Brownian bridge expansions for Lévy area approximations and particular values of the Riemann zeta function
BBROWNIAN BRIDGE EXPANSIONS FOR LÉVY AREA APPROXIMATIONSAND PARTICULAR VALUES OF THE RIEMANN ZETA FUNCTION
JAMES FOSTER AND KAREN HABERMANN
Abstract.
We study approximations for the Lévy area of Brownian motion which are basedon the Fourier series expansion and a polynomial expansion of the associated Brownian bridge.Comparing the asymptotic convergence rates of the Lévy area approximations, we see thatthe approximation resulting from the polynomial expansion of the Brownian bridge is moreaccurate than the Kloeden–Platen–Wright approximation, whilst still only using independentnormal random vectors. We then link the asymptotic convergence rates of these approximationsto the limiting fluctuations for the corresponding series expansions of the Brownian bridge.Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processesfor the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to givea stand-alone derivation of the values of the Riemann zeta function at even positive integers. Introduction
One of the well-known applications for expansions of the Brownian bridge is the strong or L ( P ) approximation of stochastic integrals. Most notably, the second iterated integrals of Brownianmotion are required by high order strong numerical methods for general stochastic differentialequations (SDEs), as discussed in [3, 15, 22]. Due to integration by parts, such integrals can beexpressed in terms of the increment and Lévy area of Brownian motion. The approximation ofmultidimensional Lévy area is well-studied, see [4, 5, 8, 10, 11, 16, 17, 21, 24], with the majority ofthe algorithms proposed being based on a Fourier series expansion or the standard piecewise linearapproximation of Brownian motion. Some alternatives include [4, 8, 17] which consider methodsassociated with a polynomial expansion of the Brownian bridge.In this paper, we compare the approximations of Lévy area based on the Fourier series expansionand on a polynomial expansion of the Brownian bridge. We particularly observe their convergencerates and link those to the fluctuation processes associated with the different expansions of theBrownian bridge. The fluctuation process for the polynomial expansion is studied in [12], and ourstudy of the fluctuation process for the Fourier series expansion allows us, at the same time, todetermine the fluctuation process for the Karhunen–Loève expansion of the Brownian bridge. Asan attractive side result, we extend the required analysis to obtain a stand-alone derivation of thevalues of the Riemann zeta function at even positive integers. Throughout, we denote the positiveintegers by N and the non-negative integers by N .Let us start by considering a Brownian bridge ( B t ) t ∈ [0 , in R with B = B = 0 . This is the uniquecontinuous-time Gaussian process with mean zero and whose covariance function K B is given by,for s, t ∈ [0 , ,(1.1) K B ( s, t ) = min( s, t ) − st . Mathematics Subject Classification. a r X i v : . [ m a t h . P R ] F e b J. FOSTER AND K. HABERMANN
We are concerned with the following three expansions of the Brownian bridge. The Karhunen–Loèveexpansion of the Brownian bridge, see Loève [18, p. 144], is of the form, for t ∈ [0 , ,(1.2) B t = ∞ (cid:88) k =1 kπt ) kπ (cid:90) cos( kπr ) d B r . The Fourier series expansion of the Brownian bridge, see Kloeden–Platen [15, p. 198] or Kahane [14,Sect. 16.3], yields, for t ∈ [0 , ,(1.3) B t = 12 a + ∞ (cid:88) k =1 ( a k cos(2 kπt ) + b k sin(2 kπt )) , where, for k ∈ N ,(1.4) a k = 2 (cid:90) cos(2 kπr ) B r d r and b k = 2 (cid:90) sin(2 kπr ) B r d r . A polynomial expansion of the Brownian bridge in terms of the shifted Legendre polynomials Q k on the interval [0 , of degree k , see [9, 12], is given by, for t ∈ [0 , ,(1.5) B t = ∞ (cid:88) k =1 (2 k + 1) c k (cid:90) t Q k ( r ) d r , where, for k ∈ N ,(1.6) c k = (cid:90) Q k ( r ) d B r . These expansions are summarised in Table 1 in Appendix A and they are discussed in more detailin Section 2. For an implementation of the corresponding approximations for Brownian motion asChebfun examples into MATLAB, see Trefethen et al. [6, 23].We remark that the polynomial expansion (1.5) can be viewed as a Karhunen–Loève expansion ofthe Brownian bridge with respect to the weight function w on (0 , given by w ( t ) = t (1 − t ) . Thisapproach is employed in [9] to derive the expansion along with the standard optimality property ofKarhunen–Loève expansions. In this setting, the polynomial approximation of ( B t ) t ∈ [0 , is optimalamong truncated series expansions in a weighted L ( P ) sense corresponding to the non-constantweight function w . To avoid confusion, we still adopt the convention throughout to reserve the termKarhunen–Loève expansion for (1.2), whereas (1.5) is referred to as the polynomial expansion.Before we investigate the approximations of Lévy area based on the different expansions of theBrownian bridge, we first analyse the fluctuations associated with the expansions. The fluctuationprocess for the polynomial expansion is studied and characterised in [12], and these results arerecalled in Section 2.3. The fluctuation processes ( F N, t ) t ∈ [0 , for the Karhunen–Loève expansionand the fluctuation processes ( F N, t ) t ∈ [0 , for the Fourier series expansion are defined as, for N ∈ N ,(1.7) F N, t = √ N (cid:32) B t − N (cid:88) k =1 kπt ) kπ (cid:90) cos( kπr ) d B r (cid:33) , and(1.8) F N, t = √ N (cid:32) B t − a − N (cid:88) k =1 ( a k cos(2 kπt ) + b k sin(2 kπt )) (cid:33) . ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 3
The scaling by √ N in the process ( F N, t ) t ∈ [0 , is the natural scaling to use because increasing N by one results in the subtraction of two additional Gaussian random variables. We use E to denotethe expectation with respect to Wiener measure P . Theorem 1.1.
The fluctuation processes ( F N, t ) t ∈ [0 , for the Karhunen–Loève expansion convergein finite dimensional distributions as N → ∞ to the collection ( F t ) t ∈ [0 , of independent Gaussianrandom variables with mean zero and variance E î (cid:0) F t (cid:1) ó = ® π if t ∈ (0 , if t = 0 or t = 1 . The fluctuation processes ( F N, t ) t ∈ [0 , for the Fourier expansion converge in finite dimensionaldistributions as N → ∞ to the collection ( F t ) t ∈ [0 , of zero-mean Gaussian random variableswhose covariance structure is given by, for s, t ∈ [0 , , E (cid:2) F s F t (cid:3) = ® π if s = t or s, t ∈ { , } otherwise . The difference between the fluctuation result for the Karhunen–Loève expansion and the fluctuationresult for the polynomial expansion, see [12, Theorem 1.6] or Section 2.3, is that there the variancesof the independent Gaussian random variables follow the semicircle π (cid:112) t (1 − t ) whereas here theyare constant on (0 , , see Figure 1. The limit fluctuations for the Fourier series expansion furtherexhibit endpoints which are correlated. Figure 1.
Table showing basis functions and fluctuations for the Brownian bridge expansions.As pointed out in [12], the reason for considering convergence in finite dimensional distributionsfor the fluctuation processes is that the limit fluctuations neither have a realisation as processes in C ([0 , , R ) , nor are they equivalent to measurable processes. J. FOSTER AND K. HABERMANN
We prove Theorem 1.1 by studying the covariance functions of the Gaussian processes ( F N, t ) t ∈ [0 , and ( F N, t ) t ∈ [0 , given in Lemma 2.2 and Lemma 2.3 in the limit N → ∞ . The key ingredient isthe following limit theorem for sine functions, which we see concerns the pointwise convergence forthe covariance function of ( F N, t ) t ∈ [0 , . Theorem 1.2.
For all s, t ∈ [0 , , we have lim N →∞ N (cid:32) min( s, t ) − st − N (cid:88) k =1 kπs ) sin( kπt ) k π (cid:33) = ® π if s = t and t ∈ (0 , otherwise . The proof of Theorem 1.2 is split into an on-diagonal and an off-diagonal argument. We start byproving the convergence on the diagonal away from its endpoints by establishing locally uniformconvergence, which ensures continuity of the limit function, and by using a moment argument toidentify the limit. As a consequence of the on-diagonal convergence, we obtain the next corollarywhich then implies the off-diagonal convergence in Theorem 1.2.
