# Generalization ofh-Convex Stochastic Processes and Some Classical Inequalities

Authors: Hao Zhou; Muhammad Shoaib Saleem; Mamoona Ghafoor; Jingjng Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The field of stochastic processes is essentially a branch of probability theory, treating probabilistic models that evolve in time. It is best viewed as a branch of mathematics, starting with the axioms of probability and containing a rich and fascinating set of results following from those axioms. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. In this paper, the concept of generalizedh-convex stochastic processes is introduced, and some basic properties concerning generalized h-convex stochastic processes are developed. Furthermore, we establish Jensen and Hermite–Hadamard and Fejér-type inequalities for this generalization.


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## Body

## 1. Introduction

Stochastic processes are a branch of probability theory, treating probabilistic models that evolve in time [1–3]. It is a branch of mathematics, starting with the axioms of probability and containing a rich and fascinating set of results following from those axioms [4]. Although the results are applicable to many areas [5], they are best understood initially in terms of their mathematical structure and interrelationships [6].There are various ways to define stochastic monotonicity and convexity for stochastic processes [7], and they are of great importance in optimization, especially in optimal designs, and also useful for numerical approximations when there exist probabilistic quantities in the literature [8]. We also refer [9–12] for detailed survey about the importance and interesting properties of stochastic models.The idea of convex functions was put forward, and many generalizations were made in this area [13]. h-Convex [14] and ϕ-convex functions [15] are famous generalizations of convex functions. Later, the theory of stochastic processes has developed very rapidly and has found application in a large number of fields. The study on convex stochastic processes was initiated in [16] by B. Nagy in 1974. After that, Nikodem in 1980 introduced the convex stochastic processes in his article [17]. Following this line of investigation, Skowronski described the properties of Jensen-convex and Wright-convex stochastic processes in [18, 19]. Some interesting properties of convex and Jensen-convex processes are also presented in [20, 21]. Assume a Jensen-convex stochastic process ξ:I×Ω⟶ℝ which is mean-square continuous in I, where I⊆ℝ is an interval. Then, for every a1, a2∈Ia1<a2,(1)ξa1+a22≤1a2−a1∫a1a2ξudu≤ξa1+ξa22a.e.,is known as the Hermite–Hadamard-type inequality for the convex stochastic process [22].The aim of this paper is to introduce the notion of the generalizedh-convex stochastic process and to extend the classical Hermite–Hadamard inequality for convex stochastic processes to generalized h-convex stochastic processes.


## 2. Novelty and Significance

The study of convex functions makes them special because of their interesting properties as maximum are attained at the boundary point, and moreover, any local minimum is global one. So, this topic of research got the attention of many researchers of different areas because of their enormous applications in optimization theory. As far as convex stochastic processes are concerned, there is a lot of work in the last few decades. In [23], the authors investigated the gradient descent optimality for strongly convex stochastic optimization. A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solve a convex stochastic optimization problem in terms of terminal wealth with budget constraints, see, e.g., [24]. For more details related to this work, see [25–31].


