < B < s Ž< Bt <. t G 0 . In fact, if ta s inf t ) 0 : < Bt < s a 4 for some a ) 0, then the equality is attained both in Ž1.2. and Ž1.3.. Consequently, by letting q1 in Ž1.3., we see that E
ž
ta
H0
1 < Bt <
dt s `.
/
Ž 1.4.
This explains why q must be strictly less than 1 when dealing with t ¬ < Bt
t
žH
0
F Ž < Bt < . dt y ct
/
Ž 1.5.
GORAN PESKIR ˇ
246
over all stopping times t for B with finite expectation. Here F: R ª R is a suitable function, and c ) 0 is given and fixed. By the Ito ˆ formula and the optional sampling theorem, we find that Ž1.5. obtains the form E Ž H Ž < Bt < . y c < Bt < 2 . ,
Ž 1.6.
where F s H0r2 Žwith H Ž0. s H9Ž0. s 0.. This further equals `
H0
Ž H Ž x . y cx 2 . dP< B < Ž x . , t
Ž 1.7.
where P < Bt < denotes the distribution law of < Bt <. Thus if x#Ž c . denotes a minimum Žresp. maximum. point of the map x ¬ H Ž x . y cx 2 on Rq, then
t # Ž c . s inf t ) 0 : < Bt < s x# Ž c . 4
Ž 1.8.
is the optimal stopping time for the problem Ž1.5.. This enables us to evaluate the minimum Z#Ž c . Žresp. the maximum Z*Ž c .. in Ž1.5. exactly. By definition of the minimum, it then follows E
t
žH
0
F Ž < Bt < . dt G sup Ž c E Ž t . q Z# Ž c . .
/
Ž 1.9.
c)0
whenever t is a stopping time for B. Similarly, by definition of the maximum, we get E
ž
t
H0
F Ž < Bt < . dt F inf Ž c E Ž t . q Z* Ž c . .
/
c)0
Ž 1.10.
for all stopping times t for B. Supposing that the supremum in Ž1.9. Žresp. the infimum in Ž1.10.. is attained at some c# ) 0, the right-hand side in Ž1.9. Žresp. Ž1.10.. defines a function of EŽt .. Since the stopping time t #Ž c . is optimal for the problem Ž1.5. regardless of c ) 0, it is clear that the inequalities Ž1.9. and Ž1.10. are sharp Žthe equality may be attained in a nontrivial manner.. In this paper we apply this method and make explicit computations only for the function F Ž x . s x r with r ) y1, and this yields the inequalities Ž1.2. and Ž1.3., respectively. We want to mention that the cases of other functions F, including the case in which B may start at any point, could be treated similarly. We believe that the presentation of our method above enables one to recognize the constraints on the function F needed to derive the inequalities Ž1.9. and Ž1.10. in a precise manner. For instance, if F is continuous and the map x ¬ H Ž x . y cx 2 is bounded from above Žresp. below. on Rq, then Ž1.9. Žresp. Ž1.10.. holds. Below we will see that the continuity condition on F may be relaxed Žby treating the case in
247
OPTIMAL STOPPING FOR BROWNIAN PATHS
which F Ž x . s x r for y1 - r - 0.. In such a case one should use Ito]Tanaka’s formula Žrather than Ito’s ˆ ˆ formula.. It should be noted, too Žparticularly when applying the optional sampling theorem., that the stopping times appearing throughout may be assumed to be bounded. The results in this paper are similar in nature and can be compared with the results in w4x. In some sense the optimal stopping appearing here could be called a Wald’s type optimal stopping for the integral of Brownian paths. Despite their appealing simplicity, we are unaware of similar results.
2. OPTIMAL STOPPING INEQUALITIES In this section we present the main results of the paper. Throughout B s Ž Bt . t G 0 is standard Brownian motion defined on a probability space Ž V, F , P .. We begin by giving a proof of the inequalities Ž1.1.. To the best of our knowledge, these inequalities have not been written down earlier, but we believe that they are known. THEOREM 2.1. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequalities are satisfied: A p E Ž t 1qp r2 . F E
t
žH
0
< Bt < p dt F Bp E Ž t 1qp r2 .
/
Ž 2.1.
for all stopping times t for B, and all p ) 0, where A p and Bp are numerical constants. The best possible ¨ alues for A p and Bp are AUp s 2 Ž z 2qp . BUp s 2 Ž zU2qp .
2q p
2q p
rŽ 2 q p . Ž 1 q p .
Ž 2.2.
rŽ 2 q p . Ž 1 q p . ,
Ž 2.3.
where z 2q p denotes the smallest positi¨ e zero of the confluent hypergeometric function x ¬ M ŽyŽ2 q p .r2, 1r2, x 2r2., and zU2q p denotes the largest positi¨ e zero of the parabolic cylinder function x ¬ D 2q p Ž x . Ž see w1x.. Ž In particular, z 2 n Ž resp. zU2 n . is the smallest Ž resp. largest . positi¨ e zero of the Hermite polynomial x ¬ He2 nŽ x . for n G 1.. Proof. Let p ) 0 be given and fixed. By Ito’s ˆ formula we have < Bt < 2q p s Ž 2 q p .
t
H0 < B <
1qp
s
s Ž 2 q p . Mt q C p
sign Ž Bs . dBs q 2y1 Ž 2 q p . Ž 1 q p . t
H0 < B < s
p
ds,
t
H0 < B < s
p
ds
Ž 2.4.
GORAN PESKIR ˇ
248
where M s Ž Mt . t G 0 is a continuous local martingale started at zero under P, and C p s 2y1 Ž2 q p .Ž1 q p .. Let w M x s Žw M x t . t G 0 denote the quadratic variation of M, and let t be any stopping time for B. In view of Ž2.1. and monotone convergence theorem, it is no restriction to assume that t is bounded by a constant l ) 0, so that we have EŽ w M xt .
