Generalization of h-Convex Stochastic Processes and Some Classical Inequalities

)e 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 h-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.


Introduction
Stochastic processes are a branch of probability theory, treating probabilistic models that evolve in time [1][2][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].
ere 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][10][11][12] for detailed survey about the importance and interesting properties of stochastic models. e 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. e 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. en, for every a 1 , a 2 ∈ I (a 1 < a 2 ), is known as the Hermite-Hadamard-type inequality for the convex stochastic process [22]. e aim of this paper is to introduce the notion of the generalized h-convex stochastic process and to extend the classical Hermite-Hadamard inequality for convex stochastic processes to generalized h-convex stochastic processes.

Novelty and Significance
e 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][26][27][28][29][30][31].

Preliminaries
Let (Ω, 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 ξ(t) is a random variable. e stochastic process ξ: I × Ω ⟶ R is known as for all v°∈ I, where μ − lim represents the limit in the probability.
for all v°∈ I, where E[ξ(v)] represents an expectation value of random variable ξ(v). Clearly in probability, mean-square continuity implies continuity, but converse is not true.
A stochastic process ξ is known as mean-square dif- for every v°∈ I. Now, we would like to recall the concept of the meansquare integral. For the definition and basic properties, see [32,33].
Assume a stochastic process ξ: I × Ω ⟶ R with E[ξ(t) 2 ] < ∞. 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 [a 1 , a 2 ], a 1 � v°< v 1 < . . . < v n � a 2 , and for all en, we write Now, we shall present some definitions and generalizations of the convex stochastic process.
Definition 3 (see [15]). e function η: Definition 4 (see [14]). A function h: J ⟶ R, J⊆R, is said to be a supermultiplicative function if for all u, v ∈ J. Now, we are ready to give our main definition.
Definition 5 (generalized h-convex stochastic process). h and η are fixed like above. We say that a stochastic process ξ: for all u, v ∈ I and α ∈ [0, 1]. In (10), if we take η(u, v) � u − v and h(α) � α, we obtain the convex stochastic process. We observe that, by taking u � v in (10), we get for any u ∈ I and α ∈ [0, 1], which implies that for any u ∈ I. Also, if we take α � 1 and h(1) � 1 in (10), we get for any u, v ∈ I. (13) obviously implies (12). We observe that if ξ: I ⟶ R is a convex stochastic process and η: ξ(I) × ξ(I) ⟶ R is an arbitrary bifunction that satisfies for any a, b ∈ I and h(α) ≥ α, then for any u, v ∈ I and α ∈ [0, 1], we have showing that ξ is a generalized h-convex stochastic process.
Example 1. Every generalized h-convex function gives an example of a generalized h-convex stochastic process.
Example 2. Let ξ: I × Ω ⟶ R be a convex stochastic process. For every k ≤ 1, consider the function Note that h k (α) ≥ α for all α ∈ (0, 1). Also, take η(x, y) � x − y. Moreover, for every u, v ∈ I and α ∈ (0, 1), the following inequality is satisfied: en, ξ is a generalized h k -convex stochastic process. e paper is organized as follows: in Section 4, 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.

Basic Results
stochastic process (2) If η is nonnegatively homogeneous, then for any c ≥ 0, Proof.

Jensen-Type Inequality
We will use the next lemma to derive the Jensen-type inequality for the generalized h-convex stochastic process.
Proof. Since η is nondecreasing, nonnegatively sublinear in the first variable, from Lemma 1, we have Finally, small calculations yield (26).
Proof. By using the definition of the generalized h-stochastic convexity of ξ 1 , change of variable, and the assumption that ξ 2 is symmetric about a 1 + a 2 /2, we have Adding (35) and (36), we get Mathematical Problems in Engineering 5 and by changing the variable u � αa 1 + (1 − α)a 2 , we obtain 1 , a 2 ]) and a nonnegative function h: (0, 1) ⟶ R. Also, suppose that ξ 2 : [a 1 , a 2 ] ⟶ R + is integrable and symmetric about a 1 + a 2 /2. en, Proof. Using the definition of the generalized h-convex stochastic process, change of variable, and the assumption that ξ 2 is symmetric about a 1 + a 2 /2, we get By changing the variable u � αa 2 + (1 − α)a 1 , we obtain □ Corollary 1. By setting η(x, y) � x − y and h(α) � α in (34 and 39) and then combining, we obtain the classical Hermite-Hadamard-Fejér-type inequality as

Ostrowski-Type Inequality
In order to prove Ostrowski-type inequality for the generalized h-convex stochastic process, the following lemma is required.
Lemma 2 (see [37]). Let ξ: I × Ω ⟶ R be a stochastic process which is mean-square differentiable on I°and its derivative ξ ′ be mean-square integrable on [a 1 , a 2 ], where a 1 , a 2 ∈ I with a 1 < a 2 ; then, the following equality holds: for each u ∈ [a, b].

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][39][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). e 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 Mathematical Problems in Engineering 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
e authors do not have any conflicts of interest.