## 3. Preliminaries

LetΩ,P be a probability space. A function ξ:Ω⟶ℝ is a random variable if it is -measurable. A function ξ:I×Ω⟶ℝ, where I⊆ℝ is an interval, is a stochastic process if for every t∈I, the function ξt is a random variable.The stochastic processξ:I×Ω⟶ℝ is known as(1)
Stochastically continuous inI if(2)μ−limv⟶v°ξv=ξv°,for allv°∈I, where μ−lim represents the limit in the probability.(2)
Mean-square continuous inI if(3)limv⟶v°ξv−ξv°2=0,for all v°∈I, where ξv represents an expectation value of random variable ξv.Clearly in probability, mean-square continuity implies continuity, but converse is not true.A stochastic processξ is known as mean-square differentiable in I if there is a stochastic process ξ′ (derivative of ξ) such that(4)limv⟶v°ξv−ξv°v−v°−ξ′v°2=0,for every v°∈I.Now, we would like to recall the concept of the mean-square integral. For the definition and basic properties, see [32, 33].Assume a stochastic processξ:I×Ω⟶ℝ with ξt2<∞. A random variable ζ:Ω⟶ℝ is said to be the mean-square integral of the process ξ on a,b if for every normal sequence of partitions of a1,a2, a1=v°<v1<…<vn=a2, and for all Θk∈vk−1,vk, we have(5)limn⟶∞∑k=1nξΘkvk−vk−1−ζ.2=0.Then, we write(6)ζ.=∫abξsdsa.e.Now, we shall present some definitions and generalizations of the convex stochastic process.Definition 1.
(see [34]). A stochastic process ξ:I×Ω⟶ℝ is known as generalized convex with respect to a bifunction η:ξI×ξI⟶ℝ if(7)ξαu+1−αv≤ξv+αηξu,ξva.e,for all u,v∈I and α∈0,1.Definition 2.
(see [35]). Let h:0,1⟶ℝ be a function which is nonnegative and h≠0. A stochastic process ξ:I×Ω⟶ℝ is said to be h-convex if(8)ξαu+1−αv≤hαξu+h1−αξva.e,for every u,v∈I and α∈0,1.Definition 3.
(see [15]). The function η:ξI×ξI⟶ℝ is said to be(1)
Nonnegatively homogeneous ifηγu,γv=γηu,v for all u,v∈ℝ and γ≥0(2)
Additive ifηu1,v1+ηu2,v2=ηu1+u2,v1+v2 for all u1,u2,v1,v2∈ℝDefinition 4.
(see [14]). A function h:J⟶ℝ, J⊆ℝ, is said to be a supermultiplicative function if(9)huv≥huhv,for all u,v∈J.
Now, we are ready to give our main definition.Definition 5.
(generalizedh-convex stochastic process). h and η are fixed like above. We say that a stochastic process ξ:I×Ω⟶ is called generalized h-convex if(10)ξαu+1−αv≤ξv+hαηξu,ξva.e,for all u,v∈I and α∈0,1.
In (10), if we take ηu,v=u−v and hα=α, we obtain the convex stochastic process. We observe that, by taking u=v in (10), we get(11)hαηξu,ξu≥0,for any u∈I and α∈0,1, which implies that(12)ηξu,ξu≥0,for any u∈I. Also, if we take α=1 and h1=1 in (10), we get(13)ξu−ξv≤ηξu,ξv,for any u,v∈I. (13) obviously implies (12).
We observe that ifξ:I⟶ℝ is a convex stochastic process and η:ξI×ξI⟶ℝ is an arbitrary bifunction that satisfies(14)ηa,b≥a−b,for any a,b∈I and hα≥α, then for any u,v∈I and α∈0,1, we have(15)ξαu+1−αv≤ξv+αξu−ξv≤ξv+hαηξu,ξv,showing that ξ is a generalized h-convex stochastic process.Example 1.
Every generalizedh-convex function gives an example of a generalized h-convex stochastic process.Example 2.
Letξ:I×Ω⟶ℝ be a convex stochastic process. For every k≤1, consider the function(16)hk:0,1⟶ℝ,x↦xk.
Note thathkα≥α for all α∈0,1. Also, take ηx,y=x−y. Moreover, for every u,v∈I and α∈0,1, the following inequality is satisfied:(17)ξαu+1−αv≤αξu+1−αξv≤ξv+hkαηξu,ξva.e.
Then,ξ is a generalized hk-convex stochastic process.
The paper is organized as follows: in Section4, we will derive some basic results for the generalized h-convex stochastic process. In Section 5, Jensen-type inequality will be proved, whereas in Section 6, Hermite–Hadamard and Fejér-type inequalities will be established. Finally, in Section 7, we will derive Ostrowski-type inequality for generalized h-convex stochastic processes.


