#### 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 generalized -convex stochastic processes is introduced, and some basic properties concerning generalized -convex stochastic processes are developed. Furthermore, we establish Jensen and Hermite–Hadamard and Fejér-type inequalities for this generalization.

#### 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]. -Convex [14] and -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 which is mean-square continuous in , where is an interval. Then, for every , ,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 generalized -convex stochastic process and to extend the classical Hermite–Hadamard inequality for convex stochastic processes to generalized -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

Let be a probability space. A function is a random variable if it is -measurable. A function , where is an interval, is a stochastic process if for every , the function is a random variable.

The stochastic process is known as(1)Stochastically continuous in if for all , where represents the limit in the probability.(2)Mean-square continuous in iffor all , where represents an expectation value of random variable .

Clearly in probability, mean-square continuity implies continuity, but converse is not true.

A stochastic process is known as mean-square differentiable in if there is a stochastic process (derivative of ) such thatfor every .

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 process with . A random variable is said to be the mean-square integral of the process on if for every normal sequence of partitions of , , and for all , we have

Then, we write

Now, we shall present some definitions and generalizations of the convex stochastic process.

*Definition 1. *(see [34]). A stochastic process is known as generalized convex with respect to a bifunction iffor all and .

*Definition 2. *(see [35]). Let be a function which is nonnegative and . A stochastic process is said to be -convex iffor every and .

*Definition 3. *(see [15]). The function is said to be(1)Nonnegatively homogeneous if for all and (2)Additive if for all

*Definition 4. *(see [14]). A function , , is said to be a supermultiplicative function iffor all .

Now, we are ready to give our main definition.

*Definition 5. *(generalized -convex stochastic process). and are fixed like above. We say that a stochastic process is called generalized -convex iffor all and .

In (10), if we take and , we obtain the convex stochastic process. We observe that, by taking in (10), we getfor any and , which implies thatfor any . Also, if we take and in (10), we getfor any . (13) obviously implies (12).

We observe that if is a convex stochastic process and is an arbitrary bifunction that satisfiesfor any and , then for any and , we haveshowing that is a generalized -convex stochastic process.

*Example 1. *Every generalized -convex function gives an example of a generalized -convex stochastic process.

*Example 2. *Let be a convex stochastic process. For every , consider the functionNote that for all . Also, take . Moreover, for every and , the following inequality is satisfied:Then, is a generalized -convex stochastic process.

The paper is organized as follows: in Section 4, we will derive some basic results for the generalized -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 -convex stochastic processes.

#### 4. Basic Results

Proposition 1. *Consider two generalized -convex stochastic processes such that*(1)*If is additive, then is -convex stochastic process*(2)*If is nonnegatively homogeneous, then for any , is a generalized -convex stochastic process*(3)*If is linear, then is a generalized -convex stochastic process*

*Proof. *The proof of the proposition is straightforward.

Proposition 2. *If is a generalized -convex stochastic process and , then we have almost everywhere*

*Proof. *For any , we have , for some , andsince is a generalized -convex stochastic process, soSince is arbitrary,

#### 5. Jensen-Type Inequality

We will use the next lemma to derive the Jensen-type inequality for the generalized -convex stochastic process.

Lemma 1. *Assume a generalized -convex stochastic process . For and , we have**Also, when for , , and , the following inequality holds almost everywhere:*

Theorem 1. *(Jensen-type inequality). Assume a generalized -convex stochastic process , and let be nondecreasing, nonnegatively sublinear in the first variable. If for such that , then we have almost everywherewherefor all .*

*Proof. *Since is nondecreasing, nonnegatively sublinear in the first variable, from Lemma 1, we haveFinally, small calculations yield (26).

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

Theorem 2. *Assume a mean-square integrable generalized -convex stochastic process . Then, for any , the following inequality holds almost everywhere:*

*Proof. *Let and ; then,Integrating the above inequality with respect to’’ over ,Now,Similarly,Adding (30) and (31),Combining (29) and (32), we get (27).

*Remark 1. *If we take and , then Theorem 1 reduces to the Hermite–Hadamard inequality for stochastic convexity in [22].

*Definition 6. *(see [36]). A stochatic process is said to be symmetric with respect to on iffor any .

Theorem 3. *Assume a generalized -convex stochastic process with bounded above on and a nonnegative function . Also, suppose that is integrable and symmetric about . Then,*

*Proof. *By using the definition of the generalized -stochastic convexity of , change of variable, and the assumption that is symmetric about , we haveAdding (35) and (36), we getand by changing the variable , we obtain

Theorem 4. *Assume a generalized -convex stochastic process with bounded above on and a nonnegative function . Also, suppose that is integrable and symmetric about . Then,*

*Proof. *Using the definition of the generalized -convex stochastic process, change of variable, and the assumption that is symmetric about , we getBy changing the variable , we obtain

Corollary 1. *By setting and in (34 and 39) and then combining, we obtain the classical Hermite–Hadamard–Fejér-type inequality as*

#### 7. Ostrowski-Type Inequality

In order to prove Ostrowski-type inequality for the generalized -convex stochastic process, the following lemma is required.

Lemma 2 (see [37]). *Let be a stochastic process which is mean-square differentiable on and its derivative be mean-square integrable on , where , with ; then, the following equality holds:for each .*

Theorem 5. *Assume a mean-square stochastic process such that (the derivative of ) is mean-square integrable on , where , with , and consider a nonnegative function which is supermultiplicative such that, for every , . If is a generalized -convex stochastic process on and for every , then*

*Proof. *By using Lemma 2 and the definition of generalized -stochastic convexity of , we get

#### 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.

#### Data Availability

All data required for this research are included within this paper.

#### Conflicts of Interest

The authors do not have any conflicts of interest.

#### Authors’ Contributions

All authors contributed equally to this paper.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).