AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/457051 457051 Research Article Karhunen-Loève Expansion for the Second Order Detrended Brownian Motion http://orcid.org/0000-0001-5927-141X Zhou Yongchun 1 Ai Xiaohui 2 Lv Minghao 1 http://orcid.org/0000-0001-5267-6764 Tian Boping 1 Romaguera Salvador 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001 China hit.edu.cn 2 Department of Mathematics, Northeast Forestry University, Harbin 150040 China nefu.edu.cn 2014 1672014 2014 08 03 2014 29 05 2014 28 06 2014 16 7 2014 2014 Copyright © 2014 Yongchun Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on the norm in the Hilbert Space L2[0,1], the second order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the space spanned by nonlinear function subspace. Karhunen-Loève expansion for this process is obtained together with the relationship of that of a generalized Brownian bridge. As applications, Laplace transform, large deviation, and small deviation are given.

1. Introduction

Let X={X(t),0t1} be a centered and continuous Gaussian process on [0,1] with covariance function (1)KX(t,s)=EX(t)X(s).

The Karhunen-Loève expansion of X is given by the (convergent in mean squares) series (2)X(t)=k=1ηkλkfk(t), where {ηk,k1} is a sequence of i.i.d. N(0,1) random variables and {λk,k1} is at most the countable set of eigenvalues of Fredholm integral operator (3)TXf(t)=01KX(t,s)f(s)ds{fk(t),k1} and forms an orthogonal sequence in L2[0,1] and 01KX(t,t)dt<.

Deheuvels et al. in  provided the Karhunen-Loève expansions for the processes that are related with Brownian motion. The Karhunen-Loève expansion for detrended Brownian motion has been studied by Ai et al. . Note that the detrended Brownian motion in  can be viewed as projection to a constant function subspace in L2[0,1]. That is, (4)01W^1(t)2dt=minc1,c201(W(t)-c1-c2t)2dt.

To generalize the projection idea into nonlinear detrended process, now we consider (5)minc1,c2,c301(W(t)-c1-c2t-c3t2)2dt and the optimal constant cj satisfy (6)cj01(W(t)-c1-c2t-c3t2)2dt=0,j=1,2,3.

It is easy to obtain (7)c1=901W(s)ds-3601W(s)sds+3001W(s)s2ds,c2=-3601W(s)ds+19201W(s)sds-18001W(s)s2ds,c3=3001W(s)ds-18001W(s)sds+18001W(s)s2ds.

Let (8)A=(aij)3×3=(9-3630-36192-18030-180180); we have (9)cj=i=13aij01si-1W(s)ds,j=1,2,3.

Now we can define the second order detrended process (10)W^2(t)=W(t)-j=13cjtj-1=W(t)+(-9+36t-30t2)01W(s)ds+(36-192t+180t2)01W(s)sds+(-30+180t-180t2)01W(s)s2ds.

2. Main Results

We give the following lemma that provides the explicit covariance function.

Lemma 1.

For convenience, we add KX(s,t) into formula (11), that is (11)KX(s,t)=E(W^2(t)W^2(s))=ts-p,q=13apq(tp-tp+1p(p+1))sq-1-i,j=13aij(si-si+1i(i+1))tj-1+p,q=13i,j=13aijapqp+i+2(p+1)(p+i+1)(i+1)tj-1sq-1, where aij,apq,i,j,p,q=1,2,3 is given in (8).

Proof.

