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Systems Engineering

Systems Engineering Procedia 2(2011)213-221

Tail Conditional Variance of Portfolio and Applications in Financial

Engineering

Chun-fu Jiang* Yu-kuan Yang

Shenzhen University, Shenzhen 518060, P R China

Abstract

The optimal portfolio selection is an important issue in financial engineering. It is well-known that downside risk measures such as TCE and CVaR only characterize the tail expectation, and pay no attention to the tail variance beyond the VaR. This is an important deficiency of measuring the extreme financial risk in engineering management, especially for insurance industry and portfolio management. In this paper, we study the optimization portfolio model based on tail conditional variance (TCV) motivated by TCE. We obtain the TCV risk of a portfolio and the explicit solution of optimal portfolio under the assumption of multivariate student t distribution. Finally, we also give an example of empirical study on China Stock Market.

© 2011 Published by Elsevier B.V.

Keywords: Optimal portfolio; tail conditional variance; risk measure; financial engineering; student t distribution;

1. Introduction

The optimal portfolio selection is a classical problem in the financial engineering. It is well recognize that the measure of risk have a crucial role in portfolio optimization under uncertainty. The primitive measure of risk, the variance of returns of a portfolio, is employed in the classic mean-variance model by Markowitz. After that many new risk measures were proposed and applied in portfolio optimization. In recent years, the quantile-based downside risk measures have received much attention from practitioners and researchers. Besides the typical value-at-risk (VaR) (see Yamai and Yoshiba [1], Alexander and Baptista [2]), such measures include the tail conditional expectation (TCE) and the expected shortfall (ES) defined in Benati [3]. Although VaR has undoubtedly become a standard risk measure and has been written into industry regulation, VaR's popularity does not mean that it is a sound measure. VaR ignores the magnitude of extreme or rare losses. In addition, it has been shown in Alexander et al. [4] that the problem of minimizing VaR of a portfolio can have multiple local minimizes. As response to these deficiencies of VaR, the notation of coherent risk measure was introduced in Artzner et al. [5], which is an important breakthrough for a comprehensive theory in financial risk measurement. In fact, VaR is also severely criticized that it not a coherent measure of risk due to its lack of subadditivity. That is, VaR associated with a combination of two portfolios may larger than the sum of the VaRs of the individual portfolio (see Acerbi and Tasche [6], Tasche [7], Kalkbrener [8] ).

* Corresponding author. Tel.: +86-755-26538922; fax: +86-755-26538959. E-mail address: jiangcf@szu.edu.cn.

2211-3819 © 2011 Published by Elsevier B.V. doi:10.1016/j.sepro.2011.10.026

To remedy the problems inherent in VaR, Artzner et al. [9] further proposed the use of expected shortfall, or ES for short (see also Acerbi and Tasche [8], Kamdem [10]), which is defined as the conditional expectation of loss beyond the VaR level. Other alternative measures to ES include conditional value-at-risk (CVaR) in Rockafellar and Uryasev [11, 12], tail conditional expectation (TCE) in Artzner et al. [5] and worst conditional expectation (WCE) in Benati [13]. It is noteworthy that CVaR as a coherent risk measure coincides with ES and TCE under the assumption of continuity distribution. CVaR now is rather appealing in portfolio optimization due to its attractive properties such as convexity and continuity with respect to portfolio weights, see for example [14-19]. Nevertheless, CVaR is also criticized by Lim et al. [20] for being fragile in portfolio optimization due to estimation errors.

These risk measures mentioned above undoubtedly make remarkable improvement on VaR. However they are only the linear probability weighted combination of losses beyond VaR. Some researchers argue that the higher orders of moments of the loss distribution should be considered in order to describe it comprehensively and decrease the extreme loss. For example, Cheng and Wang [21] recently proposed a new coherent risk measure based on p-norms with application in realistic portfolio optimization (see also Cheng and Wang [22,23] ). Landman [24] proposed using tail conditional variance (TCV), which characterizes the variance of the loss distribution beyond some critical value. They also establish correspondingly a portfolio selection model.

