Representation of Stochastic Process by Means of Stochastic Integrals

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1 Inernaional Journal of Mahemaics Research. ISSN Volume 5, Number 4 (2013), pp Inernaional Research Publicaion House hp:// Represenaion of Sochasic Process by Means of Sochasic Inegrals Dr. T. Vasanhi and Mrs. M. Geeha Associae Professor Of Mahemaics, ADM College For Women, Nagapainam Assisan Professor Of Mahemaics, Saradha Gangadharan College, Pondicherry Absrac This paper produces an affirmaive soluion o Kolmogorov concerning he normal disribuion. Keywords: Sochasic inegral, Kolmogorov problem. 1.Inroducion Le X() be a sochasic process for 0; he random variable X( ) X( ) is called he incremen of he process X() over he inerval [, ]. A process X() is said o be homogeneous if he disribuion funcion of he incremen X( + τ) X() depends only on he lengh τ of he inerval bu is independen of. Two inervals are said o be non overlapping inervals if hey have no common inerior poin. A process X() is called a process wih independen incremens if he incremens over non overlapping inervals are independen. A process is said o be coninuous a he poin if for any > 0 lim P( X( + τ) X() > ε) = 0. a process is coninuous in an inerval [A, B] if i is coninuous in every poin of [A, B]. Le X() be a homogeneous and coninuous process wih independen incremens and le us denoe he characerisic funcion of he incremen X( + τ) X() by f(u, τ). I is known ha f(u, τ) is infiniely divisible and ha f(u, τ) = f(u, 1). Le b be a funcion defined in [A, B] and υ a non negaive funcion in [A, B]. Le us consider for each ineger n a subdivision of [A, B] (D ) A =, <, <.. <, = B (n = 1,2,.. ) and assume ha Max,, 0 as n

2 386 Dr. T. Vasanhi and Mrs. M. Geeha In each subinerval,,, selec a poin, (k = 1,2,.., n). Le us form he sums S = b, X υ, X υ, If he sequence S converges in probabiliy o a random variable S, and is his limi is independen of he choice of he subdivision (D ) and he poins,, hen say ha S is a sochasic inegral in he sense of convergence in probabiliy and wrie S = b()dxυ(). The following heorem gives a condiion ensuring he exisence of he sochasic inegral in he sense of convergence in probabiliy. Theorem 1.1 Le X() be a homogeneous and coninuous process wih independen incremens defined for 0. Suppose ha he funcion b is coninuous in [A, B] and υ is a non decreasing, non negaive and lef coninuous funcion in [A, B]. Then he sochasic inegral b()dxυ() (1.1) exiss in he sense of convergence in probabiliy and is characerisic funcion h is given by log h(u) = log f(u b())dυ() (1.2) Le b and υ be as in heorem 1.1. I is well known ha here exiss a finie Borel measure V wih suppor conained in [A, B] such ha 0 if < A V((, )) = υ() υ(a) if A B υ(b) υ(a) if > B Pu υ () = V(b (, )) hen υ is a non decreasing, non negaive and lef coninuous funcion. Furher we pu, C = min b() and D = max b()

3 Represenaion of Sochasic Process by Means of Sochasic Inegrals 387 By heorem 1.1 he sochasic inegral dxυ () (1.3) exiss and is characerisic funcion h is given by log h (u) = log f(u b())dυ () by he ransform formula for inegrals, we have log f(u b())dυ() = log f(u )dυ (). Theorem 1.2 Le X(), b and υ be as in Theorem 1.1. Then he inegrals (1.1) and (1.3) are idenically disribued. 2. Represenaion Theorem Theorem 2.1 Le X() be a homogeneous and coninuous process wih independen incremens defined for 0 and le he Levy canonical represenaion of he characerisic funcion of X(0) X(1) be given by a, σ, M and N. Le υ be a non decreasing, non negaive and lef coninuous funcion in [A, B]. Then he Levy canonical represenaion for he characerisic funcion of he sochasic inegral dxυ() is given by he following formulas: a = a + (1 ) σ = σ dυ() (,) M (x) = N x dυ() (,) (,) N (x) = M x dυ() (,) x (1 + (x) )(1 + x d(m( x) + N(x)) + dυ(); ) (,) + M x dυ() (,) (,) + N x dυ() (,) (2.1) (2.2) (2.3) (x < 0) (2.4) (x > 0) (2.5) Lemma 2.1 The funcion g is an infiniely divisible characerisic funcion if, and only if, i can be