Consider (12)W^2(t)=W(t)-j=13cjtj-1,0t1 and W^2(t) is a mean zero Gaussian process; we obtain (13)E(W^2(t)W^2(s))=EW^2(t)W^2(s)=E(W(t)-j=13cjtj-1)(W(s)-q=13cqsq-1)=E(W(t)-i,j=13aij(01ui-1W(u)du)tj-1)·E(W(s)-p,q=13apq(01vp-1W(v)dv)sq-1). We notice that (14)E(W(t)W(s))=ts,(15)E(W(t)01vp-1W(v)dv)=E(01W(t)W(v)vp-1dv)=01(tv)vp-1dv=0tvpdv+t1tvp-1dv=tp-tp+1p(p+1),(16)E(01ui-1W(u)du)(01vp-1W(v)dv)=01ui-1E(W(u)01vp-1W(v)dv)du=01ui-1(up-up+1p(p+1))du=p+i+2(p+1)(p+i+1)(i+1). Substituting (16), (17), and (19) into (15), we derive (17)E(W^2(t)W^2(s))=ts-p,q=13apq(tp-tp+1p(p+1))sq-1-i,j=13aij(si-si+1i(i+1))tj-1+p,q=13i,j=13aijapqp+i+2(p+1)(p+i+1)(i+1)tj-1sq-1.

Lemma 2 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

If t[0,1], ςj(t)=k=1ωkλk,jek,j(t), j=1,2,, then the condition (18)[0,1]ς12(t)dt=law[0,1]ς22(t)dt is equivalent to the identity (19)λk,1=λk,2k1.

In the following, we will give some preliminaries, notions, and facts that are needed in Theorem 3. For v>-1, Jv(·) is Bessel function  with index v and the positive zeros of Jv(·) are infinite sequence 0<zv,1<zv,2<. When v=3/2, v=5/2, the positive zeros of J3/2,k, J5/2,k are z3/2,k, z5/2,k, k=1,2,, and they are in such a way that (20)0<z3/2,1<z5/2,1<z3/2,2<.

Now we can state one of the main results of this paper.

Theorem 3.

For the second order detrended Brownian motion W^2(t) and a generalized Brownian bridge B2(t) with n=2 in , (21)B2(t)=B(t)-136t(60t2+18t-67)B(1)-t(60t2-96t+11)01B(s)ds+10t(12t2-18t+1)01B(s)sds.

One has the distribution identities (22)01W^2(t)2dt=law01B2(t)2dt=lawk1ηk24z3/2,k2+k1ηk*24z5/2,k2, where {ηk,k1} and {ηk*,k1} denote two independent sequences of independently and identically distributed N(0,1) random variables.

Proof.

By straightforward induction based on the equation and splitting the integration range from t, we get (23)λf(t)=0tsf(s)ds+tt1f(s)ds-p,q=13apq(tp-tp+1p(p+1))01sq-1f(s)ds-i,j=13aijtj-101(si-si+1i(i+1))f(s)ds+p,q=13i,j=13aijapqp+i+2(p+1)(p+i+1)(i+1)tj-1×01sq-1f(s)ds. By differentiation of both sides of (23) with respect to t, we have (24)λf(t)=t1f(s)ds-p,q=13apq1-tpp01sq-1f(s)ds-i=1,j=23(j-1)aijtj-201(si-si+1i(i+1))f(s)ds+p,q=13i=1,j=23aijapq(p+i+2)(j-1)(p+1)(p+i+1)(i+1)tj-2×01sq-1f(s)ds. By differentiation of both sides of (24) with respect to t, we have (25)λf′′(t)+f(t)=i,p,q=13ai3apq2(p+i+2)(p+1)(p+i+1)(i+1)01sq-1f(s)ds-2i=13ai301(si-si+1i(i+1))f(s)ds+q=13a1q01sq-1f(s)ds+(q=13a2q01sq-1f(s)ds)t+(q=13a3q01sq-1f(s)ds)t2. We can simplify this equation to (26)λf′′(t)+f(t)+b1+b2t+b3t2=0, where (27)b1=-i,p,q=13ai3apq2(p+i+2)(p+1)(p+i+1)(i+1)01sq-1f(s)ds+2i=13ai301(si-si+1i(i+1))f(s)ds-q=13a1q01sq-1f(s)ds,(28)b2=-q=13a2q01sq-1f(s)ds,(29)b3=-q=13a3q01sq-1f(s)ds.