## 4. Basic Results

Proposition 1.
Consider two generalizedh-convex stochastic processes ξ1,ξ2:I×Ω⟶ℝ such that(1)
Ifη is additive, then ξ1+ξ2:I⟶ℝ is ηh-convex stochastic process(2)
Ifη is nonnegatively homogeneous, then for any γ≥0, γξ1:I×Ω⟶ℝ is a generalized h-convex stochastic process(3)
Ifξ2 is linear, then ξ1°ξ2 is a generalized h-convex stochastic processProof.
The proof of the proposition is straightforward.Proposition 2.
Ifξ:a,b⟶ℝ is a generalized h-convex stochastic process and hα≤k, then we have almost everywhere(18)maxu∈a,bξu≤maxξb,ξb+kηξa,ξb.Proof.
For anyu∈a,b, we have u=αa+1−αb, for some α∈0,1, and(19)ξu=ξαa+1−αb.since ξ is a generalized h-convex stochastic process, so(20)ξu≤ξb+hαηξa,ξb≤maxξb,ξb+kηξa,ξb.
Sinceu is arbitrary,(21)maxu∈a,bξu≤maxξb,ξb+kηξa,ξb.


## 5. Jensen-Type Inequality

We will use the next lemma to derive the Jensen-type inequality for the generalizedh-convex stochastic process.Lemma 1.
Assume a generalizedh-convex stochastic process ξ:I×Ω⟶ℝ. For u1,u2∈I and α1+α2=1, we have(22)ξα1u1+α2u2≤ξu2+hα1ηξu2,ξu2a.e.
Also, whenn>2 for u1,u2,…,un∈I, ∑i=1nαi=1, and Ti=∑j=1αj, the following inequality holds almost everywhere:(23)ξ∑i=1nαiui=ξTn−1∑i=1n−1αiTn−1ui+αnun≤ξun+hTn−1ηξ∑i=1n−1αiTn−1ui,ξun.Theorem 1.
(Jensen-type inequality). Assume a generalizedh-convex stochastic process ξ:I×Ω⟶ℝ, and let η:A×B⟶ℝ be nondecreasing, nonnegatively sublinear in the first variable. If Ti=∑j=1iαj for i=1,2,…,n such that Tn=1, then we have almost everywhere(24)ξ∑i=1nαiui≤ξun+∑i=1n−1hTiηξui,ui+1,…,un,where(25)ηξui,ui+1,…,un=ηηξui,ui+1,…,un−1,ξunandηξu=ξu,for all u∈I.Proof.
Sinceη is nondecreasing, nonnegatively sublinear in the first variable, from Lemma 1, we have(26)ξ∑i=1nαiui≤ξun+hTn−1ηξ∑i=1n−1αiTn−1ui,ξun=ξun+hTn−1ηξTn−2Tn−1∑i=1n−2αiTn−2ui+αn−1Tn−1un−1,ξun≤ξun+hTn−1ηξun−1+hTn−2Tn−1×ηξ∑i=1n−2αiTn−2ui,ξun−1,ξun≤ξun+hTn−1ηξun−1,ξun+hTn−2ηηξ∑i=1n−2αiTn−2ui,ξun−1,ξun≤…≤ξun+hTn−1ηξun−1,ξun+hTn−2ηηξun−2,ξun−1,ξun+…+hT1ηη…ηηξu1,ξu2,ξu3…,ξun−1,ξun=ξun+hTn−1ηξun−1,un+hTn−2ηξun−2,un−1,un+⋯+hT1ηξu1,u2,…,un−1,un.
Finally, small calculations yield (26).