4 388 Dr. T. Vasanhi and Mrs. M. Geeha wrien in he form log g(u) = iau + σ 2 u + r(u, x)dm(x) + r(u, x) dn(x) where a, σ are real consans; M and n are non decreasing in he inervals (, 0) and (0, ) respecively, wih M( ) = N( ) = 0 x dm(x) < and x dn(x) < for every ε > 0 and r(u, x) = e 1 (iux (1 + x )) (2.6) Proof of heorem 2.1 Wih ou loss of generaliy le us assume ha A 0 B. Firs we assume ha here exiss a number > 0 such ha is a poin of coninuiy of υ and υ( ) υ( ) = 0 (2.7) The characerisic funcion of (2.1) is denoed by h. Then by heorem 1.1 we have log h(u) = log f(u )dυ() Now le us define a funcion s by s(u, x, ) = r(u, x) r(u, x) where in view of (2.6) s(u, x, ) = i(1 )x u (1 + (x) )(1 + x ) (2.8) Since s(u, x, ) = o(x ) as x 0 and s(u, x, ) = o(1) as x he funcion s is inegrable wih respec o M and N. By Lemma 2.1 and he definiion of s we have, log f(u) = iau σ 2 (u) + (r(u, x) + s(u, x, ))dm(x) + r(u, x) + s(u, x, ) dn(x) (2.9) By virue of (2.8) and (2.9) le us obain log h(u) = iau dv() σ 2 u dv()

5 Represenaion of Sochasic Process by Means of Sochasic Inegrals s(u, x, )dm(x) + s(u, x, ) dn(x) dυ() + r(u, x)dm(x) + r(u, x) dn(x) dυ() Using he definiions of a and σ we can wrie his relaion in he form log h(u) = ia σ + r(u, x)dn(x)dυ() 2 (u) + r(u, x)dm(x)dυ() (2.10) and in view of (2.7), Le us have log h(u) = ia σ 2 (u) + r(u, x)dm(x)dυ() + r(u, x)dm(x)dυ() + r(u, x)dn(x)dυ() + r(u, x)dn(x)dυ() (2.11) Decomposing he hird erm on he righ hand side of (2.11) we ge for every ε > 0. I = r(u, x)dm(x)dυ() = r(u, x)dm(x)dυ() + + r(u, x)dn(x)dυ() = I + I Applying L Hospial s rule wice we find

6 390 Dr. T. Vasanhi and Mrs. M. Geeha r(u, x) lim x = (u) 2 Hence here is a consan C such ha for fixed u and [A, ] r(u, x) C (u) x Therefore le us ge for I he esimaion (ε 0 +) I C u dυ() Furher we can ransform I in he following way. so le us obain as ε 0 x dm(x) = o(1) I = r(u, x)d M x dυ() = r(u, x)d M x dυ() I = r(u, x)d M x dυ() Transforming he fourh, fifh and sixh erms of (2.11) in a similar manner o he hird one we ge log h(u) = ia σ 2 (u) + r(u, x)d M x dυ() + r(u, x)d + r(u, x)d M x dυ() N x dυ() + r(u, x)d N x dυ() Finally, using he definiion of M and N, we can rewrie his relaion in he form,

