Remarks on strongly convex stochastic processes

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1 Aeqat. Math. 86 (01), c The Athor(s) 01. This article is pblished with open access at Springerlink.com /1/ pblished online November 7, 01 DOI /s Aeqationes Mathematicae Remarks on strongly convex stochastic processes Dawid Kotrys Abstract. Strongly convex stochastic processes are introdced. Some well-known reslts concerning convex fnctions, like the Hermite Hadamard ineqality, Jensen ineqality, Khn theorem and Bernstein Doetsch theorem are extended to strongly convex stochastic processes. Mathematics Sbject Classification (1991). Primary 6A51; Secondary 6D15, 9B6, 60G99. Keywords. Strongly convex stochastic process, Hermite Hadamard ineqality, Jensen ineqality, Bernstein-Doetsch theorem, Khn theorem, mean-sqare integral. 1. Introdction In 1980 Nikodem [7] considered convex stochastic processes. In 1995 Skowroński [9] obtained some frther reslts on convex stochastic processes, which generalize some known properties of convex fnctions. Moreover, in a recent paper [], the present athor showed the Hermite Hadamard-type ineqality for convex stochastic processes. Let I R be an interval. Recall that a fnction f : I R is called strongly convex with modls c>0, if f ( λx +(1 λ)y ) λf(x)+(1 λ)f(y) cλ(1 λ)(x y) for any x, y I and λ [0, 1] (cf. [,8]). Obviosly, every strongly convex fnction is convex. Observe also that, for instance, affine fnctions are not strongly convex. In this paper we propose the generalization of convexity of this kind for stochastic processes. Let (Ω, A, P) be an arbitrary probability space. A fnction X : Ω R is called a random variable, if it is A-measrable. A fnction X : I Ω R is called a stochastic process, if for every t I the fnction X(t, ) is a random variable.

2 9 D. Kotrys AEM Let C :Ω R denote a positive random variable. We say that a stochastic process X : I Ω R is strongly convex with modls C( ) > 0, if the ineqality X ( λ+(1 λ)v, ) λx(, )+(1 λ)x(v, ) C( )λ(1 λ)( v) (a.e.) (1) is satisfied for all λ [0, 1] and, v I. If the above ineqality is assmed only for λ = 1, then the process X is strongly Jensen-convex with modls C( ) or strongly midconvex with modls C( ). If the above ineqality holds for a fixed nmber λ (0, 1), then we say that the process X is strongly λ-convex with modls C( ). Obviosly, by omitting the term C( )λ(1 λ)( v) in ineqality (1), we immediately get the definition of a convex stochastic process introdced by Nikodem in 1980 [7]. On the other hand, we derive it from (1) in a limit case, when C( ) 0. The main sbject of this paper is to extend some well-known reslts concerning convex fnctions to strongly convex stochastic processes. We obtain the conterparts of the Hermite Hadamard ineqality, Jensen ineqality, Khn theorem and Bernstein Doetsch theorem. In the deterministic case most of the presented reslts redce to the properties of strongly convex fnctions obtained recently in [1] and [6]. Note also that the related reslts for convex stochastic processes can be fond in [,7,9].. Jensen-type ineqality In this section we present a Jensen-type ineqality for strongly convex stochastic processes. In [] the following proposition was shown. Proposition 1. Let X : I Ω R be a convex stochastic process and t 0 int I. Then there exists a random variable A :Ω R sch that X is spported at t 0 by the process A( )(t t 0 )+X(t 0, ). That is for all t I. X(t, ) A( )(t t 0 )+X(t 0, ) (a.e.) We begin or investigations with an easy bt very sefl lemma. Of corse, its version for strongly convex fnctions is well-known (cf. e.g. []). Lemma. A stochastic process X : I Ω R is strongly λ-convex (strongly convex, respectively) with modls C( ) if and only if the stochastic process Y : I Ω R defined by Y (t, ) :=X(t, ) C( )t is λ-convex (convex, respectively).