## 6. Hermite–Hadamard and Fejér-Type Inequalities

Theorem 2.
Assume a mean-square integrable generalizedh-convex stochastic process ξ:a1,a2×Ω⟶ℝ. Then, for any a1,a2∈Ia1<a2, the following inequality holds almost everywhere:(27)ξa1+a22−h1/2a2−a1∫a1a2ηξa1+a2−u,ξudu≤1a2−a1∫a1a2ξudu≤ξa1+ξa22+12ηξa1,ξa2+ηξa2,ξa1∫01hαdα.Proof.
Letx=αa1+1−αa2 and y=1−αa1+αa2; then,(28)ξa1+a22=ξx+y2=ξ12αa1+1−αa2+121−αa1+αa2≤ξ1−αa1+αa2+h12×ηξαa1+1−αa2,ξ1−αa1+αa2.
Integrating the above inequality with respect to’α’ over 0,1,(29)ξa1+a22≤1a2−a1∫a1a2ξudu+h1/2a2−a1∫a1a2ηξa1+a2−u,ξuduξa1+a22−h1/2a2−a1∫a1a2ηξa1+a2−u,ξudu≤1a2−a1∫a1a2ξudu.
Now,(30)∫a1a2ξudu=a2−a1∫01ξαa1+1−αa2dα≤a2−a1ξa2+∫01hαηξa1,ξa2dα1a2−a1∫a1a2ξudu≤ξa2+∫01hαηξa1,ξa2dα.
Similarly,(31)1a2−a1∫a1a2ξudu≤ξa1+∫01hαηξa2,ξa1dα.
Adding (30) and (31),(32)1a2−a1∫a1a2ξudu≤ξa1+ξa22+12ηξa1,ξa2+ηξa2,ξa1∫01hαdα.
Combining (29) and (32), we get (27).Remark 1.
If we takeηx,y=x−y and hα=α, then Theorem 1 reduces to the Hermite–Hadamard inequality for stochastic convexity in [22].Definition 6.
(see [36]). A stochatic process ξ:a,b⟶ℝ is said to be symmetric with respect to a+b/2 on a,b if(33)ξx=ξa+b−x,for any a≤x≤b.Theorem 3.
Assume a generalizedh-convex stochastic process ξ1:a1,a2×Ω⟶ℝ with η bounded above on ξ1a1,a2×ξ1a1,a2 and a nonnegative function h:0,1⟶ℝ. Also, suppose that ξ2:a1,a2⟶ℝ+ is integrable and symmetric about a1+a2/2. Then,(34)∫a1a2ξ1uξ2udu≤ξ1a1+ξ1a22∫a1a2ξ2udu+ηξ1a1,ξ1a2+ηξ1a2,ξ1a12ha2−a1×∫a1a2ha2−uξ2udu.Proof.
By using the definition of the generalizedh-stochastic convexity of ξ1, change of variable, and the assumption that ξ2 is symmetric about a1+a2/2, we have(35)∫a1a2ξ1uξ2u,du≤a2−a1∫01ξ1a2+hαηξ1a1,ξ1a2×ξ2αa1+1−αa2dα=a2−a1∫01ξ1a2ξ2αa1+1−αa2dα+ηξa1,ξa2∫01hαξ2αa1+1−αa2dα,(36)∫a1a2ξ1u,ξ2u,du≤a2−a1∫01ξ1a1+hαηξ1a2,ξ1a1×ξ21−αa1+αa2dα=a2−a1∫01ξ1a1ξ2αa1+1−αa2dα+ηξ1a2,ξ1a1∫01hαξ2αa1+1−αa2dα.
Adding (35) and (36), we get(37)2∫a1a2ξ1uξ2udu≤a2−a1ξ1a1+ξ1a2×∫01ξ2αa1+1−αa2dα+a2−a1ηξ1a1,ξ1a2+ηξ1a2,ξ1a1∫01hαξ2αa1+1−αa2dα,and by changing the variable u=αa1+1−αa2, we obtain(38)∫a1a2ξ1uξ2udu≤ξ1a1+ξ1a22∫a1a2ξ2udu+ηξ1a1,ξ1a2+ηξ1a2,ξ1a12ha2−a1×∫a1a2ha2−uξ2udu.Theorem 4.
Assume a generalizedh-convex stochastic process ξ1:a1,a2×Ω⟶ℝ with η bounded above on ξ1a1,a2×ξ1a1,a2 and a nonnegative function h:0,1⟶ℝ. Also, suppose that ξ2:a1,a2⟶ℝ+ is integrable and symmetric about a1+a2/2. Then,(39)ξ1a1+a22∫a1a2ξ2u−h12∫a1a2ηξ1a1+a2−u,ξuξ2udu≤∫a1a2ξuξ2udu.Proof.
Using the definition of the generalizedh-convex stochastic process, change of variable, and the assumption that ξ2 is symmetric about a1+a2/2, we get(40)ξ1a1+a22=ξ1αa1−αa1+a1+a2+αa2−αa22=ξ1αa1+1−αa2+αa2+1−αa12≤ξ1αa2+1−αa1+h12×ηξ1αa2+1−αa1,ξ1αa2+1−αa1.
By changing the variableu=αa2+1−αa1, we obtain(41)ξ1a1+a22∫a1a2ξ2udu=ξ1a1+a22∫01ξ2αa2+1−αa1a2−a1dα≤∫01ξ1αa2+1−αa1ξ2αa2+1−αa1a2−a1dα+h12∫01ηξ1αa1+1−αa2,ξ1αa2+1−αa1×ξ2αa2+1−αa1a2−a1dα=∫a1a2ξ1uξ2udu+h12×∫a1a2ηξ1a1+a2−u,ξ1uξ2udu.Corollary 1.
By settingηx,y=x−y and hα=α in (34 and 39) and then combining, we obtain the classical Hermite–Hadamard–Fejér-type inequality as(42)ξ1a1+a22∫a1a2ξ2udu≤∫a1a2ξ1uξ2udu≤ξ1a1+ξ1a22∫a1a2ξ2udu.


