EQUIDISTANT SAMPLING FOR THE MAXIMUM OF A BROWNIAN MOTION WITH DRIFT ON A FINITE HORIZON

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1 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO WITH DRIFT O A FIITE HORIZO A.J.E.M. JASSE AD J.S.H. VA LEEUWAARDE Abstract. A Brownian motion observed at equidistant sampling points renders a random walk with normally distributed increments. For the difference between the expected maximum of the Brownian motion and its sampled version, an expansion is derived with coefficients in terms of the drift, the Riemann zeta function and the normal distribution function. Keywords: Gaussian random walk; maximum; Riemann zeta function; Euler- Maclaurin summation; equidistant sampling of Brownian motion; finite horizon. AMS 2 Subject Classification: M6, 3B4, 6G5, 6G5, 65B5. Introduction Let {Bt)} t denote a Brownian motion with drift coefficient µ and variance parameter σ 2, so that Bt) = µt + σw t), ) with {W t)} t a Wiener process standard Brownian motion). Without loss of generality, we set B) =, σ = and consider the Brownian motion on the interval [, ]. When we sample the Brownian motion at time points n, n =,,..., the resulting process is a random walk with normally distributed increments, also known as the Gaussian random walk. The fact that Brownian motion evolves in continuous space and time leads to great simplifications in determining its properties. In contrast, the Gaussian random walk, that moves only at equidistant points in time, is an object much harder to study. Although it is obvious that, for, the behavior of the Gaussian random walk can be characterized by the continuous time diffusion equation, there are many effects to take into account for finite. This paper deals with the expected maximum of the Gaussian random walk and, in particular, its deviation from the expected maximum of the underlying Brownian motion. This relatively simple characteristic already turns out to have an intriguing description: In Section 2 we derive an expansion with coefficients in terms of the Riemann zeta function and the derivatives of) the normal distribution function. Some historical remarks follow, and the proof is presented in Section 3. Date: October 2, 28.

2 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO 2 2. Main result and discussion The expected maximum of the Gaussian random walk is by Spitzer s identity see [9, 4]) given by E max Bn/) = n=,..., n EB+ n/), 2) where B + t) = max{, Bt)}. The difference between the Gaussian random walk and the Brownian motion decreases with the number of sampling points. In particular, the monotone convergence theorem, in combination with a Riemann sum approximation of the right-hand side of 2), gives see []) E max t Bt) = n= t EB+ t)dt. 3) The mean sampling error, as a function of the number of sampling points is then given by E µ) = t EB+ t)dt n= n EB+ n/). 4) Since Bt) is normally distributed with mean µt and variance t one can compute EB + t) = µtφµ ) t /2 t) + e 2 µ2t, 5) 2π where Φx) = 2π x e 2 u2 du. Substituting 5) into 4) yields where E µ) = gt)dt gn/), 6) n= gt) = µφµ t) + 2πt e 2 µ2t. 7) We are then in the position to present our main result. Theorem. The difference in expected maximum between {Bt)} t and its associated Gaussian random walk obtained by sampling {Bt)} t at equidistant points, for µ/ < 2 π, is given by E µ) = ζ/2) 2g) µ 2π 4 2π r= p k= ζ /2 r) /2) r r!2r + )2r + 2) B 2k g 2k ) ) 2k)! 2k µ ) 2r+2 + O/ 2p+2 ), 8) with O uniform in µ, ζ the Riemann zeta function, p some positive integer, B n the Bernoulli numbers, and g k) defined as the kth derivative of g in 7).

3 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO 3 E µ) shows up in a range of applications. Examples are sequentially testing for the drift of a Brownian motion [7], corrected diffusion approximations [7], simulation of Brownian motion [, 5], option pricing [3], queueing systems in heavy traffic [2, 3, 5], and the thermodynamics of a polymer chain [8]. The expression in 8) for E µ) involves terms c j j/2 with c = ζ/2) 2π, c 2 = µ 2µΦ µ) + 2φµ) 4, c 3 = ζ /2)µ2 2, c 4 = φµ) 2π 24, 9) φx) = e x2 /2 / 2π and c j = for j = 6,, 4,.... The first term c has been identified by Asmussen, Glynn & Pitman [], Thm. 2 on p. 884, and Calvin [5], Thm. on p. 6, although Calvin does not express c in terms of the Riemann zeta function. The second term c 2 was derived by Broadie, Glasserman & Kou [3], Lemma 3 on p. 77, using extended versions of the Euler-Maclaurin summation formula presented in []. To the best of the authors knowledge, all higher terms appear in the present paper for the first time. The distribution of the maximum of Brownian motion with drift on a finite interval is known to be see Shreve [8], p. 297) x µt ) P max Bt) x) = Φ e 2µx Φ t T T and integration thus yields x µt ), x, ) T E max t T Bt)) = 2µ 2Φµ T ) ) + Φµ T )µt + φµ T ) T. ) A combination of ) and 8) leads to a full characterization of the expected maximum of the Gaussian random walk. ote that the mean sampling error for the Brownian motion defined in ) on [, T ], sampled at equidistant points, is given by σ T E µ T /σ). When the drift µ is negative, results can be obtained for the expected all-time maximum. That is, for the special case µ <, σ =, T = and, one finds that lim E µ ) is equal to ζ/2) + 2π 4 µ µ2 ζ /2 r) 2π r!2r + )2r + 2) r= ) µ 2 r, 2) for 2 π < µ <. ote that 2) follows from Theorem. The result, however, was first derived by Pollaczek [6] in 93 see also []). Apparently unaware of this fact, Chernoff [7] obtained the first term ζ/2)/ 2π, Siegmund [7], Problem.2 on p. 227, obtained the second term /4 and Chang & Peres [6], p. 8, obtained the third term ζ /2)/2 2π. The complete result was rediscovered by the authors in [9], and more results for the Gaussian random walk were presented in [9, ], including series representations for all cumulants of the all-time maximum. 3. Proof of Theorem We shall treat separately the cases µ <, µ > and µ =. The proof for µ < in Subsection 3. largely builds upon Euler-Maclaurin summation and 2