1r2
sE
t
žH
0
1r2
< Bt < 2Ž pq1. dt
F lE
/
ž max < B < 0FtF l
2Ž pq1.
t
s K l pq 2 - `
/ Ž 2.5.
for some K ) 0. Therefore, by Doob’s optional sampling theorem Žsee w3x, w5x., E Ž Mt . s 0.
Ž 2.6.
Thus from Ž2.4. we obtain the following identity: E
ž
t
H0 < B <
p
t
< < 2qp . dt s Cy1 p E Bt
/
Ž 2.7.
Recall Burkholder]Davis’ inequalities: d q E Ž t q r2 . F E < Bt < q
1 - q - ` and
if
E < Bt < q F Dq E Ž t q r2 .
E Ž t q r2 . - ` Ž 2.8.
for all q ) 0,
Ž 2.9.
with the best possible values for Dq and d q given by Žsee w2x.
¡Ž z . d s~ ¢Ž z . ¡Ž z . D s~ ¢Ž z .
q
q
q
U q
q
U q
q
q
q
q
if
2Fq-`
if
1-qF2
if
2Fq-`
if
0 - q F 2,
Ž 2.10.
Ž 2.11.
where z q and zUq are such as in the statement of the theorem. Setting q s p q 2, the claims of the theorem clearly follow from Ž2.7. ] Ž2.11.. The proof is complete. In the next theorem we show how the left-hand inequality in Ž2.1. extends if the quantity EŽt 1q p r2 . is replaced by a function of EŽt ..
OPTIMAL STOPPING FOR BROWNIAN PATHS
249
THEOREM 2.2. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequality is satisfied: E
t
žH
0
< Bt < p dt G
/
2
Ž2 q p. Ž1 q p.
Ž Et .
1q pr2
Ž 2.12.
for all stopping times t for B and all p ) 0. The constant 2rŽ2 q p .Ž1 q p . is the best possible. Proof. Given p ) 0, consider the following optimal stopping problem: V Ž c . s inf E t
t
žH
0
< Bt < p dt y ct ,
/
Ž 2.13.
where c ) 0 is given and fixed, and the infimum is taken over all stopping times t for B having finite expectation. Let t be any such stopping time for B. Then in view of Ž2.12., it is no restriction to assume that t is bounded, so that by Ž2.7. we have E
ž
t
H0 < B < t
p
< Bt < 2qp y ct dt y ct s E Cy1 p
/
ž
/
< Bt < 2q p y c < Bt < 2 s E Cy1 p
ž
s
`
y1 2qp p x
H0 Ž C
/
Ž 2.14.
y cx 2 . dP< Bt < Ž x .
where C p s 2y1 Ž2 q p .Ž1 q p . and P < Bt < denotes the distribution law of < Bt <. 2qp Denote f Ž x . s Cy1 y cx 2 for x ) 0. Then it is easily verified that f p x attains its minimum on Rq at x#Ž c . s Ž cŽ1 q p ..1r p and f Ž x#Ž c .. s ypŽ1 q p . 2r p c 1q2r prŽ2 q p .. Hence from Ž2.14. we find V Ž c. s
yp Ž 1 q p .
Ž2 q p.
2rp
c 1q 2r p .
Ž 2.15.
and the optimal stopping time in Ž2.13. Žthe one at which the infimum is attained. is
t # s inf t ) 0 : < Bt < s x# Ž c . 4 .
Ž 2.16.
ŽObserve by passing to a limit that Ž2.7. holds for t * as well. From Ž2.13. we get E
t
žH
0
< Bt < p dt G sup Ž cE Ž t . q V Ž c . . .
/
c)0
Ž 2.17.
GORAN PESKIR ˇ
250
Denote g Ž c . s cEŽt . q V Ž c . for c ) 0. Then it is easily verified that g attains its maximum on Rq at c# s Ž Et . p r2rŽ1 q p . and g Ž c#. s Ž2rŽ2 q p .Ž1 q p ..Ž Et .1q p r2 . Inserting this into Ž2.17., we get Ž2.12.. The proof is complete. Remark 2.3. 1. Since for t # defined in Ž2.16. we have EŽt #. s 2 Ž . E < Bt# < 2 s x# c s Ž cŽ1 q p .. 2r p for all c ) 0, by taking c s c# s p r2 Ž Et . rŽ1 q p ., we see that each ta s inf t ) 0 : < Bt < s a 4 with a ) 0 is optimal in Ž2.17., and therefore in Ž2.12. as well Žthe equality is attained at ta for all a ) 0.. ŽThis is also verified directly by the Ito]Tanaka formula ˆ and the optional sampling theorem.. 2. The proof presented above can be applied in the case in which Brownian motion B starts at any x g R. In this way we can extend Ž2.12. by proving the following inequality: E
t
žH
0
< Bt q x < p dt G
/
2
Ž2 q p. Ž1 q p.
ž Ž EŽt . q x
2 1q pr2
.
y < x < 2qp ,
/
Ž 2.18. which is valid for all stopping times t for B, all p ) 0, and all x g R. Moreover, the constant 2rŽ2 q p .Ž1 q p . is the best possible for all x g R. In fact, if ta, x s inf t ) 0 : < Bt q x < s a 4 with a ) 0, then the equality in Ž2.18. is attained at ta, x for all a ) 0 and all x g R. 3. Note that Ž2.12. can be obtained more directly by using Jensen’s inequality Žobserve in Ž2.7. that E < Bt < 2q p s EŽ< Bt < 2 .1q p r2 G Ž E < Bt < 2 .1q p r2 s Ž Et .1q p r2 whenever t is bounded and then pass to the limit.. The main value of the proof given above lies in its applicability to the other functionals F Ždifferent from x ¬ < x < p . for which the convexity of x ¬ H Ž'x . with H0 s F Žand thus Jensen’s inequality as well. fails. The same facts hold for Ž2.18.. See also Remark 2.4 in w4x. EXAMPLE 2.4. From Ž2.1. and Ž2.12. it is clear that the quantity EŽt 1q p r2 . on the right-hand side in Ž2.1. cannot generally be replaced by Ž Et .1q p r2 . Here we exhibit a class of stopping times that violate such an inequality. For simplicity we only consider the case p s 1, but other cases could be treated similarly. So we shall show that the inequality E
t
žH
0
< Bt < dt F C Ž Et . 3r2
/
Ž 2.19.
cannot be satisfied for all stopping times t for B with a fixed numerical constant C ) 0.