The aim of this paper is to establish a portfolio selection model by considering tail conditional expectation and tail conditional variance beyond VaR simultaneously. We also will find the analytic closed solution of the associated optimal portfolio model under the multivariate student-t distribution of returns. As an application-oriented research and along a new derivation way, our approach to deriving the optimal portfolio can be regarded as the improvement on Landman [25], in which the optimal solution is based on the assumption of full rank of the constraint matrix and needs the manual partition of the constraint matrix.

This paper is organized as follows. In the next section, we provide the tail conditional variance (TCV) of a portfolio under the multivariate student-t distribution. In section 3, we formulate the optimal portfolio model under TMV defined by TCV and TCE, and derive the closed solution of the model. Section 4 illustrates some examples of Chinese market.

2. The tail conditional variance of portfolio

Consider a portfolio selection problem with y assets. The random return of the j-th asset is denoted by r, and the returns vector is denoted by r = (r1,r2,-rn) . Let /u = E(r) and Z = Var(r) be expected returns vector and covariance matrix respectively. Let coi be the fraction of wealth invested in asset i. We call the investment weight vector a = (ax,®1,---®n) a portfolio. The return of portfolio is defined by ra = r'a . The expected return and the variance risk of portfolio are, respectively, given by jua = ju'a and Var(rc)

First, we recall some so-called downside risks based on quantile such as VaR, TCE, ES. Let x = -ra be the loss of the portfolio. Given a confidence level 1 -a, ae (0,}), the VaR of the portfolio a is defined as

VoRa (X) = inf{x | P(X < x) > 1 - a}. The tail conditional expectation (TCE) and tail conditional variance are defined respectively as the following

TCEa (X) = E(XX > VoRa (X)), }

TCVa (X) = E ((X - TCEa (X))21X > VoRa (X)).

Assume that the vector of asset returns r has the multivariate elliptically distribution with the density function

f(x) =|Z|-1/2 n((x-^)'S-1(x-M)), where g is an non-negative real function. We write r ~ En Z, g). In particular, if the function g is defined by

n+v V+Y )

n (C) = Cv,n (1 + C / v)~, Cvw =-(ITL=,

r(v /2)V(v^)n

then the return vector r follows the multivariate student-t distribution with the density

m = (1 + (lU^y. (2)

T(v/2^| S |(v^)n I v J

And we denote r ~ tn (u, S, v) for such random vector r.

The elliptically distribution is a more generalized distribution family than the normal. Besides student-t distribution, it also includes fat tail distributions such as the mixture of normal distribution, symmetric stable distribution and Laplace distribution etc, which are widely used in financial data modeling and portfolio risk measuring. For example, under the assumption of elliptically distribution, we can obtain the VaR of a portfolio as

VaRa (X) = -rnU + qgaJ JVrn, TCEa(X) = -J u + Kg V JV®, (3)

where qg,K g are constants associated with the density of elliptically distribution g and the confidence level. In particular, under student-t distribution, for the calculation results of qg and K'a, we refer to the literature [10].

Theorem 1. Suppose that the vector of returns r = (r1, r2,—rn) follows a multivariate student-t distribution. Then under a confidence level 1 - a, the tail conditional variance of the portfolio ® = J j, • • j)' is given by

TCVg(-j) = (Tav -K,v)>'Sj (4)

r( v-1)

— v 2(q 2 + v)-^

T,.„ =

1) ( v V

2 r / v+1 \ f \2 ( , i o A

qa ,v K^ v 2_v + 1 v - 2 v v

V 2 ' 2 '2' ql

a )(v - 2)

and2F1 (a,P\y,6) is the hypergeometric function.

Proof. First, For the sake of expression, we denote the TCE of the portfolio ® by TCEa (-j), where -j

means the loss of the portfolio. Since the TCE is obviously equal to the ES of a portfolio in the case of multivariate student-t distribution, from [10] we see that

TCEa (-J) = -Ju + es'a = -Ju + es'a v J B |, (5)

where S = BB' is a Cholesky decomposition of B . According to the definition of tail conditional variance in eq. (1), thus we can obtain

TCVa (-j) = E[(-rm - TCEa (-rj)2 |-rj > VaRa ] = L - (TCEa (-r®))2, (6)

L = E(rJ | r® < -VaRa) = — I S | 2 f (Jx)2g((x-¡)S-1(x- u)'dx.