4 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO 4 the result in Section 4 of [9] on the expected value of the all-time maximum of the Gaussian random walk. The result for µ > in Subsection 3.2 then follows almost immediately due to convenient symmetry properties of Φ. Finally, in Subsection 3.3, the issue of uniformity in µ is addressed and the result for µ = is established in two ways: First by taking the limit µ and subsequently by a direct derivation that uses Spitzer s identity 4) for µ = and an expression for the Hurwitz zeta function. 3.. The negative-drift case. Set µ = γ with γ >. We have from 6) { E µ) = gt)dt We compute by partial integration gt)dt = { gt)dt Furthermore, with β = γ/, gn/) = n= } gn/) n= n=+ gn/) γφ γ t)dt + t /2 e 2 γ2t dt 2π }. 3) = 2γ + γ = 2γ. 4) n= γφ γ n/ ) ) + e 2 γ2 n/ 2πn/ = e 2 β2 n βφ β ) n) = EM, 5) 2πn n= with EM as in 4.) of [9]. From 4), 5) and [9], 4.25), it follows that gt)dt n= gn/) = ζ/2) 2π γ 4 2π r= ζ /2 r) /2) r r!2r + )2r + 2) γ ) 2r+2. 6) This handles the first term on the right-hand side of 3). For the second term, we use Euler-Maclaurin summation see De Bruijn [4], Sec. 3.6, pp. 4-42) for the series n=+ gn/). With fx) = gx/), x, 7)

5 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO 5 we have for p =, 2,... fn) = f) + n=+ fn) n= [ M = f) + lim M p + k= M B 2k 2k)! fx)dx + 2 f) f 2k ) M) f 2k ) ) f 2p) x) B 2p x x ) dx 2p)! p = 2 f) + fx)dx k= ] ) B 2k 2k)! f 2k ) ) + R p,, 8) where B n t) denotes the nth Bernoulli polynomial, B n = B n ) denotes the nth Bernoulli number, and R p, = Since f l) x) = g l) x/)/ l+, we thus obtain where n=+ gn/) = 2 g) + R p, = 2p f 2p) x) B 2p x x ) dx. 9) 2p)! k= gx)dx p B 2k 2k)! 2k g2k ) ) + R p,, 2) g 2p) x) B 2p x x ) dx. 2) 2p)! From the definition of g in 7) it is seen that g 2p) is smooth and rapidly decaying, hence R p, = O/ 2p ). Since R p, = B 2p+2 2p + 2)! 2p+2 g2p+) ) + R p+,, 22) we even have R p, = O/ 2p+2 ). Therefore, from 2), gt)dt n=+ gn/) = 2 g) + p k= B 2k 2k)! 2k g2k ) ) + O/ 2p+2 ). 23) Combining 6) and 23) completes the proof, aside from the uniformity issue, for the case that µ = γ <.

6 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO The positive-drift case. The analysis so far was for the case with negative drift µ = γ with γ >. The results can be transferred to the case that µ > as follows. ote first from Φ x) = Φx) that gt) = µ Φ µ t) + 2πt) /2 exp 2 µ)2 t). Therefore, by 6) E µ) = E µ), 24) since the term µ vanishes from the right-hand side of 6). Then use the result already proved with µ < instead of µ. This requires replacing gt) from 7) by µφ µ t) + 2πt) /2 e 2 µ)2 t 25) and µ by µ everywhere in 8). The term 2gt) µ then becomes 2 µφ µ ) t) + 2πt) /2 exp 2 µ)2 t) µ) = 2 µφµ ) t) + 2πt) /2 exp 2 µ2 t) µ, 26) which is in the form 2gt) µ with g from 7). ext we compute g t) = 2 2π t 3/2 e 2 µ2 t = d [ µφ µ ] t) + 2πt) /2 e 2 µ)2 t. 27) dt Finally, the infinite series with the ζ-function involves µ quadratically. Thus writing down 8) with µ < instead of µ turns the right-hand side into the same form with g given by 7). This completes the proof of Theorem for µ The zero-drift case. We shall first establish the uniformity in µ < of the error term O in 8), for which we need that R p, = 2p g 2p) x) B 2p x x ) dx 28) 2p)! can be bounded uniformly in µ < as O 2p ). Write ν = 2 µ2, and observe from 27) and ewton s formula that for k =, 2,... g k) t) = 2 2π = )k 2 2π e νt ) d k [ t 3/2 e νt] dt k ) k n= n n )νk n t 3/2 n. 29) Hence, g 2p) t) > and g 2p ) ) < for p =, 2,.... Therefore, with C an upper bound for B 2p x x ) /2p)!, 3)