251
OPTIMAL STOPPING FOR BROWNIAN PATHS
For this consider the hitting times of the square root boundaries: 1r2 Tc s inf t ) 0 : < Bt < s c Ž t q 1 . 4
Ž 2.20.
for c ) 0. Then it is well known that EŽTcq . - ` if and only if c - c0 Ž q ., where c0 Ž q . is the smallest positive zero of the confluent hypergeometric function c ¬ M Žyq, 1r2, c 2r2. Žsee w6x.. In particular, it is easily verified that c 0 Ž1. s 1 ) c 0 Ž3r2. s 0.84 . . . . Choose now 0 - c n c 0 Ž3r2.. Denote Tn s Tc n and T0 s Tc 0 Ž3r2. . Then EŽTn3r2 . - ` for any n G 1 given and fixed, so that E
Tn
žH
0
1r2
< Bt < 4 dt
2
F E Ž c n4 Ž Tn q 1 . Tn .
/
1r2
F c n2 E Ž Tn3r2 q '2 Tn q Tn1r2 .
F c n2 Ž 2 q '2 . Ž 1 q E Ž Tn3r2 . . - `.
Ž 2.21.
Therefore, by Ito’s ˆ formula Žapplied to < Bt < 3 . and Doob’s optional sampling theorem, we find E
Tn
žH
0
< Bt < dt s
/
1 3
E < BT n < 3 s
c n3 3
E Ž Tn q 1 .
3r2
F C Ž ETn .
3r2
.
Ž 2.22.
So, if Ž2.19. would be satisfied, then c n3 3
E Ž Tn3r2 . F
c n3 3
E Ž Tn q 1 .
3r2
Ž 2.23.
for all n G 1. Letting n ª `, we would get c03 Ž 3r2 . 3
E Ž T03r2 . F C Ž ET0 .
3r2
- `,
Ž 2.24.
since c 0 Ž3r2. - 1. However, this contradicts the fact that EŽT03r2 . s `. Thus Ž2.19. cannot be satisfied for all Tn with n G 1. The proof of the claim is complete. Motivated by the preceding example, we change the function of the Brownian path in Ž2.12. so that we obtain the following extension of this inequality. The proof is similar to the proof of Theorem 2.2, and we only sketch the most important parts for the convenience of the reader. THEOREM 2.5. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequality is satisfied: E
ž
t
H0
1 < Bt <
q
dt F
/
2
Ž2 y q. Ž1 y q.
Ž Et .
1y qr2
Ž 2.25.
GORAN PESKIR ˇ
252
for all stopping times t for B and all 0 F q - 1. The constant 2rŽ2 y q .Ž1 y q . is the best possible. Proof. Given 0 F q - 1, consider the following optimal stopping problem: t 1 W Ž c . s sup E Ž 2.26. q dt y ct , 0 < Bt < t
žH
/
where c ) 0 is given and fixed, and the supremum is taken over all Žand stopping times t for B having finite expectation. By Ito]Tanaka’s ˆ . Ž w x . occupation times formula see 5 , pp. 208]209 , we find < Bt < 2y q s Ž 2 y q .
t
H0 < B <
1yq
s
sign Ž Bs . dBs
q 2y1 Ž 2 y q . Ž 1 y q . s Ž 2 y q . Mt q C q
1
t
H0
t
yq
H0 < B <
< Bs < q
s
ds
Ž 2.27.
ds,
where M s Ž Mt . t G 0 is a continuous local martingale started at zero under P, and C q s 2y1 Ž2 y q .Ž1 y q .. Let t be any stopping time for B having finite expectation. Then it is no restriction to assume that t is bounded, so that in exactly the same way as in the proof of Theorem 2.1 ŽDoob’s optional sampling theorem., we find EŽ Mt . s 0. Thus from Ž2.27. we obtain t 1 y1 < < 2y q . E Ž 2.28. q dt s C q E Bt 0 < Bt <
žH
/
Hence we find that E
ž
t
H0
1 < Bt < q
< Bt < 2y q y ct s E Cy1 < Bt < 2yq y c < Bt < 2 dt y ct s E Cy1 q q
/
ž
s
`
H0 Ž C
/
y1 2yq q x
ž
y cx 2 . dP< Bt < Ž x . ,
/
Ž 2.29.
where P < Bt < denotes the distribution law of < Bt <. 2yq Denote f Ž x . s Cy1 y cx 2 for x ) 0. Then it is easily verified that f q x attains its maximum on Rq at x*Ž c . s Ž cŽ1 y q ..y1 r q and f Ž x*Ž c .. s qc1y 2r qrŽ2 y q .Ž1 y q . 2r q. Hence from Ž2.23. we find q W Ž c. s c 1y 2r q , Ž 2.30. 2rq Ž2 y q. Ž1 y q.
OPTIMAL STOPPING FOR BROWNIAN PATHS
253
and the optimal stopping time in Ž2.26. Žthe one at which the infimum is attained. is
t * s inf t ) 0 : < Bt < s x* Ž c . 4 .
Ž 2.31.