We solve the inhomogeneous second differential equation to obtain (30)f(t)=c1costλ+c2sintλ+2λb3-b1-b2t-b3t2.

We substitute f(t) into (28) and (29) to obtain (31)(λsin1λ+6λcos1λ-12λλsin1λ+6λ)c1+(-λcos1λ+6λsin1λiiiiiiiiiiiiii+12λλcos1λ-12λλ+λ)c2=0,(-2λsin1λ-14λcos1λ+30λλsin1λ-16λ)c1+(2λcos1λ-14λsin1λiiiiiiiiiiiiii-30λλcos1λ+30λλ-3λ)c2=0.

In order that there are nonzero choices for c1,c2, the determinant of the above two equations has to be zero, which can be written as (32)D11D22-D12D21=0, where (33)D11=λsin1λ+6λcos1λ-12λλsin1λ+6λ,D12=-λcos1λ+6λsin1λ+12λλcos1λ-12λλ+λ,D21=-2λsin1λ-14λcos1λ+30λλsin1λ-16λ,D22=2λcos1λ-14λsin1λ-30λλcos1λ+30λλ-3λ.

We obtain, after some simplification, (34)24λ2λ+4λλ=(24λ2-λ)sin1λ+(24λ2λ-8λλ)cos1λ.

Then λ0 is an eigenvalue if and only if (34) holds. We therefore obtain (35)D(λ)=-720((24λ-7/2-λ-5/2)sinλ1/2iiiiiiiiiiii+(24λ-4-8λ-3)cosλ1/2-24λ-4-4λ-3), with D(0)=1.

According to the trigonometric function formula (36)sin1λ=2sin12λcos12λ,cos1λ=2cos212λ-1=1-2sin212λ,

we can observe that (37)D11D22-D12D21=-12πλJ3/2(12λ)J5/2(12λ)=0, where J3/2(z), J5/2(z) are Bessel functions as follows: (38)J3/2(z)=2π·zπ(sinzz2-coszz),J5/2(z)=2π·zπ((-1z+3z3)sinz-3z2cosz), which gives two sequences of eigenvalues of (37), namely, (2z3/2,k)-2 and (2z5/2,k)-2.

Similarly, we can obtain the two eigenvalues (2z3/2,k)-2, (2z5/2,k)-2 corresponding to those of integral operator of a generalized Brownian bridge B2(t). Note that the integral operator is (39)01K2(s,t)f(s)ds. Actually, in Lemma 2, we have the distribution identities (40)01W^2(t)2dt=law01B2(t)2dt=lawk1ηk24z3/2,k2+k1ηk*24z5/2,k2.

Remark 4.

From (11) and (22), we derive that (41)01KX(t,t)dt=01E(W^2(t)2)dt=E01W^2(t)2dt=k114z3/2,k2+k114z5/2,k2=140+156=3140 by using the Rayleigh’s formula, for v=3/2 and v=5/2 (see, e.g., [3, (1.91), page 77] and [6, page 502]).

To check (41), from (11), we infer that (42)01KX(t,t)dt=01[t-p,q=13apq(tqp-tp+qp(p+1))iiiiiiiiiiiiiiii-i,j=13aij(tji-ti+ji(i+1))iiiiiiiiiiiiiiii+p,q=13i,j=13aijapqiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiip,q=13apq×p+i+2(p+1)(p+i+1)(i+1)tj+q]dt=3140 which is in agreement with (41).

3. Applications

In this section, the relevant applications of Karhunen-Loève expansion are given.

Proposition 5.

For each θR, one has (43)Eexp(-θ2201W^2(t)2dt)={-720((24θ-7-θ-5)sinθ+(-24θ-8+8θ-6)cosθiiiiiiiiiiiiiiiiiiiiiii+24θ-8+4θ-6)}-1/2.

Proof.