^ Jx<VaRa}

Changing variable y = B-1 (x - u), and dx =| B \ dy, where | B \ denotes the determinant of B, then we get

L = -f ' VR (J'By + a'uygdy^dx.

a J {J By<-J u-VaRa

Let H be a rotation which sends®'B to (| JB |,0,0, — ,0). Let y = zH . And if we let z1 be the first component of z , and | z |= z12 +1 z* |2 with z* e Rn-1, thus it follows that

L = - LB< ' VR }(J B|z1 +®'u)2 g (|z|2)dz . Now we divide L into three part as

L = - f ' VR }(|J B|2 z12 + 21 ®' B|(u'®)z1 + (Ju)2)g(|z|2)dz = A+ L2 + (Ju)2, (7)

L = — f |J B |2 z,2 g (| z |2)dz, L, = — f 21 ®' B |(uJ z,g (| z |2)dz.

1 a J|JB|z1 <-Ju-VaRa V 1 2 a J{\JB\zl<-Ju-VaRa } 1 1 M 17

By using spherical variable z* = % e Sn_2, we get

I CB |2| SY

—C j—VoRa

Prn—2 j |CB| z,2n[(z,2 + r2)2]dz,

J 0 J —w

where | Sn_2 | is the surface measure of the unit sphere in R"~2, and | Sn-2|= 2n 2 /r^) Replace the variable z1 by -z1 and let u = z12 + r2. Thus, we obtain that

IC B I2

Y—3 f, C

a r( w21)

B|2 tf 2

cn.vx L+" LT zi2 (c—zi2)211+cV I dcdzi

iT~ Cnvv 2 j; zi2 (c—zi2)2 (c+v)— 2 dcdzi

ar(W2i) 'av'zi

n— i

|CB |2 r(V+i)tf2 Jr+»

v2 j zi2 (v + zi2) 2 dzi,

where the last line is due to the following integral from Gradshteyn and Ryzhik [26, p316]

I(zi) = j" (d — zi2 )2 (v + c) 2 de = (zi2 + v)~2 B

f V +1 n + i

where B(a,P) is the Euler Beta function. Also let

v+1 1 1 v+1

z»+w 2 ( 2 \----1 r+w — / \—-—

h(y,v) = jy zi (zi + v) 2 dzi = -jy2 y2 (y + -) 2 dy.

Note that the integral in Gradshteyn and Ryzhik [26, p316], that is, if

xj—1 eM—vi p- vi

arg—

<n, Re v1 > Re j> 0, then

+w ' , e- p -1 f 1

■p.—-¿-vd=-2 Fi\vi, vi — J vi —J+1;-—

(1 + Px) 1 v, — J | Pe

where 2 Fl (a,P\y,9) is the hypergeometric function. From (8) (10) (11), we gets

r( v+1)

aVtfr( -2-)(v — 2)

v 2 F 21 1

^ , 10 ^ v +1 v — 2 v v

2 2 qa

®' Z®.

Similarly, we have

L2 = —(®J) I®' B^ Cnv j+"jz+2w z, (c — z,2)2 Ii + -I 2 dedz1

Y-1 f* e Y~

Recalling the integral in eq. (9), we can obtain

¿2 r(^

2 aVnTQ The theorem follows from (5) (6) (7) (12) and (13).

(®J)CB\£aC j+w —-(e — qa,v) 2 I 1 +

2a Jqa, n — 1 I

v v—1

v2 (+v )2 (cj) c b\

3. Tail Mean-Variance Optimization Portfolio Model in Financial Engineering

In the classic mean-variance (MV) model, the mean-variance risk can be expressed by

MV (X) = E (X) + War (X)

= E (-r®) + XVar (-r®) = -/® + X®y S®, where A> 0 is a constant depended on the investors preferences. We can derive the optimal portfolio by minimizing the mean-variance risk MV(X) Similarly, we can define the tail mean-variance risk by

TMVa (X) = TCEa (X) + ATCVa (X). (14)

Under multivariate student-t distribution, the portfolio TCE risk from eq. (5) is

p(v-1) v - v-1

TCEa(X)=-M(+v~2 ia+v)2 (15)

Substituting (4) and (15) into (14), we have

TMVa (X) = -/® + \a(£a + l\a®'S® ,

a = eS'a,v , a = eS'a,v (<7a - eS'a,v ) + Ta,v.