Corollary 1.3.
For all t ∈ (0 , , we have lim N →∞ N ∞ (cid:88) k = N +1 cos(2 kπt ) k π = 0 . Moreover, and of interest in its own right, the moment analysis we use to prove the on-diagonalconvergence in Theorem 1.2 leads to a stand-alone derivation of the result that the values of theRiemann zeta function ζ : C \ { } → C at even positive integers can be expressed in terms of theBernoulli numbers B n as, for n ∈ N , ζ (2 n ) = ( − n +1 (2 π ) n B n n )! , see Borevich and Shafarevich [2]. In particular, the identity(1.9) ∞ (cid:88) k =1 k = π , that is, the resolution to the Basel problem posed by Mengoli [19] is a consequence of our analysisand not a prerequisite for it.We turn our attention to studying approximations of second iterated integrals of Brownian motion.For d ≥ , let ( W t ) t ∈ [0 , denote a d -dimensional Brownian motion and let ( B t ) t ∈ [0 , given by B t = W t − tW be its associated Brownian bridge in R d . We denote the independent componentsof ( W t ) t ∈ [0 , by ( W ( i ) t ) t ∈ [0 , , for i ∈ { , . . . , d } , and the components of ( B t ) t ∈ [0 , by ( B ( i ) t ) t ∈ [0 , ,which are also independent by construction. We now focus on approximations of Lévy area. Definition 1.4.
The Lévy area of the d -dimensional Brownian motion W over the interval [ s, t ] is the antisymmetric d × d matrix A s,t with the following entries, for i, j ∈ { , . . . , d } , A ( i,j ) s,t := 12 Ç (cid:90) ts Ä W ( i ) r − W ( i ) s ä d W ( j ) r − (cid:90) ts Ä W ( j ) r − W ( j ) s ä d W ( i ) r å . For an illustration of Lévy area for a two-dimensional Brownian motion, see Figure 2.
Remark 1.5.
Given the increment W t − W s and the Lévy area A s,t , we can recover the seconditerated integrals of Brownian motion using integration by parts as, for i, j ∈ { , . . . , d } with i (cid:54) = j , (cid:90) ts Ä W ( i ) r − W ( i ) s ä d W ( j ) r = 12 Ä W ( i ) t − W ( i ) s ä Ä W ( j ) t − W ( j ) s ä + A ( i,j ) s,t . ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 5 𝑊 ሺ2ሻ 𝑊 ሺ1ሻ Figure 2.
Lévy area is the chordal area between independent Brownian motions.We consider the sequences { a k } k ∈ N , { b k } k ∈ N and { c k } k ∈ N of Gaussian random vectors, wherethe coordinate random variables a ( i ) k , b ( i ) k and c ( i ) k are defined for i ∈ { , . . . , d } by (1.4) and (1.6),respectively, in terms of the Brownian bridge ( B ( i ) t ) t ∈ [0 , . Using the random coefficients arising fromthe Fourier series expansion (1.3), we obtain the Kloeden–Platen approximation in [15], and theKloeden–Platen–Wright approximation in [16, 24] of Brownian Lévy area, which we shall compareto an approximation in terms of the random coefficients from the polynomial expansion (1.5).These Lévy area approximations are summarised in Table 2 in Appendix A and have the followingasymptotic convergence rates. Theorem 1.6 (Asymptotic convergence rates of Lévy area approximations) . For n ∈ N , we set N = 2 n and define approximations (cid:98) A n , (cid:101) A n and s A n of the Lévy area A , by, for i, j ∈ { , . . . , d } , (cid:98) A ( i,j ) n := 12 Ä a ( i )0 W ( j )1 − W ( i )1 a ( j )0 ä + π n − (cid:88) k =1 k Ä a ( i ) k b ( j ) k − b ( i ) k a ( j ) k ä , (1.10) (cid:101) A ( i,j ) n := π n − (cid:88) k =1 k Å a ( i ) k Å b ( j ) k − kπ W ( j )1 ã − Å b ( i ) k − kπ W ( i )1 ã a ( j ) k ã , (1.11) s A ( i,j )2 n := 12 Ä W ( i )1 c ( j )1 − c ( i )1 W ( j )1 ä + 12 n − (cid:88) k =1 Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä . (1.12) Then (cid:98) A n , (cid:101) A n and s A n are antisymmetric d × d matrices and, for i (cid:54) = j and as N → ∞ , we have E ï (cid:16) A ( i,j )0 , − (cid:98) A ( i,j ) n (cid:17) ò ∼ π Å N ã , E ï (cid:16) A ( i,j )0 , − (cid:101) A ( i,j ) n (cid:17) ò ∼ π Å N ã , E ï (cid:16) A ( i,j )0 , − s A ( i,j )2 n (cid:17) ò ∼ Å N ã . J. FOSTER AND K. HABERMANN
The asymptotic convergence rates in Theorem 1.6 are phrased in terms of N since the number ofGaussian random vectors required to define the above Lévy area approximations is N or N − ,respectively. Of course, it is straightforward to define the polynomial approximation s A n for n ∈ N ,see Theorem 5.4.Intriguingly, the convergence rates for the approximations resulting from the Fourier series and thepolynomial expansion correspond exactly with the areas under the limit variance function for eachfluctuation process, which are (cid:90) π d t = 1 π and (cid:90) π » t (1 − t ) d t = 18 . We provide heuristics demonstrating how this correspondence arises at the end of Section 5.By adding an additional Gaussian random matrix that matches the covariance of the tail sum,it is possible to derive high order Lévy area approximations with O ( N − ) convergence in L ( P ) .Wiktorsson [24] proposed this approach using the Kloeden–Platen–Wright approximation (1.11)and this was recently improved by Mrongowius and Rößler in [21] who use the approximation (1.10)obtained from the Fourier series expansion (1.3).We expect that an O ( N − ) polynomial-based approximation is possible using the same techniques.While this approximation should be slightly less accurate than the Fourier approach, we expectit to be easier to implement due to both the independence of the coefficients { c k } k ∈ N and thecovariance of the tail sum having a closed-form expression, see Theorem 5.4. Moreover, this typeof approach has already been studied in [4, 7, 8] with Brownian Lévy area being approximated by(1.13) Û A ( i,j )0 , := 12 Ä W ( i )1 c ( j )1 − c ( i )1 W ( j )1 ä + λ ( i,j )0 , , where the antisymmetric d × d matrix λ , is normally distributed and designed so that Û A , has thesame covariance structure as the Brownian Lévy area A , . Davie [4] as well as Flint and Lyons [7]generate each ( i, j ) -entry of λ , independently as λ ( i,j )0 , ∼ N (cid:0) , (cid:1) for i < j . In [8], it is shownthat the covariance structure of A , can be explicitly computed conditional on both W and c .By matching the conditional covariance structure of A , , the work [8] obtains the approximation λ ( i,j )0 , ∼ N Å ,