7 we have EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO 7 R p, C 2p = C 2p g 2p) t)dt = C 2p g2p ) ) e ν 2p 2 2 ) 2p 2 3 2π n n )ν2p 2 n, 3) n= which is bounded in ν > when p =, 2... is fixed. This settles the uniformity issue and thus the case µ = by letting µ. A direct derivation of the result 8) for the case µ = is also possible. When ζs, x) is the analytic continuation to C \ { } of the function ζs, x) = n + x) s, Res) >, x R, 32) x<n n> x then for s C \ { } and p =, 2,... with 2p + > Res), there holds see Borwein, Bradley & Crandall [2], Section 3, for similar expressions) ζs, x) = n + x) s x + ) s s 2 x + ) s Combination of p ) s B2k 2k s x + ) s 2k+ + O s 2p ). 33) k= E ) = 2 2π /2 k= ) n /2 n= 34) and 33) with s = /2, x = and replaced by, leads to E ) = ζ/2) 2π 2 2π 2 p ) /2 B 2k 2k + O 2p 2 ). 35) 2π 2k ote that ) 2 /2 B 2k = B 2k π 2k 2k)! h2k ) t) ; ht) =, 36) t= 2πt and so 35) corresponds to 8) with µ =, indeed. References [] Asmussen, S., Glynn, P. and Pitman, J. 995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Annals of Applied Probability 5: [2] Borwein, J.M., Bradley, D.M., Crandall, R.E. 2). Computational strategies for the Riemann zeta function. Journal of Computational and Applied Mathematics 2: [3] Broadie, M., Glasserman, P. and Kou, S.G. 999). Connecting discrete and continuous path-dependent options. Finance and Stochastics 3: [4] De Bruijn,.G. 98). Asymptotic Methods in Analysis. Dover Publications, ew York. [5] Calvin, J. 995). Average performance of nonadaptive algorithms for global optimization. Journal of Mathematical Analysis and Applications 9: [6] Chang, J.T. and Peres, Y. 997). Ladder heights, Gaussian random walks and the Riemann zeta function. Annals of Probability 25:

8 EQUIDISTAT SAMPLIG FOR THE MAXIMUM OF A BROWIA MOTIO 8 [7] Chernoff, H. 965). Sequential test for the mean of a normal distribution IV discrete case). Annals of Mathematical Statistics 36: [8] Comtet, A. and Majumdar, S.. 25). Precise asymptotics for a random walker s maximum. Journal of Statistical Mechanics: Theory and Experiment, P63. [9] Janssen, A.J.E.M. and van Leeuwaarden, J.S.H. 27). On Lerch s transcendent and the Gaussian random walk. Annals of Applied Probability 7: [] Janssen, A.J.E.M. and van Leeuwaarden, J.S.H. 27). Cumulants of the maximum of the Gaussian random walk. Stochastic Processes and Their Applications 7: [] Janssen, A.J.E.M. and van Leeuwaarden, J.S.H. 28). Back to the roots of queueing theory and the works of Erlang, Crommelin and Pollaczek. Statistica eerlandica 62: [2] Janssen, A.J.E.M., van Leeuwaarden, J.S.H. and Zwart, B. 28). Corrected diffusion approximations for a multi-server queue in the Halfin-Whitt regime. Queueing Systems 58: [3] Jelenković, P., Mandelbaum, A. and Momčilović, P. 24). Heavy traffic limits for queues with many deterministic servers. Queueing Systems 47: [4] Kingman, J.F.C. 962). Spitzer s identity and its use in probability theory. Journal of the London Mathematical Society 37: [5] Kingman, J.F.C. 965). The heavy traffic approximation in the theory of queues. In: Proc. Symp. on Congestion Theory, eds. W.L. Smith and W. Wilkinson University of orth Carolina Press, Chapel Hill) pp [6] Pollaczek, F. 93). Über zwei Formeln aus der Theorie des Wartens vor Schaltergruppen. Elektrische achrichtentechnik 8: [7] Siegmund, D. 985). Corrected Diffusion Approximations in Certain Random Walk Problems. Springer-Verlag, ew York. [8] Shreve, S.E. 24). Stochastic Calculus for Finance II. Continuous-Time Models. Springer-Verlag, ew York. [9] Spitzer, F.L. 956). A combinatorial lemma and its application to probability theory. Transactions of the American Mathematical Society 82: Philips Research. Digital Signal Processing Group, High Tech Campus 36, 5656 AE Eindhoven, The etherlands address: a.j.e.m.janssen@philips.com Eindhoven University of Technology and EURADOM. P.O. Box MB Eindhoven, The etherlands address: j.s.h.v.leeuwaarden@tue.nl