7 Represenaion of Sochasic Process by Means of Sochasic Inegrals 391 log h(u) = ia σ 2 (u) + r(u, x)d M (x) + r(u, x)d N (x) Le us complee he proof by showing ha a, σ, M and N saisfy he condiion of Lemma 2.1. Obviously a and σ are real consans and σ 0. By definiion i is easily seen ha M and N are non decreasing in he inervals (, 0) and (0, ), respecively, having he properies M ( ) = N ( ) = 0 For every ε > 0 we obain he inequaliy x dn (x) = x dn(x)dυ() + dυ() x dn(x) + dυ() x dm(x)dυ() x dm(x) < Analogously, le us ge x dm (x) < Then Lemma 2.1 shows he saemen provided ha (2.7) is valid. Now le us urning o he general case n which (2.7) need no be rue. Pu for n max ( 1/A, 1/B) υ() if 1 n υ () = υ( 1 n) if 1 n < 1 n υ() if > 1 n Obviously, we have lim υ () = υ() (2.12) and he funcions υ saisfying (2.7) are non decreasing, non negaive and lef coninuous. Hence le us can apply he firs par of he proof o he sochasic inegrals dxυ () (2.13) and obain represenaion of a, σ, M and N by formulas analogous o (2.2) (2.5) using Helly s second heorem, we ge

8 392 Dr. T. Vasanhi and Mrs. M. Geeha lim log f(u )dυ() = log f(u )dυ() (2.14) Because log f(u ) considered as funcion of is coninuous and bounded and by (2.12) he sequence υ converges weakly o v. Le us denoe he characerisic funcion of (2.13) by h. In view of heorem 1.1. relaion (2.14) is equivalen lim h (u) = h(u) Using he known fac ha under his circumsance a a, σ σ, M M and N N ( sands for weak convergence) he saemen follows. 3.ON A PROBLEM OF KOLMOGOROV CONCERNING THE NORMAL DISTRIBUTION The following heorem gives an affirmaive answer o a quesion posed by A. N. Kolmogorov. Theorem3.1 Le F be an infiniely divisible (i.d.) disribuion funcion and F (x) == Φ(x) = e du for x < 0; hen F (x) Φ(x) For he proof le us sar by collecing ogeher several lemmas. Lemma 3.2 Le F be i. d. and e df(x) <, y > 0. Then he Levy represenaion of f() = e df(x) log f(iy) = ay + σ 2 y + r(y, u)dm(u) + r(y, u) dn(u) where r(y, u) = e 1 + yu 1 + u holds for y > 0. Lemma 3.3 Under he assumpions of lemma 3.2 log f(iy) increases a leas exponenially, if M(U) 0.

9 Represenaion of Sochasic Process by Means of Sochasic Inegrals 393 Proof Using he formula u r(y, u) = e 1 + yu y 1 + u we obain for η > 0 r(y, u) dn(u) = y l(yu)udn(u) y dn(u) 1 + u + r(y, u) dn(u) u where l(υ) = υ (e 1 + υ) > 0 whence we ge he esimaion r(y, u) dn(u) y dn(u) 1 + u dn(u) = L (y) (say) u Now we choose < p < q < 0 and assume M (q) > M (p). Then we obain from lemma 3.2 log f(iy) ay + σ 2 y + e 1 y u 1 + u dm(u) + L (y) This proves he asserion. Lemma 3.4 Le Φ (x) 2Φ(x), x < 0 = 1, x > 0 hen he corresponding characerisic funcion (c. f.) is φ () = e 1 i 2 π e dw wih he asympoic behaviour φ (iy) = e 2 + oe (y ). Proof We have o calculae

10 394 Dr. T. Vasanhi and Mrs. M. Geeha f() = 2 π e e du = 2 π e e () du The second inegral can be evaluaed by means of conour inegraion saring wih 0 = e () dς + ( ) + ( ) + ( ) and leing Z. Equaion (4) is an easy consequence of (3). Lemma 3.5 Le f() be he c. f. of an i. d. disribuion F(x). Then F(0) = 0, F(ε) > 0 if and only if in he corresponding Levy represenaion characerized by (â, σ, M, N). We have σ = 0, M(u) 0(u < 0), udn(u) <, a + u dn(u) 1 + u = 0 In his case we have log f(iy) = (e 1) dn(u) Proof Puing G(x) = 2F (x) 1, x > 0, we represen F by F(x) = 1 2 [Φ (x) + G(x)]. We denoe he c. f. of G(x) by g() wih he propery g(iy) < 1 (y > 0) and obain by lemma 3 log f(iy) = e df(x) = 1 2 [φ (iy) + g(iy)] = e 1 + Oe i.e log f(iy) = y 2 + Oe By lemma 1 we also have he represenaion (1). We are going o compare he asympoic (y ) behaviour of (1) and (5). From Lemma 2 we immediaely ge he conclusion M(u) 0( u < 0 ).