120 + 120 (cid:16)(cid:0) c ( i )1 (cid:1) + (cid:0) c ( j )1 (cid:1) (cid:17) ã , where the entries (cid:8) λ ( i,j )0 , (cid:9) i In Section 2, we provide an overview of the three expansionswe consider for the Brownian bridge, and we characterise the associated fluctuation processes ( F N, t ) t ∈ [0 , and ( F N, t ) t ∈ [0 , . Before discussing their behaviour in the limit N → ∞ , we initiatethe moment analysis used to prove the on-diagonal part of Theorem 1.2 and we extend the analysisto determine the values of the Riemann zeta function at even positive integers in Section 3. Theproof of Theorem 1.2 follows in Section 4, where we complete the moment analysis and establish alocally uniform convergence to identify the limit on the diagonal, before we deduce Corollary 1.3, ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 7 which then allows us to obtain the off-diagonal convergence in Theorem 1.2. We close Section 4 byproving Theorem 1.1. In Section 5, we compare the asymptotic convergence rates of the differentapproximations of Lévy area, which results in a proof of Theorem 1.6.2. Series expansions for the Brownian bridge We discuss the Karhunen–Loève expansion as well as the Fourier expansion of the Brownian bridgemore closely, and we derive expressions for the covariance functions of their Gaussian fluctuationprocesses.In our analysis, we frequently use a type of Itô isometry for Itô integrals with respect to a Brownianbridge, and we include its statement and proof for completeness. Lemma 2.1. Let ( B t ) t ∈ [0 , be a Brownian bridge in R with B = B = 0 , and let f, g : [0 , → R be integrable functions. Setting F (1) = (cid:82) f ( t ) d t and G (1) = (cid:82) g ( t ) d t , we have E ñÇ (cid:90) f ( t ) d B t å Ç (cid:90) g ( t ) d B t åô = (cid:90) f ( t ) g ( t ) d t − F (1) G (1) . Proof. For a standard one-dimensional Brownian motion ( W t ) t ∈ [0 , , the process ( W t − tW ) t ∈ [0 , has the same law as the Brownian bridge ( B t ) t ∈ [0 , . In particular, the random variable (cid:82) f ( t ) d B t is equal in law to the random variable (cid:90) f ( t ) d W t − W (cid:90) f ( t ) d t = (cid:90) f ( t ) d W t − W F (1) . Using a similar expression for (cid:82) g ( t ) d B t and applying the usual Itô isometry, we deduce that E ñÇ (cid:90) f ( t ) d B t å Ç (cid:90) g ( t ) d B t åô = (cid:90) f ( t ) g ( t ) d t − F (1) (cid:90) g ( t ) d t − G (1) (cid:90) f ( t ) d t + F (1) G (1)= (cid:90) f ( t ) g ( t ) d t − F (1) G (1) , as claimed. (cid:3) The Karhunen–Loève expansion. Mercer’s theorem, see [20], states that for a continuoussymmetric non-negative definite kernel K : [0 , × [0 , → R there exists an orthonormal basis { e k } k ∈ N of L ([0 , which consists of eigenfunctions of the Hilbert–Schmidt integral operatorassociated with K and whose eigenvalues { λ k } k ∈ N are non-negative and such that, for s, t ∈ [0 , ,we have the representation K ( s, t ) = ∞ (cid:88) k =1 λ k e k ( s ) e k ( t ) , which converges absolutely and uniformly on [0 , × [0 , . For the covariance function K B definedby (1.1) of the Brownian bridge ( B t ) t ∈ [0 , , we obtain, for k ∈ N and t ∈ [0 , , e k ( t ) = √ kπt ) and λ k = 1 k π . The Karhunen–Loève expansion of the Brownian bridge is then given by B t = ∞ (cid:88) k =1 √ kπt ) Z k where Z k = (cid:90) √ kπr ) B r d r , J. FOSTER AND K. HABERMANN which after integration by parts yields the expression (1.2). Applying Lemma 2.1, we can computethe covariance functions of the associated fluctuation processes ( F N, t ) t ∈ [0 , . Lemma 2.2. The fluctuation process ( F N, t ) t ∈ [0 , for N ∈ N is a zero-mean Gaussian processwith covariance function N C N where C N : [0 , × [0 , → R is given by C N ( s, t ) = min( s, t ) − st − N (cid:88) k =1 kπs ) sin( kπt ) k π . Proof. From the definition (1.7), we see that ( F N, t ) t ∈ [0 , is a zero-mean Gaussian process. Hence,it suffices to determine its covariance function. By Lemma 2.1, we have, for k, l ∈ N , E ñÇ (cid:90) cos( kπr ) d B r å Ç (cid:90) cos( lπr ) d B r åô = (cid:90) cos( kπr ) cos( lπr ) d r = ® if k = l otherwiseand, for t ∈ [0 , , E ñ B t (cid:90) cos( kπr ) d B r ô = (cid:90) t cos( kπr ) d r = sin( kπt ) kπ . Therefore, from (1.1) and (1.7), we obtain that, for all s, t ∈ [0 , , E î F N, s F N, t ó = N (cid:32) min( s, t ) − st − N (cid:88) k =1 kπs ) sin( kπt ) k π (cid:33) , as claimed. (cid:3) Consequently, Theorem 1.2 is a statement about the pointwise convergence of the function N C N in the limit N → ∞ .For our stand-alone derivation of the values of the Riemann zeta function at even positive integersin Section 3, it is further important to note that since, by Mercer’s theorem, the representation(2.1) K B ( s, t ) = min( s, t ) − st = ∞ (cid:88) k =1 kπs ) sin( kπt ) k π converges uniformly for s, t ∈ [0 , , the sequence { C N } N ∈ N converges uniformly on [0 , × [0 , tothe zero function. It follows that, for all n ∈ N ,(2.2) lim N →∞ (cid:90) C N ( t, t ) t n d t = 0 . The Fourier expansion. Whereas for the Karhunen–Loève expansion the sequence ® (cid:90) cos( kπr ) d B r ´ k ∈ N of random coefficients is formed by independent Gaussian random variables, it is crucial to observethat the random coefficients appearing in the Fourier expansion are not independent. Integratingby parts, we can rewrite the coefficients defined in (1.4) as(2.3) a = 2 (cid:90) B r d r = − (cid:90) r d B r and b = 0 as well as, for k ∈ N ,(2.4) a k = − (cid:90) sin(2 kπr ) kπ d B r and b k = (cid:90) cos(2 kπr ) kπ d B r . ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 9 Applying Lemma 2.1, we see that(2.5) E (cid:2) a (cid:3) = 4 Ç (cid:90) r d r − å = 13 and, for k, l ∈ N ,(2.6) E [ a k a l ] = E [ b k b l ] = k π if k = l otherwise . Since the random coefficients are Gaussian random variables with mean zero, by (2.3) and (2.4),this implies that, for k ∈ N , a ∼ N Å , ã and a k , b k ∼ N Å , k π ã . For the remaining covariances of these random coefficients, we obtain that, for k, l ∈ N ,(2.7) E [ a k b l ] = 0 , E [ a a k ] = 2 (cid:90) sin(2 kπr ) kπ r d r = − k π and E [ a b k ] = 0 . Using the covariance structure of the random coefficients, we determine the covariance functionsof the fluctuation processes ( F N, t ) t ∈ [0 , defined in (1.8) for the Fourier series expansion. Lemma 2.3. The fluctuation process ( F N, t ) t ∈ [0 , for N ∈ N is a Gaussian process with meanzero and whose covariance function is N C N where C N : [0 , × [0 , is given by C N ( s, t ) = min( s, t ) − st + s − s t − t − N (cid:88) k =1 cos(2 kπ ( t − s ))2 k π . Proof. Repeatedly applying Lemma 2.1, we compute that, for t ∈ [0 , ,(2.8) E [ B t a ] = − (cid:90) t r d r + (cid:90) t d r = t − t as well as, for k ∈ N ,(2.9) E [ B t a k ] = − (cid:90) t sin(2 kπr ) kπ d r = cos(2 kπt ) − k π and E [ B t b k ] = sin(2 kπt )2 k π . From (2.5) and (2.8), it follows that, for s, t ∈ [0 , , E ïÅ B s − a ã Å B t − a ãò = min( s, t ) − st + s − s t − t , whereas (2.7) and (2.9) imply that E (cid:34) a N (cid:88) k =1 a k cos(2 kπt ) − B s N (cid:88) k =1 a k cos(2 kπt ) (cid:35) = − N (cid:88) k =1 cos(2 kπs ) cos(2 kπt )2 k π as well as E (cid:34) B s N (cid:88) k =1 b k sin(2 kπt ) (cid:35) = N (cid:88) k =1 sin(2 kπs ) sin(2 kπt )2 k π . It remains to observe that, by (2.6) and (2.7), E (cid:34)(cid:32) N (cid:88) k =1 ( a k cos(2 kπs ) + b k sin(2 kπs )) (cid:33) (cid:32) N (cid:88) k =1 ( a k cos(2 kπt ) + b k sin(2 kπt )) (cid:33)(cid:35) = N (cid:88) k =1 cos(2 kπs ) cos(2 kπt ) + sin(2 kπs ) sin(2 kπt )2 k π . Using the identity(2.10) cos(2 kπ ( t − s )) = cos(2 kπs ) cos(2 kπt ) + sin(2 