ŽNote that Ž2.28. holds for t * as well.. From Ž2.26. we get E
ž
1
t
H0
< Bt < q
dt F inf Ž cE Ž t . q W Ž c . . .
/
c)0
Ž 2.32.
Denote g Ž c . s cEŽt . q W Ž c . for c ) 0. Then it is easily verified that g attains its minimum on Rq at c# s 1rŽ1 y q .Ž Et . q r2 and g Ž c#. s Ž2rŽ2 y q .Ž1 y q ..Ž Et .1y q r2 . Inserting this into Ž2.32., we get Ž2.25.. The proof is complete. Remark 2.6. 1. In exactly the same way as in part 1 of Remark 2.3 above, we can verify that each ta s inf t ) 0 : < Bt < s a 4 with a ) 0 is optimal in Ž2.32., and therefore in Ž2.25. as well Žthe equality is attained at ta for all a ) 0.. ŽThis is also obtained directly by Ito]Tanaka’s formula ˆ and the optional sampling theorem.. Finally, letting q1 in Ž2.25. with t s ta , we see that Ž1.4. holds for all a ) 0. This explains why q in Theorem 2.5 must be strictly less than 1. 2. The proof above can be applied in the case in which Brownian motion B starts at any x g R. In this way we can extend Ž2.25. by proving the following inequality: E
t
žH
0
1 < Bt q x < q
dt F
/
2
Ž EŽt . q x Ž2 y q. Ž1 y q. ž
2 1y qr2
.
y < x < 2yq ,
/
Ž 2.33. which is valid for all stoppping times t for B, all 0 F q - 1, and all x g R. Moreover, the constant 2rŽ2 y q .Ž1 y q . is the best possible for all x g R Žthe equality is attained at ta, x s inf t ) 0 : < Bt q x < s a 4 for all a ) 0 and all x g R.. 3. Note that Ž2.25. can be obtained more directly by using Jensen’s inequality Žobserve in Ž2.28 . that E < Bt < 2y q s E Ž < Bt < 2 . 1y q r 2 F Ž E < Bt < 2 .1y q r2 s Ž Et .1y q r2 whenever t is bounded, and then pass to the limit.. The main value of the proof given above lies in its applicability to the other functionals F Ždifferent from x ¬ < x
254
GORAN PESKIR ˇ
From Ž2.28. in the proof of Theorem 2.5 and Ž2.8. ] Ž2.11. in the proof of Theorem 2.1, we obtain the following result. COROLLARY 2.7. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequalities are satisfied: A q E Ž t 1y q r2 . F E
ž
1
t
H0
< Bt < q
dt F Bq E Ž t 1yq r2 .
/
Ž 2.34.
for all stopping times t for B and all 0 F q - 1, where A q and Bq are numerical constants. The best possible ¨ alues for A q and Bq are AUq s 2 Ž zU2yq . BUq s 2 Ž z 2yq .
2y q
2y q
rŽ 2 y q . Ž 1 y q .
Ž 2.35.
rŽ 2 y q . Ž 1 y q . ,
Ž 2.36.
where zU2y q denotes the largest positi¨ e zero of the parabolic cylinder function x ¬ D 2y q Ž x ., and z 2yq denotes the smallest positi¨ e zero of the confluent hypergeometric function x ¬ M ŽyŽ2 y q .r2, 1r2, x 2r2. Ž see w1x..
REFERENCES 1. M. Abramowicz and I. A. Stegun, ‘‘Handbook of Mathematical Functions,’’ Dover, New York, 1970. 2. B. Davis, On the L p norms of stochastic integrals and other martingales, Duke Math. J. 43 Ž1976., 697]704. 3. J. L. Doob, ‘‘Stochastic Processes,’’ Wiley, New York, 1953. 4. S. E. Graversen and G. Peskir, ˇ On Wald-type optimal stopping for Brownian motion, J. Appl. Probab. 34 Ž1997., 66]73. 5. D. Revuz and M. Yor, ‘‘Continuous Martingales and Brownian Motion,’’ Springer-Verlag, Berlin, 1991. 6. L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38 Ž1967., 1912]1914.
222, 244]254 Ž1998.
AY985937
Optimal Stopping Inequalities for the Integral of Brownian Paths Goran Peskir* ˇ Uni¨ ersity of Aarhus and Uni¨ ersity of Zagreb Submitted by Sandy L. Zabell Received November 18, 1996
Let B s Ž Bt . t G 0 be a Brownian motion started at x g R. Given a stopping time t for B and a real valued map F, we show how one can optimally bound: E
žH
t
0
F Ž < Bt < . dt
/
in terms of EŽt .. The method of proof relies upon solving the optimal stopping problem where one minimizes or maximizes E
žH
t
0
F Ž < Bt < . dt y ct
/
over all stopping times t for B with c ) 0. By Ito’s ˆ formula and the optional sampling theorem, this problem simplifies to the form where explicit computations are possible. The method is quantitatively demonstrated through the example of F Ž x . s x r , with r ) y1. This yields some new sharp inequalities. Q 1998 Academic Press
Key Words: Brownian motion; integral of Brownian path; optimal stopping; Burkholder]Davis’ inequality; Ito]Tanaka formula; occupation times formula; ˆ Doob’s optional sampling theorem; hitting time; square root boundary.
1. DESCRIPTION OF THE PROBLEM AND METHOD OF PROOF Let Ž Bt . t G 0 be standard Brownian motion. For simplicity in this section, we assume that B starts at zero, but most of the considerations can be extended to the case in which B starts at any x g R. Our starting point is * E-mail: [email protected]. http:rrwww.mi.aau.dkr; goran 244 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
OPTIMAL STOPPING FOR BROWNIAN PATHS
245
Theorem 2.1, where we prove the following inequalities: t
A p E Ž t 1qp r2 . F E
žH
0
< Bt < p dt F Bp E Ž t 1qp r2 .