## 7. Ostrowski-Type Inequality

In order to prove Ostrowski-type inequality for the generalizedh-convex stochastic process, the following lemma is required.Lemma 2 (see [37]).
Letξ:I×Ω⟶ℝ be a stochastic process which is mean-square differentiable on I° and its derivative ξ′ be mean-square integrable on a1,a2, where a1, a2∈I with a1<a2; then, the following equality holds:(43)ξu−1a2−a1∫a1a2ξvdv=u−a12a2−a1∫01αξ′αu+1−αa1dα−a2−u2a2−a1∫01αξ′αu+1−αb1dαa.e.,for each u∈a,b.Theorem 5.
Assume a mean-square stochastic processξ:I×Ω⟶ℝ such that ξ′ (the derivative of ξ) is mean-square integrable on a1,a2, where a1, a2∈I with a1<a2, and consider a nonnegative function h:0,1⟶ℝ which is supermultiplicative such that, for every α, hα>α. If ξ′ is a generalized h-convex stochastic process on I and ξ′u≤M for every u, then(44)ξu−1a2−a1∫a1a2ξvdv≤Mu−a12+a2−u2a2−a1∫01hydy+u−a12a2−a1ηξ′u,ξ′a1+a2−u2a2−a1ηξ′u,ξ′a2∫01h2ydy.Proof.
By using Lemma2 and the definition of generalized h-stochastic convexity of ξ′, we get(45)ξu−1a2−a1∫a1a2ξvdv≤u−a12a2−a1∫01yξ′yu+1−ya1dy+a2−u2a2−a1∫01yξ′yu+1−ya2dy≤u−a12a2−a1∫01yξ′a1+hyηξ′u,ξ′a1dy+a2−u2a2−a1∫01yξ′a2+hyηξ′u,ξ′a2dy≤Mu−a12a2−a1∫01hydy+u−a12a2−a1∫01h2yηξ′u,ξ′a1dy+Ma2−u2a2−a1∫01hydy+a2−u2a2−a1∫01h2yηξ′u,ξ′a2dy≤Mu−a12+a2−u2a2−a1∫01hydy+u−a12a2−a1ηξ′u,ξ′a1×∫01h2ydy+a2−u2a2−a1ηξ′u,ξ′a2∫01h2ydy=Mu−a12+a2−u2a2−a1∫01hydy+u−a12a2−a1ηξ′u,ξ′a1+a2−u2a2−a1ηξ′u,ξ′a2∫01h2ydy.


## 8. Conclusion

Stochastic processes have many applications in statistics, which obviously lead to lots of other domains, for example, Kolmogorov–Smirnoff test on the equality of distributions [38–40] (the test statistic is derived from a Brownian bridge, which is a Brownian motion conditioned to have certain values at the endpoints of an interval of time). The other applications include sequential analysis [41, 42] (this is the rigorous way you can stop an A/B test dynamically. Stopping rules are obtained by approximating discrete problems with their continuous time analogs, in which the sufficient statistic process follows a stochastic differential equation) and quickest detection [43, 44] (my stock story is that you are using a gold mine which gives you a random amount of gold per day, but at some point, it will get depleted, and you want to know when this happens as quickly as possible. Decision rules are again based on properties of hitting times of random processes). In this paper, we have introduced generalized h-conevex stochastic processes and proved Jensen, Hermite–Hadamard, and Fejér-type inequalities. Our results are applicable because applying the convex function to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable.



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