11 Represenaion of Sochasic Process by Means of Sochasic Inegrals 395 So we have o direc our aenion o he funcion N (u) (u > 0), which always has he propery We have o consider wo cases. u dn(u) <. α) udn(u) <. The funcion l(u) in (2) is increasing, hence, puing H(u) = υdn(υ)(nb: H(+0) = ), We have for η > 0 R (y) =: l(yu)udn(u) > l(υ)dh υ y e [H(η) H(y )] Therefore, by assumpion α), lim R (y) =. Now we conclude from (1) and log f(iy) = σ 2 y + y a + R (y) u 1 + u (e 1) dn(u) + u 1 + u dn(u) + dn(u) = σ 2 y + yr (y)(1 + o(1)) This formula mus be compared wih (5). Le us firs assume ha lim y R (y) = 0. Then i follows σ = 1, and we have he conradicion yr (y) = Oe On he oher hand, if we have lim y R (y) = l > 0, hen we ge σ 2 + l = 1 2 Now we noe ha (as l(v)v < с, v > 0)

12 396 Dr. T. Vasanhi and Mrs. M. Geeha yr (y) = y l(yu) yu u dn(u) cy u dn(u), Therefore l can be made arbirarily small. Hence we again obain σ = 1 and he wrong equaion (6). So assumpion α) is false. In his case (1) yields β) udn(u) <. log f(iy) = σ 2 y + y a + u 1 + u dn(u) + S (y) + (e 1) dn(u) where S (y) = (e 1) dn(u) = y e 1 d υdn(u) yu The inegrand is increasing, i. e. y υdn(u) S (y) 0 Comparing (7) and (5), we now immediaely obain σ = 1 and a + u 1 + u dn(u) + 1 y S (y) = Oe Puing we ge from (9) lim y S (y) = s a + u 1 + u dn(u) + S = 0 Bu (8) shows, has vdn(v) can be chosen arbirarily small. Therefore a + u 1 u dn(u) = 0

13 Represenaion of Sochasic Process by Means of Sochasic Inegrals 397 From Lemma 4 and (7) we now conclude ha f(iy) = e f(iy) where f() is he c. f. of a disribuion funcion F(x) wih F(0) = 0, F(ε) > 0. In view of (5) his is possible only for f(iy) 1, i. e. N(u) = N(u) 0. Finally we obain a = 0 from (10). The proof is complee. 4.References [1] Lukacs, E. (1969). A characerizaion of sable process. J. Appl. Prob, [2] Lukacs, E. (1970) Characerizaion heorems for cerain sochasic processes. Rev. Inerne. Sais. Ins. 38, [3] Lukacs, E. (1968). Characerisic Funcions, 2 nd Ediion, Griffin, London. [4] Lukacs, E. (1968). Sochasic Convergence, 2 nd Ediion, Academic press, New York [5] Prakasa Rao. B.L.S (1968) On a characerizaion of symmeric sable processes wih finie mean, Vol. 39, No.5, [6] Riedel M. (1976) On he onesided ails of infiniely divisible disribuions. Mah. Nachr. 70, [7] Riedel. M. (1980) Represenaion of he characerisic funcion of a sochasic inegral. J. Appl. Prob. 17, [8] Rossberg. H.G (1974) On a Problem of Kolmogorov concerning he normal disribuion. Theor. Prob. Appl. 19, [9] Riedel. M. (1980) Characerizaion of sable processes by idenically disribued sochasic inegrals. Adv. Appl. Prob. 12, [10] Widder, D.V (1946): The Laplace Transform. Princeon Universiy Press, Princeon, NJ.

14 398 Dr. T. Vasanhi and Mrs. M. Geeha