/
Ž 1.1.
for all stopping times t for B, and all p ) 0, where A p and Bp are numerical constants. Although the best values for the constants A p and Bp in Ž1.1. are also identified below, in most of the cases it is much easier to evaluate EŽt . rather than EŽt 1q p r2 .. In this paper we shall answer the question on how the inequality Ž1.1. can be optimally modified if the quantity EŽt 1q p r2 . is replaced with a function of EŽt .. It turns out that the left-hand inequality in Ž1.1. admits such a modification. In Theorem 2.2 we prove that E
ž
t
H0 < B <
p
t
2
dt G
/
Ž2 q p. Ž1 q p.
Ž Et .
1q pr2
Ž 1.2.
for all stopping times t for B and all p ) 0, and, moreover, we show that 2rŽ2 q p .Ž1 q p . is the best possible constant. The analogue of such an inequality does not extend to the right-hand side in Ž1.1.. In Example 2.4 we show that the hitting times of the square root boundaries violate its validity. For this reason we are forced to change the functional of the Brownian path in Ž1.1. from t ¬ < Bt < to t ¬ < Bt
ž
t
H0
1 < Bt <
q
dt F
/
2
Ž2 y q. Ž1 y q.
Ž Et .
1y qr2
Ž 1.3.
for all stopping times t for B and all 0 F q - 1, and, moreover, we again show that 2rŽ2 y q .Ž1 y q . is the best possible constant. The optimality of the constant in Ž1.2. and Ž1.3. is achieved through the hitting time of any positive number by the reflected Brownian motion < B < s Ž< Bt <. t G 0 . In fact, if ta s inf t ) 0 : < Bt < s a 4 for some a ) 0, then the equality is attained both in Ž1.2. and Ž1.3.. Consequently, by letting q1 in Ž1.3., we see that E
ž
ta
H0
1 < Bt <
dt s `.
/
Ž 1.4.
This explains why q must be strictly less than 1 when dealing with t ¬ < Bt
t
žH
0
F Ž < Bt < . dt y ct
/
Ž 1.5.
GORAN PESKIR ˇ
246
over all stopping times t for B with finite expectation. Here F: R ª R is a suitable function, and c ) 0 is given and fixed. By the Ito ˆ formula and the optional sampling theorem, we find that Ž1.5. obtains the form E Ž H Ž < Bt < . y c < Bt < 2 . ,
Ž 1.6.
where F s H0r2 Žwith H Ž0. s H9Ž0. s 0.. This further equals `
H0
Ž H Ž x . y cx 2 . dP< B < Ž x . , t
Ž 1.7.
where P < Bt < denotes the distribution law of < Bt <. Thus if x#Ž c . denotes a minimum Žresp. maximum. point of the map x ¬ H Ž x . y cx 2 on Rq, then
t # Ž c . s inf t ) 0 : < Bt < s x# Ž c . 4
Ž 1.8.
is the optimal stopping time for the problem Ž1.5.. This enables us to evaluate the minimum Z#Ž c . Žresp. the maximum Z*Ž c .. in Ž1.5. exactly. By definition of the minimum, it then follows E
t
žH
0
F Ž < Bt < . dt G sup Ž c E Ž t . q Z# Ž c . .
/
Ž 1.9.
c)0
whenever t is a stopping time for B. Similarly, by definition of the maximum, we get E
ž
t
H0
F Ž < Bt < . dt F inf Ž c E Ž t . q Z* Ž c . .
/
c)0
Ž 1.10.
for all stopping times t for B. Supposing that the supremum in Ž1.9. Žresp. the infimum in Ž1.10.. is attained at some c# ) 0, the right-hand side in Ž1.9. Žresp. Ž1.10.. defines a function of EŽt .. Since the stopping time t #Ž c . is optimal for the problem Ž1.5. regardless of c ) 0, it is clear that the inequalities Ž1.9. and Ž1.10. are sharp Žthe equality may be attained in a nontrivial manner.. In this paper we apply this method and make explicit computations only for the function F Ž x . s x r with r ) y1, and this yields the inequalities Ž1.2. and Ž1.3., respectively. We want to mention that the cases of other functions F, including the case in which B may start at any point, could be treated similarly. We believe that the presentation of our method above enables one to recognize the constraints on the function F needed to derive the inequalities Ž1.9. and Ž1.10. in a precise manner. For instance, if F is continuous and the map x ¬ H Ž x . y cx 2 is bounded from above Žresp. below. on Rq, then Ž1.9. Žresp. Ž1.10.. holds. Below we will see that the continuity condition on F may be relaxed Žby treating the case in
247
OPTIMAL STOPPING FOR BROWNIAN PATHS
which F Ž x . s x r for y1 - r - 0.. In such a case one should use Ito]Tanaka’s formula Žrather than Ito’s ˆ ˆ formula.. It should be noted, too Žparticularly when applying the optional sampling theorem., that the stopping times appearing throughout may be assumed to be bounded. The results in this paper are similar in nature and can be compared with the results in w4x. In some sense the optimal stopping appearing here could be called a Wald’s type optimal stopping for the integral of Brownian paths. Despite their appealing simplicity, we are unaware of similar results.
2. OPTIMAL STOPPING INEQUALITIES In this section we present the main results of the paper. Throughout B s Ž Bt . t G 0 is standard Brownian motion defined on a probability space Ž V, F , P .. We begin by giving a proof of the inequalities Ž1.1.. To the best of our knowledge, these inequalities have not been written down earlier, but we believe that they are known. THEOREM 2.1. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequalities are satisfied: A p E Ž t 1qp r2 . F E
t
žH
0
< Bt < p dt F Bp E Ž t 1qp r2 .