kπs ) sin(2 kπt ) and recalling the definition (1.8) of the fluctuation process ( F N, t ) t ∈ [0 , for the Fourier expansion,we obtain the desired result. (cid:3) By combining Corollary 1.3, the resolution (1.9) to the Basel problem and the representation (2.1),we can determine the pointwise limit of N C N as N → ∞ . We leave further considerations untilSection 4.2 to demonstrate that the identity (1.9) is really a consequence of our analysis.2.3. The polynomial expansion. As pointed out in the introduction and as discussed in detailin [9], the polynomial expansion of the Brownian bridge is a type of Karhunen–Loève expansion inthe weighted L ( P ) space with weight function w on (0 , defined by w ( t ) = t (1 − t ) .An alternative derivation of the polynomial expansion is given in [12] by considering iteratedKolmogorov diffusions. It is shown that the first component of an iterated Kolmogorov diffusion ofstep N ∈ N conditioned to return to ∈ R N in time has the same law as the stochastic process (cid:32) B t − N − (cid:88) k =1 (2 k + 1) (cid:90) t Q k ( r ) d r (cid:90) Q k ( r ) d B r (cid:33) t ∈ [0 , , where Q k is the shifted Legendre polynomial of degree k ∈ N on the interval [0 , . The polynomialexpansion (1.5) is then an immediate consequence of the result [12, Theorem 1.4] that these firstcomponents of the conditioned iterated Kolmogorov diffusions converge weakly as N → ∞ to thezero process.As for the Karhunen–Loève expansion discussed above, the sequence { c k } k ∈ N of random coefficientsdefined by (1.6) is again formed by independent Gaussian random variables. To see this, we firstrecall the following identities for Legendre polynomials [1, (12.23), (12.31), (12.32)] which in termsof the shifted Legendre polynomials read as, for k ∈ N ,(2.11) Q k = 12(2 k + 1) (cid:0) Q (cid:48) k +1 − Q (cid:48) k − (cid:1) , Q k (0) = ( − k , Q k (1) = 1 . In particular, it follows that, for all k ∈ N , (cid:90) Q k ( r ) d r = 0 , which, by Lemma 2.1, implies that, for k, l ∈ N , E [ c k c l ] = E ñÇ (cid:90) Q k ( r ) d B r å Ç (cid:90) Q l ( r ) d B r åô = (cid:90) Q k ( r ) Q l ( r ) d r = k + 1 if k = l otherwise . Since the random coefficients are Gaussian with mean zero, this establishes their independence. ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 11 The fluctuation processes ( F N, t ) t ∈ [0 , for the polynomial expansion defined by(2.12) F N, t = √ N (cid:32) B t − N − (cid:88) k =1 (2 k + 1) (cid:90) t Q k ( r ) d r (cid:90) Q k ( r ) d B r (cid:33) are studied in [12]. According to [12, Theorem 1.6], they converge in finite dimensional distributionsas N → ∞ to the collection ( F t ) t ∈ [0 , of independent Gaussian random variables with mean zeroand variance E î (cid:0) F t (cid:1) ó = 1 π » t (1 − t ) , that is, the variance function of the limit fluctuations is given by a scaled semicircle.3. Particular values of the Riemann zeta function We demonstrate how to use the Karhunen–Loève expansion of the Brownian bridge or, moreprecisely, the series representation arising from Mercer’s theorem for the covariance function of theBrownian bridge to determine the values of the Riemann zeta function at even positive integers.The analysis further feeds directly into Section 4.1 where we characterise the limit fluctuations forthe Karhunen–Loève expansion.The crucial ingredient is the observation (2.2) from Section 2, which implies that, for all n ∈ N ,(3.1) ∞ (cid:88) k =1 (cid:90) kπt )) k π t n d t = (cid:90) (cid:0) t − t (cid:1) t n d t = 1( n + 2)( n + 3) . For completeness, we recall that the Riemann zeta function ζ : C \ { } → C analytically continuesthe sum of the Dirichlet series ζ ( s ) = ∞ (cid:88) k =1 k s . When discussing its values at even positive integers, we encounter the Bernoulli numbers. TheBernoulli numbers B n , for n ∈ N , are signed rational numbers defined by an exponential generatingfunction via t e t − ∞ (cid:88) n =1 B n t n n ! , see Borevich and Shafarevich [2, Chapter 5.8]. These numbers play an important role in numbertheory and analysis. For instance, they feature in the series expansion of the (hyperbolic) tangentand the (hyperbolic) cotangent, and they appear in formulae by Bernoulli and by Faulhaber for thesum of positive integer powers of the first k positive integers. The characterisation of the Bernoullinumbers which is essential to our analysis is that, according to [2, Theorem 5.8.1], they satisfy andare uniquely given by the recurrence relations(3.2) m (cid:88) n =1 Ç m + 1 n å B n = 0 for m ∈ N . In particular, choosing m = 1 yields B = 0 , which shows that B = − . Moreover, since the function defined by t e t − t ∞ (cid:88) n =2 B n t n n ! is an even function, we obtain B n +1 = 0 for all n ∈ N , see [2, Theorem 5.8.2]. It follows from (3.2)that the Bernoulli numbers B n indexed by even positive integers are uniquely characterised bythe recurrence relations(3.3) m (cid:88) n =1 Ç m + 12 n å B n = 2 m − for m ∈ N . These recurrence relations are our tool for identifying the Bernoulli numbers when determiningthe values of the Riemann zeta function at even positive integers.The starting point for our analysis is (3.1), and we first illustrate how it allows us to compute ζ (2) .Taking n = 0 in (3.1), multiplying through by π , and using that (cid:82) (sin( kπt )) d t = for k ∈ N ,we deduce that ζ (2) = ∞ (cid:88) k =1 k = ∞ (cid:88) k =1 (cid:90) kπt )) k d t = π . We observe that this is exactly the identity obtained by applying the general result (cid:90) K ( t, t ) d t = ∞ (cid:88) k =1 λ k for a representation arising from Mercer’s theorem to the representation for the covariance function K B of the Brownian bridge.For working out the values for the remaining even positive integers, we iterate over the degree ofthe moment in (3.1). While for the remainder of this section it suffices to only consider the evenmoments, we derive the following recurrence relation and the explicit expression both for the evenand for the odd moments as these are needed in Section 4.1. For k ∈ N and n ∈ N , we set e k,n = (cid:90) kπt )) t n d t . Lemma 3.1. For all k ∈ N and all n ∈ N with n ≥ , we have e k,n = 1 n + 1 − n ( n − k π e k,n − subject to the initial conditions e k, = 1 and e k, = 12 . Proof. For k ∈ N , the values for e k, and e k, can be verified directly. For n ∈ N with n ≥ , weintegrate by parts twice to obtain e k,n = (cid:90) kπt )) t n d t = 1 − (cid:90) Å t − sin(2 kπt )2 kπ ã nt n − d t = 1 − n n ( n − (cid:90) Ç t − (sin( kπt )) k π å t n − d t = 2 − n n ( n − Å n + 1 − k π e k,n − ã = 1 n + 1 − n ( n − k π e k,n − , as claimed. (cid:3) ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 13 Iteratively applying the recurrence relation, we find the following explicit expression, which despiteits involvedness is exactly what we need. Lemma 3.2. For all k ∈ N and m ∈ N , we have e k, m = 12 m + 1 + m (cid:88) n =1 ( − n (2 m )!