(44) E exp ( - θ 2 2 0 1 W ^ 2 ( t ) 2 d t ) = E exp ( - θ 2 2 k = 1 λ k ξ k 2 ) = k = 1 ( 1 + λ k θ 2 ) - 1 / 2 = ( D ( - θ 2 ) ) - 1 / 2 = { - 720 ( ( 24 θ - 7 - θ - 5 ) sin θ + ( - 24 θ - 8 + 8 θ - 6 ) cos θ iiiiiiiiiiiiiiiiiiiiiii + 24 θ - 8 + 4 θ - 6 ) } - 1 / 2 , where λ1>λ2>>0 and k=1λk<.

Proposition 6.

If x>0, then (45)P(01W^22(t)dt>x)=1πk=1(-1)k+1×γ2k-1γ2k(e-ux/2-24u-4-4u-3)|1/2)-1iiiiiiiiiiiiiiiiiiiiii×(-24u-4-4u-3)|)1/2)-1u(-24u-4-4u-3)|)1/2|-720((24u-7/2-u-5/2)sinu1/2-24u-4-4u-3)|1/2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii+(24u-4-8u-3)cosu1/2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-24u-4-4u-3)|)1/2)-1)du, where γk=λk-1, k=1,2,.

Proof.

It can be proved by the Smirnov formula [8, 9], formula (23), and the definition of the Fredholm determinant. Similar proof method can be found from Proposition 3.3 in .

Next, we give the large deviation and small deviation probabilities of the second order detrended Brownian motion with respect to the norm in the Hilbert Space L2[0,1].

Proposition 7.

Consider x, (46)P(01W^2(t)2dt>x)=(1+o(1))(2π)1/2(2z3/2,1)-2x-1/2exp(-2z3/2,12x)·{720((-324z3/2,1-9+1328z3/2,1-7)sin2z3/2,1iiiiiiiiiiiiiiiiiiiii+(-325z3/2,1-10+926z3/2,1-8-127z3/2,1-6)iiiiiiiiiiiiiiiiiiiii×cos2z3/2,1+325z3/2,1-10+326z3/2,1-8)}-1/2.

Proof.

By Deheuvels  and Martynov , we have for all x>0(47)P(01W^2(t)2dt>x)=(1+o(1))(2π)1/2γ1-1(-D(γ1))-1/2×x-1/2exp(-γ1x2); we take D(λ) and γ1=(2z3/2,1)2 into (47), and then the proof is completed.

Proposition 8.

There exists a constant c>0 such that (48)P(01W^2(t)2dtε)=(c+o(1))ε-2exp(-18ε),asε0.

Proof.

Given two sequences ak>0 and bk>0 with (49)k1ak<,k1bk<,k1|1-akbk|<, we have, as ε0, (50)P(k1akξk2ε)=(1+o(1))(k1bkak)1/2P(k1bkξk2ε).

By the asymptotic formula for zeros of Bessel function (51)z3/2,k=(k+12)π+O(k-1),k,z5/2,k=(k+1)π+O(k-1),k, then ak=λk, b2k-1=((2k+1)π)-2, and b2k=((2k+2)π)-2, kN, which satisfy (49) and by the distribution identity 01W^22(t)dt=k1λ2k-1ηk2+k1λ2kη2k*2 and (50), there exists a constant c1, such that (52)P(01W^2(t)2dtε)=P(k1λ2k-1ηk2+k1λ2kηk*2ε)=(1+o(1))k1(bkak)1/2P(k1bkξk2ε)=(1+o(1))c1×P(k1ξ2k-12((2k+1)π)2+k1ξ2k2((2k+2)π)2ε)=(1+o(1))c1P(k1(k+2)-2ξk2επ2),asε0.

Also, for all d>-1, there exists a constant c2>0, such that, as ε0, (53)P(k1(k+d)-2ξk2επ2)=(1+o(1))c2ε-dexp(-18ε). Connecting (52) with (53), we can obtain the proposition.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Fund (71350005), Heilong Jiang Province Natural Science Fund (G200815), and the Fundamental Research Funds for the Central Universities (no. DL13BBX10).

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