/
Ž 2.1.
for all stopping times t for B, and all p ) 0, where A p and Bp are numerical constants. The best possible ¨ alues for A p and Bp are AUp s 2 Ž z 2qp . BUp s 2 Ž zU2qp .
2q p
2q p
rŽ 2 q p . Ž 1 q p .
Ž 2.2.
rŽ 2 q p . Ž 1 q p . ,
Ž 2.3.
where z 2q p denotes the smallest positi¨ e zero of the confluent hypergeometric function x ¬ M ŽyŽ2 q p .r2, 1r2, x 2r2., and zU2q p denotes the largest positi¨ e zero of the parabolic cylinder function x ¬ D 2q p Ž x . Ž see w1x.. Ž In particular, z 2 n Ž resp. zU2 n . is the smallest Ž resp. largest . positi¨ e zero of the Hermite polynomial x ¬ He2 nŽ x . for n G 1.. Proof. Let p ) 0 be given and fixed. By Ito’s ˆ formula we have < Bt < 2q p s Ž 2 q p .
t
H0 < B <
1qp
s
s Ž 2 q p . Mt q C p
sign Ž Bs . dBs q 2y1 Ž 2 q p . Ž 1 q p . t
H0 < B < s
p
ds,
t
H0 < B < s
p
ds
Ž 2.4.
GORAN PESKIR ˇ
248
where M s Ž Mt . t G 0 is a continuous local martingale started at zero under P, and C p s 2y1 Ž2 q p .Ž1 q p .. Let w M x s Žw M x t . t G 0 denote the quadratic variation of M, and let t be any stopping time for B. In view of Ž2.1. and monotone convergence theorem, it is no restriction to assume that t is bounded by a constant l ) 0, so that we have EŽ w M xt .
1r2
sE
t
žH
0
1r2
< Bt < 2Ž pq1. dt
F lE
/
ž max < B < 0FtF l
2Ž pq1.
t
s K l pq 2 - `
/ Ž 2.5.
for some K ) 0. Therefore, by Doob’s optional sampling theorem Žsee w3x, w5x., E Ž Mt . s 0.
Ž 2.6.
Thus from Ž2.4. we obtain the following identity: E
ž
t
H0 < B <
p
t
< < 2qp . dt s Cy1 p E Bt
/
Ž 2.7.
Recall Burkholder]Davis’ inequalities: d q E Ž t q r2 . F E < Bt < q
1 - q - ` and
if
E < Bt < q F Dq E Ž t q r2 .
E Ž t q r2 . - ` Ž 2.8.
for all q ) 0,
Ž 2.9.
with the best possible values for Dq and d q given by Žsee w2x.
¡Ž z . d s~ ¢Ž z . ¡Ž z . D s~ ¢Ž z .
q
q
q
U q
q
U q
q
q
q
q
if
2Fq-`
if
1-qF2
if
2Fq-`
if
0 - q F 2,
Ž 2.10.
Ž 2.11.
where z q and zUq are such as in the statement of the theorem. Setting q s p q 2, the claims of the theorem clearly follow from Ž2.7. ] Ž2.11.. The proof is complete. In the next theorem we show how the left-hand inequality in Ž2.1. extends if the quantity EŽt 1q p r2 . is replaced by a function of EŽt ..
OPTIMAL STOPPING FOR BROWNIAN PATHS
249
THEOREM 2.2. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequality is satisfied: E
t
žH
0
< Bt < p dt G
/
2
Ž2 q p. Ž1 q p.
Ž Et .
1q pr2
Ž 2.12.
for all stopping times t for B and all p ) 0. The constant 2rŽ2 q p .Ž1 q p . is the best possible. Proof. Given p ) 0, consider the following optimal stopping problem: V Ž c . s inf E t
t
žH
0
< Bt < p dt y ct ,
/
Ž 2.13.
where c ) 0 is given and fixed, and the infimum is taken over all stopping times t for B having finite expectation. Let t be any such stopping time for B. Then in view of Ž2.12., it is no restriction to assume that t is bounded, so that by Ž2.7. we have E
ž
t
H0 < B < t
p
< Bt < 2qp y ct dt y ct s E Cy1 p
/
ž
/
< Bt < 2q p y c < Bt < 2 s E Cy1 p
ž
s
`
y1 2qp p x
H0 Ž C
/
Ž 2.14.
y cx 2 . dP< Bt < Ž x .
where C p s 2y1 Ž2 q p .Ž1 q p . and P < Bt < denotes the distribution law of < Bt <. 2qp Denote f Ž x . s Cy1 y cx 2 for x ) 0. Then it is easily verified that f p x attains its minimum on Rq at x#Ž c . s Ž cŽ1 q p ..1r p and f Ž x#Ž c .. s ypŽ1 q p . 2r p c 1q2r prŽ2 q p .. Hence from Ž2.14. we find V Ž c. s
yp Ž 1 q p .
Ž2 q p.
2rp
c 1q 2r p .
Ž 2.15.
and the optimal stopping time in Ž2.13. Žthe one at which the infimum is attained. is
t # s inf t ) 0 : < Bt < s x# Ž c . 4 .
Ž 2.16.
ŽObserve by passing to a limit that Ž2.7. holds for t * as well. From Ž2.13. we get E
t
žH
0
< Bt < p dt G sup Ž cE Ž t . q V Ž c . . .
/
c)0
Ž 2.17.