(2( m − n ) + 1)!2 n k n π n and e k, m +1 = 12 m + 2 + m (cid:88) n =1 ( − n (2 m + 1)!(2( m − n ) + 2)!2 n k n π n . Proof. We proceed by induction over m . Since e k, = 1 and e k, = for all k ∈ N , the expressionsare true for m = 0 with the sums being understood as empty sums in this case. Assuming that theresult is true for some fixed m ∈ N , we use Lemma 3.1 to deduce that e k, m +2 = 12 m + 3 − (2 m + 2)(2 m + 1)4 k π e k, m = 12 m + 3 − m + 24 k π − m (cid:88) n =1 ( − n (2 m + 2)!(2( m − n ) + 1)!2 n +2 k n +2 π n +2 = 12 m + 3 + m +1 (cid:88) n =1 ( − n (2 m + 2)!(2( m − n ) + 3)!2 n k n π n as well as e k, m +3 = 12 m + 4 − (2 m + 3)(2 m + 2)4 k π e k, m +1 = 12 m + 4 − m + 34 k π − m (cid:88) n =1 ( − n (2 m + 3)!(2( m − n ) + 2)!2 n +2 k n +2 π n +2 = 12 m + 4 + m +1 (cid:88) n =1 ( − n (2 m + 3)!(2( m − n ) + 4)!2 n k n π n , which settles the induction step. (cid:3) Focusing on the even moments for the remainder of this section, we see that by (3.1), for all m ∈ N , ∞ (cid:88) k =1 e k, m k π = 1(2 m + 2)(2 m + 3) . From Lemma 3.2, it follows that ∞ (cid:88) k =1 k π (cid:32) m (cid:88) n =0 ( − n (2 m )!(2( m − n ) + 1)!2 n k n π n (cid:33) = 1(2 m + 2)(2 m + 3) . Since (cid:80) ∞ k =1 k − n converges for all n ∈ N , we can rearrange sums to obtain m (cid:88) n =0 ( − n (2 m )!(2( m − n ) + 1)!2 n (cid:32) ∞ (cid:88) k =1 k n +2 π n +2 (cid:33) = 1(2 m + 2)(2 m + 3) , which in terms of the Riemann zeta function and after reindexing the sum rewrites as m +1 (cid:88) n =1 ( − n +1 (2 m )!(2( m − n ) + 3)!2 n − ζ (2 n ) π n = 1(2 m + 2)(2 m + 3) . Multiplying through by (2 m + 1)(2 m + 2)(2 m + 3) shows that, for all m ∈ N , m +1 (cid:88) n =1 Ç m + 32 n å Ç ( − n +1 n )!(2 π ) n ζ (2 n ) å = 2 m + 12 . Comparing the last expression with the characterisation (3.3) of the Bernoulli numbers B n indexedby even positive integers implies that B n = ( − n +1 n )!(2 π ) n ζ (2 n ) , that is, we have established that, for all n ∈ N , ζ (2 n ) = ( − n +1 (2 π ) n B n n )! . Fluctuations for the trigonometric expansions of the Brownian bridge We first prove Theorem 1.2 and Corollary 1.3 which we use to determine the pointwise limits forthe covariance functions of the fluctuation processes for the Karhunen–Loève expansion and of thefluctuation processes for the Fourier series expansion, and then we deduce Theorem 1.1.4.1. Fluctuations for the Karhunen–Loève expansion. For the moment analysis initiated inthe previous section to allow us to identify the limit of N C N as N → ∞ on the diagonal awayfrom its endpoints, we apply the Arzelà–Ascoli theorem to guarantee continuity of the limit awayfrom the endpoints. To this end, we first need to establish the uniform boundedness of two familiesof functions. Lemma 4.1. The family { N C N ( t, t ) : N ∈ N and t ∈ [0 , } is uniformly bounded.Proof. Combining the expression for C N ( t, t ) from Lemma 2.2 and the representation (2.1) for K B arising from Mercer’s theorem, we see that N C N ( t, t ) = N ∞ (cid:88) k = N +1 kπt )) k π . In particular, for all N ∈ N and all t ∈ [0 , , we have (cid:12)(cid:12) N C N ( t, t ) (cid:12)(cid:12) ≤ N ∞ (cid:88) k = N +1 k π . We further observe that(4.1) lim M →∞ N M (cid:88) k = N +1 k ≤ lim M →∞ N M (cid:88) k = N +1 Å k − − k ã = lim M →∞ Å − NM ã = 1 . It follows that, for all N ∈ N and all t ∈ [0 , , (cid:12)(cid:12) N C N ( t, t ) (cid:12)(cid:12) ≤ π , which is illustrated in Figure 3 and which establishes the claimed uniform boundedness. (cid:3) Lemma 4.2. Fix ε > . The family ß N dd t C N ( t, t ) : N ∈ N and t ∈ [ ε, − ε ] ™ is uniformly bounded. ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 15 Figure 3. Profiles of t (cid:55)→ N C N ( t, t ) plotted for N ∈ { , , } along with t (cid:55)→ π . Proof. According to Lemma 2.2, we have, for all t ∈ [0 , , C N ( t, t ) = t − t − N (cid:88) k =1 kπt )) k π , which implies that N dd t C N ( t, t ) = N (cid:32) − t − N (cid:88) k =1 kπt ) kπ (cid:33) . The desired result then follows by showing that, for ε > fixed, the family (cid:40) N (cid:32) π − t − N (cid:88) k =1 sin( kt ) k (cid:33) : N ∈ N and t ∈ [ ε, π − ε ] (cid:41) is uniformly bounded, as illustrated in Figure 4. Employing a usual approach, we use the Dirichletkernel, for N ∈ N , N (cid:88) k = − N e i kt = 1 + N (cid:88) k =1 kt ) = sin (cid:0)(cid:0) N + (cid:1) t (cid:1) sin (cid:0) t (cid:1) to write, for t ∈ (0 , π ) , π − t − N (cid:88) k =1 sin( kt ) k = − (cid:90) tπ (cid:32) N (cid:88) k =1 ks ) (cid:33) d s = − (cid:90) tπ sin (cid:0)(cid:0) N + (cid:1) s (cid:1) sin (cid:0) s (cid:1) d s . Integration by parts yields − (cid:90) tπ sin (cid:0)(cid:0) N + (cid:1) s (cid:1) sin (cid:0) s (cid:1) d s = cos (cid:0)(cid:0) N + (cid:1) t (cid:1) (2 N + 1) sin (cid:0) t (cid:1) − N + 1 (cid:90) tπ cos ÅÅ N + 12 ã s ã dd s Ç (cid:0) s (cid:1) å d s . By the first mean value theorem for definite integrals, it follows that for t ∈ (0 , π ] fixed, there exists ξ ∈ [ t, π ] , whereas for t ∈ [ π, π ) fixed, there exists ξ ∈ [ π, t ] , such that − (cid:90) tπ sin (cid:0)(cid:0) N + (cid:1) s (cid:1) sin (cid:0) s (cid:1) d s = cos (cid:0)(cid:0) N + (cid:1) t (cid:1) (2 N + 1) sin (cid:0) t (cid:1) − cos (cid:0)(cid:0) N + (cid:1) ξ (cid:1) N + 1 Ç (cid:0) t (cid:1) − å . Since (cid:12)(cid:12) cos (cid:0)(cid:0) N + (cid:1) ξ (cid:1)(cid:12)(cid:12) is bounded above by one independently of ξ and as t ∈ (0 , π ) for t ∈ (0 , π ) implies that < sin (cid:0) t (cid:1) ≤ , we conclude that, for all N ∈ N and for all t ∈ (0 , π ) , N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) π − t − N (cid:88) k =1 sin( kt ) k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N (2 N + 1) sin (cid:0) t (cid:1) , which, for t ∈ [ ε, π − ε ] , is uniformly bounded by / sin (cid:0) ε (cid:1) . (cid:3) Figure 4. Profiles of t (cid:55)→ N Å π − t − N (cid:80) k =1 sin( kt ) k ã plotted for N ∈ { , , , } on [ ε, π − ε ] with ε = 0 . . Remark 4.3. In the proof of the previous lemma, we have essentially controlled the error in theFourier series expansion for the fractional part of t which is given by − ∞ (cid:88) k =1 sin(2 kπt ) kπ , see [13, Exercise on p. 4].We can now prove the convergence in Theorem 1.2 on the diagonal away from the endpoints,which consists of a moment analysis to identify the moments of the limit function as well as anapplication of the Arzelà–Ascoli theorem to show that the limit function is continuous away fromthe endpoints. Alternatively, one could prove Corollary 1.3 directly with a similar approach as inthe proof of Lemma 4.2, but integrating the Dirichlet kernel twice, and then deduce Theorem 1.2.However, as the moment analysis was already set up in Section 3 to determine the values of theRiemann zeta function at even positive integers, we demonstrate how to proceed with this approach. Proposition 4.4. For all t ∈ (0 , , we have lim N →∞ N (cid:32) t − t − N (cid:88) k =1 kπt )) k π (cid:33) = 1 π . Proof. Recall that, due Lemma 2.2 and the representation (2.1), we have, for t ∈ [0 , ,(4.2) C N ( t, t ) = t − t − N (cid:88) k =1 kπt )) k π = ∞ (cid:88) k = N +1 kπt )) k π . ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 17 By Lemma 4.1 and Lemma 4.2, the Arzelà–Ascoli theorem implies that the sequence { N C N } N ∈ N is locally uniform convergent on the diagonal away from the endpoints. It follows that the function t (cid:55)→ lim N →∞ C N ( t, t ) is continuous in t ∈ (0 , .To identify the limit function, we proceed with the moment analysis initiated in Section 3. ApplyingLemma 3.2, we see that, for m ∈ N , N ∞ (cid:88) k = N +1 e k, m k π = N ∞ (cid:88) k = N +1 k π (cid:32) m + 1 + m (cid:88) n =1 ( − n (2 m )!(2( m − n ) + 1)!2 n k n π n (cid:33) , (4.3) N ∞ (cid:88) k = N +1 e k, m +1 k π = N ∞ (cid:88) k = N +1 k π (cid:32) m + 2 + m (cid:88) n =1 ( − n (2 m + 1)!(2( m − n ) + 2)!2 n k n π n (cid:33) . (4.4)The bound (4.1) together with lim M →∞ N M (cid:88) k = N +1 k ≥ lim M →∞ N M (cid:88) k = N +1 Å k − k + 1 ã = lim M →∞ Å NN + 1 − NM + 1 ã = NN + 1 implies that(4.5) lim N →∞ N ∞ (cid:88) k = N +1 k = 1 . For n ∈ N , we further have ≤ N ∞ (cid:88) k = N +1 k n +2 ≤ N ( N + 1) ∞ (cid:88) k = N +1 k n ≤ N ∞ (cid:88) k =1 k n , and since (cid:80) ∞ k =1 k − n converges, this yields lim N →∞ N ∞ (cid:88) k = N +1 k n +2 = 0 for n ∈ N . From (4.2) as well as (4.3) and (4.4), it follows that, for all n ∈ N , lim N →∞ (cid:90) N C N ( t, t ) t n d t = lim N →∞ N ∞ (cid:88) k = N +1 e k,n k π = 1( n + 1) π . This shows that, for all n ∈ N , lim N →∞ (cid:90) N C N ( t, t ) t n d t = (cid:90) π t n d t , which by continuity of the limit function on (0 , establishes the claimed result. (cid:3) We included the on-diagonal convergence in Theorem 1.2 as a separate statement to demonstratethat Corollary 1.3 is a consequence of Proposition 4.4, which is then used to prove the off-diagonalconvergence in Theorem 1.2. Proof of Corollary 1.3. Using the identity that, for k ∈ N ,(4.6) cos(2 kπt ) = 1 − kπt )) , we obtain ∞ (cid:88) k = N +1 cos(2 kπt ) k π = ∞ (cid:88) k = N +1 k π − ∞ (cid:88) k = N +1 kπt )) k π . From (4.5) and Proposition 4.4, it follows that, for all t ∈ (0 , , lim N →∞ N ∞ (cid:88) k = N +1 cos(2 kπt ) k π = 1 π − π = 0 , as claimed. (cid:3) Proof of Theorem 1.2. If s ∈ { , } or t ∈ { , } , the result follows immediately from sin( kπ ) = 0 for all k ∈ N , and if s = t for t ∈ (0 , , the claimed convergence is given by Proposition 4.4.Therefore, it remains to consider the off-diagonal case, and we may assume that s, t ∈ (0 , aresuch that s < t . Due to the representation (2.1) and the identity kπs ) sin( kπt ) = cos( kπ ( t − s )) − cos( kπ ( t + s )) , we have min( s, t ) − st − N (cid:88) k =1 kπs ) sin( kπt ) k π = ∞ (cid:88) k = N +1 kπs ) sin( kπt ) k π = ∞ (cid:88) k = N +1 cos( kπ ( t − s )) − cos( kπ ( t + s )) k π . Since < t − s < t + s < for s, t ∈ (0 , with s < t , the convergence away from the diagonal is aconsequence of Corollary 1.3. (cid:3) Note that Theorem 1.2 states, for s, t ∈ [0 , ,(4.7) lim N →∞ N C N ( s, t ) = ® π if s = t and t ∈ (0 , otherwise , which is the key ingredient for obtaining the characterisation of the limit fluctuations for theKarhunen–Loève expansion given in Theorem 1.1. We provide the full proof of Theorem 1.1 belowafter having determined the limit of N C N as N → ∞ .4.2. Fluctuations for the Fourier series expansion. Instead of setting up another momentanalysis to study the pointwise limit of N C N as N → ∞ , we simplify the expression for C N fromLemma 2.3 and deduce the desired pointwise limit from Corollary 1.3.Using the standard Fourier basis for L ([0 , , the polarised Parseval identity and the trigonometricidentity (2.10), we can write, for s, t ∈ [0 , , min( s, t ) = (cid:90) [0 ,s ] ( r ) [0 ,t ] ( r ) d r = st + ∞ (cid:88) k =1 (cid:90) s cos(2 kπr ) d r (cid:90) t cos(2 kπr ) d r + ∞ (cid:88) k =1 (cid:90) s sin(2 kπr ) d r (cid:90) t sin(2 kπr ) d r = st − ∞ (cid:88) k =1 cos(2 kπs )2 k π − ∞ (cid:88) k =1 cos(2 kπt )2 k π + ∞ (cid:88) k =1 cos(2 kπ ( t − s ))2 k π + ∞ (cid:88) k =1 k π . Applying the identity (4.6) as well as the representation (2.1) and using the value for ζ (2) derivedin Section 3, we have ∞ (cid:88) k =1 cos(2 kπt )2 k π = ∞ (cid:88) k =1 k π − ∞ (cid:88) k =1 (sin( kπt )) k π = 112 + t − t . ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 19 Once again exploiting the value for ζ (2) , we obtain min( s, t ) − st + s − s t − t ∞ (cid:88) k =1 cos(2 kπ ( t − s ))2 k π . Using the expression for C N from Lemma 2.3, it follows that, for s, t ∈ [0 , , C N ( s, t ) = ∞ (cid:88) k = N +1 cos(2 kπ ( t − s ))2 k π . This implies that if t − s is an integer then, as a result of the limit (4.5), lim N →∞ N C N ( s, t ) = 1 π , whereas if t − s is not an integer then, by Corollary 1.3, lim N →∞ N C N ( s, t ) = 0 . This can be summarised as, for s, t ∈ [0 , ,(4.8) lim N →∞ N C N ( s, t ) = ® π if s = t or s, t ∈ { , } otherwise . We finally prove Theorem 1.1 by considering characteristic functions. Proof of Theorem 1.1. According to Lemma 2.2 as well as Lemma 2.3, the fluctuation processes ( F N, t ) t ∈ [0 , and ( F N, t ) t ∈ [0 , are zero-mean Gaussian processes with covariance functions N C N and N C N , respectively.By the pointwise convergences (4.7) and (4.8) of the covariance functions in the limit N → ∞ , forany n ∈ N and any t , . . . , t n ∈ [0 , , the characteristic functions of the Gaussian random vectors ( F N,it , . . . , F N,it n ) , for i ∈ { , } , converge pointwise as N → ∞ to the characteristic function of theGaussian random vector ( F it , . . . , F it n ) . Therefore, the claimed convergences in finite dimensionaldistributions are consequences of Lévy’s continuity theorem. (cid:3) Approximations of Brownian Lévy area In this section, we consider approximations of second iterated integrals of Brownian motion, whichare fundamental within the numerical analysis of stochastic differential equations (SDEs), see [15].Due to their presence within stochastic Taylor expansions, increments and second iterated integralsof multidimensional Brownian motion are required by high order methods for general SDEs, such asstochastic Taylor [15] and Runge–Kutta [22] methods. Currently, the only methodology for exactlygenerating the increment and second iterated integral, or equivalently the Brownian Lévy area,given by Definition 1.4, of a d -dimensional Brownian motion is limited to the case when d = 2 .This algorithm for the exact generation of Brownian increments and Lévy area is detailed in [10].Obtaining good approximations