3 Vol. 86 (01) Remarks on strongly convex stochastic processes 9 Proof. In the first part of the proof assme that X is strongly λ-convex. Fix, v I. By strong λ-convexity we get Y (λ +(1 λ)v, ) =X(λ +(1 λ)v, ) C( )(λ +(1 λ)v) λx(, )+(1 λ)x(v, ) C( ) ( λ(1 λ)( v) +(λ +(1 λ)v) ) = λx(, )+(1 λ)x(v, ) C( ) ( λ +(1 λ)v ) = λ ( X(, ) C( ) ) +(1 λ) ( X(v, ) C( )v ) = λy (, )+(1 λ)y(v, ) (a.e.). The proof of the second part is similar, so we omit it. If λ [0, 1] is arbitrarily chosen, then we obtain the lemma for strongly convex stochastic processes. From Lemma and Proposition 1 we immediately derive Corollary. If a stochastic process X : I Ω R is strongly convex with modls C( ), then for all t 0 int I, X is spported at t 0 by the process H : I Ω R of the form H(t, ) =C( )(t t 0 ) + A( )(t t 0 )+X(t 0, ). Now we present a Jensen-type theorem for strongly convex stochastic processes. Theorem 4. Let X : I Ω R be a strongly convex stochastic process with modls C( ). Then ( n ) X λ i t i, λ i X(t i, ) C( ) λ i (t i t) (a.e.) for all t 1,...,t n I, λ 1,...,λ n > 0, sch that λ λ n =1and t = λ 1 t λ n t n. Proof. Let s take t 1,...,t n I and λ 1,...,λ n > 0, sch that λ 1 + +λ n =1. We pt t = λ 1 t λ n t n. According to Corollary, we have the spport H(t, ) =C( )(t t) + A( )(t t)+x( t, ) at t. Then for each i {1,...,n} we have X(t i, ) H(t i, ) =C( )(t i t) + A( )(t i t)+x( t, ) (a.e.). Mltiplying the above ineqality by λ i and smming p all the ineqalities we have λ i X(t i, ) C( ) λ i (t i t) + A( ) λ i (t i t)+x( t, ) (a.e.).

4 94 D. Kotrys AEM Becase n λ i(t i t) =0, ( n ) X λ i t i, λ i X(t i, ) C( ) λ i (t i t ). Khn-type and Bernstein Doetsch-type reslts (a.e.). The classical reslt de to Khn (cf. [5]) states that if f : I R flfils, for some fixed λ (0, 1) and for all x, y I, the ineqality f ( λx +(1 λ)y ) λf(x)+(1 λ)f(y), i.e. f is a λ-convex fnction, then f is also Jensen-convex, which means that ( ) x + y f f(x)+f(y), x,y I. Skowroński proved in [9] thataλ-convex stochastic process is also Jensenconvex. In this section we prove the conterparts of these facts for strongly λ-convex stochastic processes. Theorem 5. Let λ (0, 1) be a fixed nmber and X : I Ω R be a strongly λ-convex stochastic process with modls C( ). ThenX is Jensen-convex with modls C( ). Proof. Assme that X is strongly λ-convex. Lemma yields that the process Y (t, ) =X(t, ) C( )t is λ-convex. By Skowroński s reslt Y is midconvex, which means that ( ) + v Y (, )+Y(v, ) Y, (a.e.). Therefore ( ) ( ) + v + v X, C( ) X(, ) C( ) + X(v, ) C( )v (a.e.) and after some rearrangement we arrive at ( + v ) X(, )+X(v, ) X, C( ) 4 ( v) (a.e.), which finishes the proof. It is well-known that a midconvex fnction f : I R is convex nder slight reglarity assmptions, like local pper bondedness at some point (Bernstein Doetsch Theorem) or measrability (Sierpiński s Theorem), also nder some other assmptions of this type (cf. [4]). Nikodem presented in [7] the conditions garanteeing the convexity of midconvex stochastic processes. Now we consider a similar problem for strongly

5 Vol. 86 (01) Remarks on strongly convex stochastic processes 95 convex processes. We wold like to recall the following definitions. A stochastic process X : I Ω R is called (i) P-pper bonded on the interval (a, b) I, iff { ( {ω lim sp } )} P Ω:X(t, ω) n =0, n t (a,b) (ii) continos in probability in interval I, if for all t 0 I we have P lim X(t, ) =X(t 0, ), t t0 where P lim denotes the limit in probability. For more details we refer the reader to [7]. Now we shall prove the following Theorem 6. If a stochastic process X : I Ω R is strongly midconvex with modls C( ) and P-pper bonded on the interval (a, b) I, then it is continos in the interval I. Proof. Being strongly a midconvex, X is also a midconvex stochastic process. Since X is P-pper bonded on the interval (a, b), it is continos in view of [7, Theorem 4]. Theorem 7. Assme that I is an open interval. A strongly midconvex stochastic process X : I Ω R with modls C( ) is continos if and only if it is strongly convex with modls C( ). Proof. To prove necessity take the process Y (t, ) = X(t, ) C( )t. By Lemma we get that Y is midconvex. Since X is continos, Y is also continos. Using Nikodem s reslt [7, Theorem 5] we arrive at that Y is convex. Using Lemma once more, we infer that X is strongly convex with modls C( ). To prove sfficiency we observe that if X is strongly convex, then X is also convex. By Nikodem s reslt [7, Theorem 5] we get its continity. By Theorems 5 and 7 we obtain immediately Corollary 8. If a process X : I Ω R is continos and strongly λ-convex with modls C( ), then it is strongly convex with modls C( ). 4. Hermite Hadamard-type ineqality It is well-known that every convex fnction f : I R satisfies the Hermite Hadamard ineqality ( ) x + y f 1 y y x x f(s)ds f(x)+f(y).