We now construct a portfolio model with respect to tail mean-variance risk, that is

Jmin -/® + S® + AÁ2a®'S® (16)

[s.t 1'®= 1.

According to theorem 1, we can find the optimal solution of model (16). For this, we first analyze the generalized problem of minimization of the goal function

f(x) = / x + /(V x' Sx + yx' Sx), (17)

subject to a linear restriction

Ax = b, (18)

where /3, y > 0, A is some real m x n matrix, b is some m x 1 vector. It is obvious that the goal function in (17) is a convex function. For the optimization problem of the goal function (17) with the linear restriction (18), we can obtain the theorem in the following. Theorem 2. The minimization problem of the goal function in (17) subject to (18) exist optimal solution

x* = S-1 A'(AS-1 A ■)-1 b -®® Z (Z' SZ)-1Z' /u

where Z = (A')1 is the orthogonal complement of A', l = y¡/ Z (Z' Z)-1 Z' / , ®* is the unique rational root of the quartic equation

®4 + 2k®3 + (f0 + k2 —^)®2 - 2kf0® + k2f = 0, 4y

Proof. With Y and Z, we denote the matrices formed by the basis for the image space of matrix A' and the zero space of matrix A', respectively. Let the rank of A be equal to k. Then we have the matrix Y e Rnxk and Z e Rnx(n-k) are orthogonal and satisfy A' Z = 0 and A 'Y being non-singular. Without loss of generality, we choose Y satisfies A 'Y = I. Let

x = Yx + Zx. (19)

Thus from the restriction condition A'x = b, we can obtain

A 'Yx + A'Zx = b,

from which x = b follows. Further, the eq. (19) can be rewritten by

x = Yb + Zx. (20)

Consequently, the minimization problem with constraints described above can simplify a minimization problem without constraints, that is, the minimization problem ming (x), where

g (x) = /'(Yb + Zx) + /y(Yb + Zx)' S(Yb + Zx) + / (Yb + Zx)' S( Yb + Zx).

where f0 = b '(A'S-1A)-1 b, k

and note that

x0 = argmin x ' Sx = S-1A( A ' S-1A)-1b,

A ' x=b

(Yb + Zx)'S(gb + ZX) > (x0)'Sx0 = f0 > 0.

In order to find the optimal solution of the problem min g (x), Let xx be the unique solution of the equation

—n(xx) = Z '/u + P[b 'g 'SYb + b 'g 'SZx +x'Z 'SYb + xx'Z 'SZx + 2^]2(Z 'SZx + Z 'SYb) = 0,

which can be rewritten by

(Z' SZX + Z' SYb) = T[((Yb + Zx)' S(Yb + Zx))2 + 2y\l , where r = (t,t,---t ) =__—Z'j. Consider X* in the form X* = X0 + y*, where y* = (y,y2,---,yn_k)'.

Since x0 = Yb + ZX0 is the optimal solution of (21), similar to the analysis of (22), thus it follows that

Z' SZx0 + Z' SYb = 0.

According to the analysis above, we know y* now is the unique solution of the following equation

(22) (23)

y* = p_T[((Yb + ZX*)'S(Yb + Zx )) 2 + 2y], where P = Z'SZ . If r = 0, then y* = 0 follows obviously. And if r ^ 0 , then the matrix P_1 is definitive. Let us present the matrix P _1 in the following block decomposition

ip -A 1

where p 1 is the first row of P 1, and P2 1 consists of the rest n _ k _ 1 rows of P 1. Thus we can derive

y* = y*(1,r')' ,

where r = P P2_1r. Substituting (25) into (24), we obtain

y* = p-V[((Yb + Zx*) ' S(gb + Zx*))~2 + 2y] = P1-1t

yjf0 + + 2r-1

( P-1^)2 P2

where the last line is due to (21), (23) and the following equality

(1, r') P(1, r')' =-

(P_T)2

By letting®* = ly1l(PP_1r), the eq. (26) can be expressed by

tP- V.

+®*2 Pa

-2y, a > 0,

which is equivalent to the following quartic equation

a4 + 2k® + (f0 + k2---)®2 _ 2kf0a + k2f0 = 0,

where f = b'(A' SA)_b, l = Ja'(Z'Z)_ A, A = _Z' j, k = l/2a-.In fact, the eq. (27) implies a is a solution of the equation (18) on the interval (0, k). Consequently, it follows from (25)-(27) that

y = P-1 V

, Pa L (P-V)

=® p- 1A.