of Brownian Lévy area in an L ( P ) sense is known to be difficult.For example, it was shown in [5] that any approximation of Lévy area which is measurable withrespect to N Gaussian random variables, obtained from linear functionals of the Brownian path,cannot achieve strong convergence faster than O ( N − ) . In particular, this result extends theclassical theorem of Clark and Cameron [3] which establishes a best convergence rate of O ( N − ) for approximations of Lévy area based on only the Brownian increments { W ( n +1) h − W nh } ≤ n ≤ N − .Therefore, approximations have been developed which fall outside of this paradigm, see [4, 8, 21, 24].In the analysis of these methodologies, the Lévy area of Brownian motion and its approximation areprobabilistically coupled in such a way that L ( P ) convergence rates of O ( N − ) can be established. We are interested in the approximations of Brownian Lévy area that can be obtained directly fromthe Fourier series expansion (1.3) and the polynomial expansion (1.5) of the Brownian bridge. Forthe remainder of the section, the Brownian motion ( W t ) t ∈ [0 , is assumed to be d -dimensional and ( B t ) t ∈ [0 , is its associated Brownian bridge.We first recall the standard Fourier approach to the strong approximation of Brownian Lévy area. Theorem 5.1 (Strong approximation of Brownian Lévy area via Fourier coefficients, [15, p. 205]) . For n ∈ N , we define a random antisymmetric d × d matrix (cid:98) A n by, for i, j ∈ { , . . . , d } , (cid:98) A ( i,j ) n := 12 Ä a ( i )0 W ( j )1 − W ( i )1 a ( j )0 ä + π n − (cid:88) k =1 k Ä a ( i ) k b ( j ) k − b ( i ) k a ( j ) k ä , where the normal random vectors { a k } k ∈ N and { b k } k ∈ N are the coefficients from the Brownianbridge expansion (1.3), that is, the coordinates of each random vector are independent and definedaccording to (1.4). Then, for i, j ∈ { , . . . , d } with i (cid:54) = j , we have E ï (cid:16) A ( i,j )0 , − (cid:98) A ( i,j ) n (cid:17) ò = 12 π ∞ (cid:88) k = n k . Remark 5.2. Using the covariance structure given by (2.5), (2.6), (2.7) and the independence ofthe components of a Brownian bridge, it immediately follows that the coefficients { a k } k ∈ N and { b k } k ∈ N are jointly normal with a ∼ N (cid:0) , I d (cid:1) , a k , b k ∼ N (cid:0) , k π I d (cid:1) , cov( a , a k ) = − k π I d and cov( a l , b k ) = 0 for k ∈ N and l ∈ N .In practice, the above approximation may involve generating the N independent random vectors { a k } ≤ k ≤ N followed by the coefficient a , which will not be independent, but can be expressed asa linear combination of { a k } ≤ k ≤ N along with an additional independent normal random vector.Without this additional normal random vector, we obtain the following discretisation of Lévy area. Theorem 5.3 (Kloeden–Platen–Wright approximation of Brownian Lévy area, see [16, 24]) . For n ∈ N , we define a random antisymmetric d × d matrix (cid:101) A n by, for i, j ∈ { , . . . , d } , (cid:101) A ( i,j ) n := π n − (cid:88) k =1 k Å a ( i ) k Å b ( j ) k − kπ W ( j )1 ã − Å b ( i ) k − kπ W ( i )1 ã a ( j ) k ã , where the sequences { a k } k ∈ N and { b k } k ∈ N of independent normal random vectors are the same asbefore. Then, for i, j ∈ { , . . . , d } with i (cid:54) = j , we have E ï (cid:16) A ( i,j )0 , − (cid:101) A ( i,j ) n (cid:17) ò = 32 π ∞ (cid:88) k = n k . Proof. As for Theorem 5.1, the result follows by direct calculation. The constant is larger because,for i ∈ { , . . . , d } and k ∈ N , E ï (cid:16) b ( i ) k − kπ W ( i )1 (cid:17) ò = 32 k π = 3 E (cid:104)(cid:0) b ( i ) k (cid:1) (cid:105) , which yields the required result. (cid:3) Finally, we give the approximation of Lévy area corresponding to the polynomial expansion (1.5).Whilst this result does appear in [17], we shall present a more intuitive derivation of the formula. ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 21 Theorem 5.4 (Polynomial approximation of Brownian Lévy area, see [17, p. 194]) . For n ∈ N ,we define a random antisymmetric d × d matrix s A n by, for n ∈ N and i, j ∈ { , . . . , d } , s A ( i,j ) n := 12 Ä W ( i )1 c ( j )1 − c ( i )1 W ( j )1 ä + 12 n − (cid:88) k =1 Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä , where the normal random vectors { c k } k ∈ N are the coefficients from the polynomial expansion (1.5),that is, the coordinates are independent and defined according to (1.6), and we set s A ( i,j )0 := 0 . Then, for n ∈ N and for i, j ∈ { , . . . , d } with i (cid:54) = j , we have E ï (cid:16) A ( i,j )0 , − s A ( i,j ) n (cid:17) ò = 18 n + 4 . Remark 5.5. By applying Lemma 2.1, the orthogonality of shifted Legendre polynomials and theindependence of the components of a Brownian bridge, we see that the coefficients { c k } k ∈ N areindependent and distributed as c k ∼ N (cid:0) , k +1 I d (cid:1) for k ∈ N . Proof. It follows from the polynomial expansion (1.5) that, for i, j ∈ { , . . . , d } with i (cid:54) = j ,(5.1) (cid:90) B ( i ) t d B ( j ) t = (cid:90) (cid:32) ∞ (cid:88) k =1 (2 k + 1) c ( i ) k (cid:90) t Q k ( r ) d r (cid:33) d (cid:32) ∞ (cid:88) l =1 (2 l + 1) c ( j ) l (cid:90) t Q l ( r ) d r (cid:33) . To simplify (5.1), we use the identities in (2.11) for shifted Legendre polynomials as well as theorthogonality of shifted Legendre polynomials to obtain that, for k, l ∈ N , (cid:90) Ç (cid:90) t Q k ( r ) d r å d Ç (cid:90) t Q l ( r ) d r å = (cid:90) Q l ( t ) (cid:90) t Q k ( r ) d r d t = 12(2 k + 1) (cid:90) Q l ( t ) ( Q k +1 ( t ) − Q k − ( t )) d t = k + 1) (cid:90) ( Q k +1 ( t )) d t if l = k + 1 − k + 1) (cid:90) ( Q k − ( t )) d t if l = k − otherwise . Evaluating the above integrals gives, for k, l ∈ N , (cid:90) Ç (cid:90) t Q k ( r ) d r å d Ç (cid:90) t Q l ( r ) d r å = k + 1)(2 k + 3) if l = k + 1 − k + 1)(2 k − if l = k − otherwise . (5.2)In particular, for k, l ∈ N , this implies that (cid:90) Ç (2 k + 1) c ( i ) k (cid:90) t Q k ( r ) d r å d Ç (2 l + 1) c ( j ) l (cid:90) t Q l ( r ) d r å = c ( i ) k c ( j ) k +1 if l = k + 1 − c ( i ) k c ( j ) k − if l = k − otherwise . Hence by the bounded convergence theorem in L ( P ) , we can simplify the Lévy area expansion (5.1)to (cid:90) B ( i ) t d B ( j ) t = 12 ∞ (cid:88) k =1 Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä . (5.3)Since W t = tW + B t for t ∈ [0 , , we have, for i, j ∈ { , . . . , d } with i (cid:54) = j , (cid:90) W ( i ) t d W ( j ) t = (cid:90) (cid:0) tW ( i )1 (cid:1) d (cid:0) tW ( j )1 (cid:1) + (cid:90) B ( i ) t d (cid:0) tW ( j )1 (cid:1) + (cid:90) (cid:0) tW ( i )1 (cid:1) d B ( j ) t + (cid:90) B ( i ) t d B ( j ) t = 12 W ( i )1 W ( j )1 − W ( j )1 (cid:90) t d B ( i ) t + W ( i )1 (cid:90) t d B ( j ) t + (cid:90) B ( i ) t d B ( j ) t , where the second line follows by integration by parts. As (cid:90) W ( i ) t d W ( j ) t = 12 W ( i )1 W ( j )1 + A ( i,j )0 , and Q ( t ) = 2 t − , the above and (5.3) imply that, for i, j ∈ { , . . . , d } , A ( i,j )0 , = 12 Ä W ( i )1 c ( j )1 − c ( i )1 W ( j )1 ä + 12 ∞ (cid:88) k =1 Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä . By the independence of the normal random vectors in the sequence { c k } k ∈ N , it is straightforwardto compute the mean squared error in approximating A , and we obtain, for n ∈ N and for i, j ∈ { , . . . , d } with i (cid:54) = j , E ï (cid:16) A ( i,j )0 , − s A ( i,j ) n (cid:17) ò = E (cid:32) ∞ (cid:88) k = n Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä (cid:33) = 14 ∞ (cid:88) k = n k + 1)(2 k + 3)= 14 ∞ (cid:88) k = n Å k + 1 − k + 3 ã = 18 n + 4 , by Remark 5.5. Similarly, as the normal random vector W and the ones in the sequence { c k } k ∈ N are independent, we have E ï (cid:16) A ( i,j )0 , − s A ( i,j )0 (cid:17) ò = E ñÅ Ä W ( i )1 c ( j )1 − c ( i )1 W ( j )1 äã ô + E (cid:32) ∞ (cid:88) k =1 Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä (cid:33) = 16 + 112 = 14 , as claimed. (cid:3) Given that we have now considered three different strong approximations of Brownian Lévy area,it is reasonable to compare their respective rates of convergence. Combining the above theorems,we obtain the following result. ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 23 Corollary 5.6 (Asymptotic convergence rates of Lévy area approximations) . For n ∈ N , weset N = 2 n so that the number of Gaussian random vectors required to define the Lévy areaapproximations (cid:98) A n , (cid:101) A n and s A n is N or N − , respectively. Then, for i, j ∈ { , . . . , d } with i (cid:54) = j and as N → ∞ , we have E ï (cid:16) A ( i,j )0 , − (cid:98) A ( i,j ) n (cid:17) ò ∼ π Å N ã , E ï (cid:16) A ( i,j )0 , − (cid:101) A ( i,j ) n (cid:17) ò ∼ π Å N ã , E ï (cid:16) A ( i,j )0 , − s A ( i,j )2 n (cid:17) ò ∼ Å N ã . In particular, the polynomial approximation of Brownian Lévy area is more accurate than theKloeden–Platen–Wright approximation, both of which use only independent Gaussian vectors. Remark 5.7. It was shown in [5] that π (cid:0) N (cid:1) is the optimal asymptotic rate of mean squaredconvergence for Lévy area approximations that are measurable with respect to N Gaussian randomvariables, obtained from linear functionals of the Brownian path.As one would expect, all the Lévy area approximations converge in L ( P ) with a rate of O ( N − ) and thus the main difference between their respective accuracies is in the leading error constant.More concretely, for sufficiently large N , the approximation based on the Fourier expansion of theBrownian bridge is roughly 11% more accurate in L ( P ) than that of the polynomial approximation.On the other hand, the polynomial approximation is easier to implement in practice as all ofthe required coefficients are independent. Since it has the largest asymptotic error constant, theKloeden–Platen–Wright approach gives the least accurate approximation for Brownian Lévy area.We observe that the leading error constants for the Lévy area approximations resulting fromthe Fourier series and the polynomial expansion coincide with the average L ( P ) error of theirrespective fluctuation processes, that is, applying Fubini’s theorem followed by the limit theoremsfor the fluctuation processes ( F N, t ) t ∈ [0 , and ( F N, t ) t ∈ [0 , defined by (1.8) and (2.12), respectively,gives lim N →∞ E ñ (cid:90) Ä F N, t ä d t ô = (cid:90) π d t = 1 π , lim N →∞ E ñ (cid:90) Ä F N, t ä d t ô = (cid:90) π » t (1 − t ) d t = 18 . To demonstrate how this correspondence arises, we close with some heuristics. For N ∈ N , weconsider an approximation of the Brownian bridge which uses N random vectors, and we denote thecorresponding approximation of Brownian motion ( W t ) t ∈ [0 , by ( S Nt ) t ∈ [0 , , where the differencebetween Brownian motion and its associated Brownian bridge is the first term in the approximation.In the Fourier and polynomial approaches, the error in approximating Brownian Lévy area is thenessentially given by (cid:90) W ( i ) t d W ( j ) t − (cid:90) S N, ( i ) t d S N, ( j ) t = (cid:90) Ä W ( i ) t − S N, ( i ) t ä d W ( j ) t + (cid:90) S N, ( i ) t d Ä W ( j ) t − S N, ( j ) t ä . If one can argue that (cid:90) S N, ( i ) t d Ä W ( j ) t − S N, ( j ) t ä = O Å N ã , which, for instance, for the polynomial approximation follows directly from (5.2) and Remark 5.5,then in terms of the fluctuation processes ( F Nt ) t ∈ [0 , defined by F Nt = √ N (cid:0) W t − S Nt (cid:1) , the error of the Lévy area approximation can be expressed as √ N (cid:90) F N, ( i ) t d W ( j ) t + O Å N ã . Thus, by Itô’s isometry and Fubini’s theorem, the leading error constant in the mean squared erroris indeed given by (cid:90) lim N →∞ E (cid:104) Ä F N, ( i ) t ä (cid:105) d t . This connection could be interpreted as an asymptotic Itô isometry for Lévy area approximations. Appendix A. Summarising tables Type of expansion Expansion of the Brownian bridge ( B t ) t ∈ [0 , Karhunen–Loève(Loève [18]) B t = ∞ (cid:88) k =1 kπt ) kπ (cid:90) cos( kπr ) d B r Fourier series(Kahane [14] orKloeden–Platen [15]) B t = 12 a + ∞ (cid:88) k =1 ( a k cos(2 kπt ) + b k sin(2 kπt )) with, for k ∈ N , a k = 2 (cid:90) cos(2 kπr ) B r d r , b k = 2 (cid:90) sin(2 kπr ) B r d r Polynomial(Foster et al. [9] andHabermann [12]) B t = ∞ (cid:88) k =1 (2 k + 1) c k (cid:90) t Q k ( r ) d r with, for k ∈ N , c k = (cid:90) Q k ( r ) d B r and Q k denoting the shifted Legendre polynomial of degree k Table 1. Table summarising the Brownian bridge expansions considered in this paper. ROWNIAN BRIDGE EXPANSIONS, LÉVY AREA AND THE RIEMANN ZETA FUNCTION 25 Type of expansion Expansion of the Brownian Lévy area A , Fourier series(Kloeden–Platen [15]) A ( i,j )0 , = 12 Ä a ( i )0 W ( j )1 − W ( i )1 a ( j )0 ä + π ∞ (cid:88) k =1 k Ä a ( i ) k b ( j ) k − b ( i ) k a ( j ) k ä with, for k ∈ N , a ( i ) k = 2 (cid:90) cos(2 kπr ) B ( i ) r d r , b ( i ) k = 2 (cid:90) sin(2 kπr ) B ( i ) r d r Fourier series(Kloeden–Platen–Wright[16]) A ( i,j )0 , = π ∞ (cid:88) k =1 k Ç a ( i ) k Ç b ( j ) k − W ( j )1 kπ å − Ç b ( i ) k − W ( i )1 kπ å a ( j ) k å Polynomial(Kuznetsov [17]) A ( i,j )0 , = 12 Ä W ( i )1 c ( j )1 − c ( i )1 W ( j )1 ä + 12 ∞ (cid:88) k =1 Ä c ( i ) k c ( j ) k +1 − c ( i ) k +1 c ( j ) k ä with, for k ∈ N , c ( i ) k = (cid:90) Q k ( r ) d B ( i ) r and Q k denoting the shifted Legendre polynomial of degree k Table 2. Table summarising the Lévy area expansions considered in this paper. References [1] George B. Arfken and Hans J. Weber. 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Email address : [email protected] Karen Habermann, Department of Statistics, University of Warwick, Coventry, CV4 7AL, UnitedKingdom. Email address ::