6 96 D. Kotrys AEM for any x, y I. This celebrated reslt plays a very important role in convex analysis. In [, Theorem ] its conterpart for convex stochastic processes was presented. Below we qote this reslt. Let s recall before that a stochastic process X : I Ω R is mean-sqare continos in the interval I, if for all t 0 I the condition lim t t0 E( X(t) X(t 0 ) ) = 0 holds. Theorem 9. If X : I Ω R is a Jensen-convex, mean-sqare continos stochastic process in the interval I, then for any, v I we have ( + v ) X, 1 v X(, )+X(v, ) X(t, )dt v (a.e.). () The integral in the statement is mean-sqare integral. For the definition and basic properties of mean-sqare integral see for example [10]. Now we wold like to prove the Hermite Hadamard ineqality for strongly convex stochastic processes. We start with a technical lemma. Lemma 10. Let X : I Ω R be the stochastic process of the form X(t, ) = C( )t,wherec :Ω R is a random variable, sch that E[C ] <. If [, v] I, then v X(t, )dt = C( ) v. Proof. By elementary properties of the expectation we have [ n ] [ E X(Θ i )(t i t i 1 ) C v n ] = E CΘ i (t i t i 1 ) C v [ ( n )] = E C Θ i (t i t i 1 ) v ( n ) = Θ i (t i t i 1 ) v E[C ]. If n, then the above expression tends to zero, becase of the definition of the Riemann integral. This finishes the proof. Theorem 11. Let X : I Ω R be a stochastic process, which is strongly Jensen-convex with modls C( ) and mean-sqare continos in the interval I. Then for any, v I we have

7 Vol. 86 (01) Remarks on strongly convex stochastic processes 97 ( ) + v (v ) X, + C( ) 1 1 v v X(t, )dt X(, )+X(v, ) ( v) C( ) 6 (a.e.). () Proof. According to the assmption the process X is strongly convex with modls C( ), so by Lemma the process Y (t, ) =X(t, ) C( )t is convex. By ineqality () we get Hence ( ) + v X, C( ) ( ) + v Y, 1 v Y (t, )dt v Y (, )+Y(v, ) ( + v (a.e.). ) 1 v ( X(t, ) C( )t ) dt v X(, ) C( ) + X(v, ) C( )v (a.e.). Frthermore ( ) ( ) + v + v X, C( ) 1 v X(t, )dt 1 v C( )t dt v v X(, )+X(v, ) C( ) ( + v ) (a.e.). By Lemma 10 we have ( ) + v X, C( ) ( + v ) 1 v v X(, )+X(v, ) 1 v X(t, )dt C( ) v C( ) ( + v ) (a.e.). Adding to all sides of the above ineqality the term C( ) 1 v v and making some simple comptation, we get ineqality (). Open Access. This article is distribted nder the terms of the Creative Commons Attribtion License which permits any se, distribtion, and reprodction in any medim, provided the original athor(s) and the sorce are credited.

8 98 D. Kotrys AEM References [1] Azócar, A., Giménez, J., Nikodem, K., Sánchez, J.L.: On strongly midconvex fnctions. Opscla Math. 1/1, 15 6 (011) [] Hiriart-Urrty, J.B., Lemaréchal, C.: Fndamentals of Convex Analysis. Springer, Berlin (001) [] Kotrys, D.: Hermite Hadamard ineqality for convex stochastic processes. Aeqationes Math. 8, (01) [4] Kczma, M.: An Introdction to the Theory of Fnctional Eqations and Ineqalities. Birkhäser, Basel (009) [5] Khn, N.: A note on t-convex fnctions. In: General ineqalities 4 (Oberwolfach, 198). International Schriftenreihe Nmerical Mathematics, vol. 71, pp Birkhäser, Basel (1984) [6] Merentes, N., Nikodem, K.: Remarks on strongly convex fnctions. Aeqationes Math. 80, (010) [7] Nikodem, K.: On convex stochastic processes. Aeqationes Math. 0, (1980) [8] Roberts, A.W., Varberg, D.E.: Convex fnctions. Academic Press, New York, London (197) [9] Skowroński, A.: On Wright-convex stochastic processes. Ann. Math. Sil. 9, 9 (1995) [10] Sobczyk, K.: Stochastic differential eqations with applications to physics and engineering. Klwer Academic Pblishers, Dordrecht (1991) Dawid Kotrys Department of Mathematics and Compter Science University of Bielsko Bia la Willowa, 4 09 Bielsko Bia la, Poland dkotrys@ath.bielsko.pl Received: Jne 1, 01 Revised: Agst 8, 01

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