Thus, we deduce by recalling eq. (20)

which proves the result.

x* = Yb + Zx* = Yb + Z ( x0 + y*) = x0 + Zy*,

4. A Case Study of Chinese Market

In this section, let us take Chinese Stock Market as a case study. We choose nine stocks from Shenzhen Stock Exchange(SZSE) and Shanghai Stock Exchange(SHSE) including China Vank A shares(CVKA), Polaroid Hotel special treatment(PLRHST), China baoan group co.,ltd(CBG), SZPRD A shares(SZPRDA),CGS Holding co.,ltd A shares(CGSHA), Konka GroupA shares(KONKAA), Shenzhen Victor Onward Textile Industrial co.,ltd A shares special treatment(SVOTIAST), Shenzhen kaifa technology co.,ltd(SKF), Shenzhen Chiwan Wharf Holding Limited A shares(SCWHLA) et al. The sample data employed was 54 weekly observations of log returns covering the period from January 2007 to June 2011. We begin the analysis by calculating the mean and variance of sample data with the result in table 1 and table 2.

Table 1. Estimation of stock week returns

CVKA PLRHST CBG SZPRDA CGSHA KONKAA SVOTIAST SKF SCWHLA

0.0044 0.0075 0.0123 0.0067 0.0090 0.0061 0.0048 0.0057 0.0016

Table 2. Covariance matrix of stock week returns

CVKA PLRHST CBG SZPRDA CGSHA KONKAA SVOTIAST SKF SCWHLA

CVKA 0.0059

PLRHST 0.0019 0.0046

CB 0.0031 0.0026 0.0078

SZPRDA 0.0021 0.0029 0.0025 0.0069

CGSHA 0.0028 0.0024 0.0029 0.0027 0.0076

KONKAA 0.0019 0.0024 0.0030 0.0025 0.0032 0.0048

SVOTIAST 0.0016 0.0019 0.0030 0.0017 0.0021 0.0023 0.0053

SKF 0.0018 0.0022 0.0030 0.0021 0.0031 0.0029 0.0022 0.0047

SCWHLA 0.0020 0.0018 0.0023 0.0017 0.0028 0.0023 0.0014 0.0021 0.0033

Denoting fi = \a, 7 = XX2a/Xla , we can obtain the optimal solution of portfolio selection model (16) under confidence level 0.9 and 0.8, and the calculation results are presented in table 3, where X = 5, v = 3.

Table 3. Optimal portfolio under different confidence level

CVKA PLRHST CBG SZPRDA CGSHA KONKAA SVOTIAST SKF SCWHLA a=0.1 0.1442 0.1398 -0.0814 0.0814 -0.0765 0.0237 0.2130 0.1239 0.4319

a=0.2 0.1463 0.1360 -0.0890 0.0813 -0.0819 0.0228 0.2156 0.1249 0.4439

Let us denote Max = TCEa(-rx), ct^ = TCVa(-rx). Note that ua,x = -j x + \aJx'Sx and cr2ax = X2ax' Sx ,

According Theorem 2, thus we can derive the formulation between TCE risk and TCV risk of the TMV optimal portfolio under the student-t distribution by

Ua,x = r0 + r^ VCT.x - X + "77= CTa,x , •4bA2,a y¡A2,a

where r0 =-p'x0, r1 =-b =-/u<Z(Z'SZ)-1 Z'ju.

In fig.1, we further present the graph of TCE and TCV of the optimal portfolio with the case of X = 5, v = 3. For the limit case of a = 1, it is easy see that the TMV model will reduce theoretically to the MV model. The result of empirical study in fig.1 demonstrates such a fact. In fig.2. by taking the risk aversion parameter X with the value

varying from 3 to 20, we present the graph of the efficient frontier under TMV portfolio selection model with the confidence level 1 -a = 0.8, 0.9 respectively.

Fig.1. TCE and TCV of the optimal portfolio

5. Copyright

Fig.2. the efficient frontier of portfolio

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Acknowledgements

This paper is partially supported by Natural Science Foundation of Guangdong Province (No. 2008276). References

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