GORAN PESKIR ˇ
250
Denote g Ž c . s cEŽt . q V Ž c . for c ) 0. Then it is easily verified that g attains its maximum on Rq at c# s Ž Et . p r2rŽ1 q p . and g Ž c#. s Ž2rŽ2 q p .Ž1 q p ..Ž Et .1q p r2 . Inserting this into Ž2.17., we get Ž2.12.. The proof is complete. Remark 2.3. 1. Since for t # defined in Ž2.16. we have EŽt #. s 2 Ž . E < Bt# < 2 s x# c s Ž cŽ1 q p .. 2r p for all c ) 0, by taking c s c# s p r2 Ž Et . rŽ1 q p ., we see that each ta s inf t ) 0 : < Bt < s a 4 with a ) 0 is optimal in Ž2.17., and therefore in Ž2.12. as well Žthe equality is attained at ta for all a ) 0.. ŽThis is also verified directly by the Ito]Tanaka formula ˆ and the optional sampling theorem.. 2. The proof presented above can be applied in the case in which Brownian motion B starts at any x g R. In this way we can extend Ž2.12. by proving the following inequality: E
t
žH
0
< Bt q x < p dt G
/
2
Ž2 q p. Ž1 q p.
ž Ž EŽt . q x
2 1q pr2
.
y < x < 2qp ,
/
Ž 2.18. which is valid for all stopping times t for B, all p ) 0, and all x g R. Moreover, the constant 2rŽ2 q p .Ž1 q p . is the best possible for all x g R. In fact, if ta, x s inf t ) 0 : < Bt q x < s a 4 with a ) 0, then the equality in Ž2.18. is attained at ta, x for all a ) 0 and all x g R. 3. Note that Ž2.12. can be obtained more directly by using Jensen’s inequality Žobserve in Ž2.7. that E < Bt < 2q p s EŽ< Bt < 2 .1q p r2 G Ž E < Bt < 2 .1q p r2 s Ž Et .1q p r2 whenever t is bounded and then pass to the limit.. The main value of the proof given above lies in its applicability to the other functionals F Ždifferent from x ¬ < x < p . for which the convexity of x ¬ H Ž'x . with H0 s F Žand thus Jensen’s inequality as well. fails. The same facts hold for Ž2.18.. See also Remark 2.4 in w4x. EXAMPLE 2.4. From Ž2.1. and Ž2.12. it is clear that the quantity EŽt 1q p r2 . on the right-hand side in Ž2.1. cannot generally be replaced by Ž Et .1q p r2 . Here we exhibit a class of stopping times that violate such an inequality. For simplicity we only consider the case p s 1, but other cases could be treated similarly. So we shall show that the inequality E
t
žH
0
< Bt < dt F C Ž Et . 3r2
/
Ž 2.19.
cannot be satisfied for all stopping times t for B with a fixed numerical constant C ) 0.
251
OPTIMAL STOPPING FOR BROWNIAN PATHS
For this consider the hitting times of the square root boundaries: 1r2 Tc s inf t ) 0 : < Bt < s c Ž t q 1 . 4
Ž 2.20.
for c ) 0. Then it is well known that EŽTcq . - ` if and only if c - c0 Ž q ., where c0 Ž q . is the smallest positive zero of the confluent hypergeometric function c ¬ M Žyq, 1r2, c 2r2. Žsee w6x.. In particular, it is easily verified that c 0 Ž1. s 1 ) c 0 Ž3r2. s 0.84 . . . . Choose now 0 - c n c 0 Ž3r2.. Denote Tn s Tc n and T0 s Tc 0 Ž3r2. . Then EŽTn3r2 . - ` for any n G 1 given and fixed, so that E
Tn
žH
0
1r2
< Bt < 4 dt
2
F E Ž c n4 Ž Tn q 1 . Tn .
/
1r2
F c n2 E Ž Tn3r2 q '2 Tn q Tn1r2 .
F c n2 Ž 2 q '2 . Ž 1 q E Ž Tn3r2 . . - `.
Ž 2.21.
Therefore, by Ito’s ˆ formula Žapplied to < Bt < 3 . and Doob’s optional sampling theorem, we find E
Tn
žH
0
< Bt < dt s
/
1 3
E < BT n < 3 s
c n3 3
E Ž Tn q 1 .
3r2
F C Ž ETn .
3r2
.
Ž 2.22.
So, if Ž2.19. would be satisfied, then c n3 3
E Ž Tn3r2 . F
c n3 3
E Ž Tn q 1 .
3r2
Ž 2.23.
for all n G 1. Letting n ª `, we would get c03 Ž 3r2 . 3
E Ž T03r2 . F C Ž ET0 .
3r2
- `,
Ž 2.24.
since c 0 Ž3r2. - 1. However, this contradicts the fact that EŽT03r2 . s `. Thus Ž2.19. cannot be satisfied for all Tn with n G 1. The proof of the claim is complete. Motivated by the preceding example, we change the function of the Brownian path in Ž2.12. so that we obtain the following extension of this inequality. The proof is similar to the proof of Theorem 2.2, and we only sketch the most important parts for the convenience of the reader. THEOREM 2.5. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequality is satisfied: E
ž
t
H0
1 < Bt <
q
dt F
/
2
Ž2 y q. Ž1 y q.
Ž Et .
1y qr2
Ž 2.25.
GORAN PESKIR ˇ
252
for all stopping times t for B and all 0 F q - 1. The constant 2rŽ2 y q .Ž1 y q . is the best possible. Proof. Given 0 F q - 1, consider the following optimal stopping problem: t 1 W Ž c . s sup E Ž 2.26. q dt y ct , 0 < Bt < t
žH
/
where c ) 0 is given and fixed, and the supremum is taken over all Žand stopping times t for B having finite expectation. By Ito]Tanaka’s ˆ . Ž w x . occupation times formula see 5 , pp. 208]209 , we find < Bt < 2y q s Ž 2 y q .
t
H0 < B <
1yq
s
sign Ž Bs . dBs
q 2y1 Ž 2 y q . Ž 1 y q . s Ž 2 y q . Mt q C q
1
t
H0
t
yq
H0 < B <
< Bs < q
s
ds
Ž 2.27.
ds,
where M s Ž Mt . t G 0 is a continuous local martingale started at zero under P, and C q s 2y1 Ž2 y q .Ž1 y q .. Let t be any stopping time for B having finite expectation. Then it is no restriction to assume that t is bounded, so that in exactly the same way as in the proof of Theorem 2.1 ŽDoob’s optional sampling theorem., we find EŽ Mt . s 0. Thus from Ž2.27. we obtain t 1 y1 < < 2y q . E Ž 2.28. q dt s C q E Bt 0 < Bt <
žH
/
Hence we find that E
ž
t
H0
1 < Bt < q
< Bt < 2y q y ct s E Cy1 < Bt < 2yq y c < Bt < 2 dt y ct s E Cy1 q q
/
ž
s
`
H0 Ž C
/
y1 2yq q x
ž
y cx 2 . dP< Bt < Ž x . ,
/
Ž 2.29.
where P < Bt < denotes the distribution law of < Bt <. 2yq Denote f Ž x . s Cy1 y cx 2 for x ) 0. Then it is easily verified that f q x attains its maximum on Rq at x*Ž c . s Ž cŽ1 y q ..y1 r q and f Ž x*Ž c .. s qc1y 2r qrŽ2 y q .Ž1 y q . 2r q. Hence from Ž2.23. we find q W Ž c. s c 1y 2r q , Ž 2.30. 2rq Ž2 y q. Ž1 y q.
OPTIMAL STOPPING FOR BROWNIAN PATHS
253
and the optimal stopping time in Ž2.26. Žthe one at which the infimum is attained. is
t * s inf t ) 0 : < Bt < s x* Ž c . 4 .
Ž 2.31.
ŽNote that Ž2.28. holds for t * as well.. From Ž2.26. we get E
ž
1
t
H0
< Bt < q
dt F inf Ž cE Ž t . q W Ž c . . .
/
c)0
Ž 2.32.
Denote g Ž c . s cEŽt . q W Ž c . for c ) 0. Then it is easily verified that g attains its minimum on Rq at c# s 1rŽ1 y q .Ž Et . q r2 and g Ž c#. s Ž2rŽ2 y q .Ž1 y q ..Ž Et .1y q r2 . Inserting this into Ž2.32., we get Ž2.25.. The proof is complete. Remark 2.6. 1. In exactly the same way as in part 1 of Remark 2.3 above, we can verify that each ta s inf t ) 0 : < Bt < s a 4 with a ) 0 is optimal in Ž2.32., and therefore in Ž2.25. as well Žthe equality is attained at ta for all a ) 0.. ŽThis is also obtained directly by Ito]Tanaka’s formula ˆ and the optional sampling theorem.. Finally, letting q1 in Ž2.25. with t s ta , we see that Ž1.4. holds for all a ) 0. This explains why q in Theorem 2.5 must be strictly less than 1. 2. The proof above can be applied in the case in which Brownian motion B starts at any x g R. In this way we can extend Ž2.25. by proving the following inequality: E
t
žH
0
1 < Bt q x < q
dt F
/
2
Ž EŽt . q x Ž2 y q. Ž1 y q. ž
2 1y qr2
.
y < x < 2yq ,
/
Ž 2.33. which is valid for all stoppping times t for B, all 0 F q - 1, and all x g R. Moreover, the constant 2rŽ2 y q .Ž1 y q . is the best possible for all x g R Žthe equality is attained at ta, x s inf t ) 0 : < Bt q x < s a 4 for all a ) 0 and all x g R.. 3. Note that Ž2.25. can be obtained more directly by using Jensen’s inequality Žobserve in Ž2.28 . that E < Bt < 2y q s E Ž < Bt < 2 . 1y q r 2 F Ž E < Bt < 2 .1y q r2 s Ž Et .1y q r2 whenever t is bounded, and then pass to the limit.. The main value of the proof given above lies in its applicability to the other functionals F Ždifferent from x ¬ < x
254
GORAN PESKIR ˇ
From Ž2.28. in the proof of Theorem 2.5 and Ž2.8. ] Ž2.11. in the proof of Theorem 2.1, we obtain the following result. COROLLARY 2.7. Let B s Ž Bt . t G 0 be standard Brownian motion. Then the following inequalities are satisfied: A q E Ž t 1y q r2 . F E
ž
1
t
H0
< Bt < q
dt F Bq E Ž t 1yq r2 .
/
Ž 2.34.
for all stopping times t for B and all 0 F q - 1, where A q and Bq are numerical constants. The best possible ¨ alues for A q and Bq are AUq s 2 Ž zU2yq . BUq s 2 Ž z 2yq .
2y q
2y q
rŽ 2 y q . Ž 1 y q .
Ž 2.35.
rŽ 2 y q . Ž 1 y q . ,
Ž 2.36.
where zU2y q denotes the largest positi¨ e zero of the parabolic cylinder function x ¬ D 2y q Ž x ., and z 2yq denotes the smallest positi¨ e zero of the confluent hypergeometric function x ¬ M ŽyŽ2 y q .r2, 1r2, x 2r2. Ž see w1x..
REFERENCES 1. M. Abramowicz and I. A. Stegun, ‘‘Handbook of Mathematical Functions,’’ Dover, New York, 1970. 2. B. Davis, On the L p norms of stochastic integrals and other martingales, Duke Math. J. 43 Ž1976., 697]704. 3. J. L. Doob, ‘‘Stochastic Processes,’’ Wiley, New York, 1953. 4. S. E. Graversen and G. Peskir, ˇ On Wald-type optimal stopping for Brownian motion, J. Appl. Probab. 34 Ž1997., 66]73. 5. D. Revuz and M. Yor, ‘‘Continuous Martingales and Brownian Motion,’’ Springer-Verlag, Berlin, 1991. 6. L. A. Shepp, A first passage problem for the Wiener process, Ann. Math. Statist. 38 Ž1967., 1912]1914.