Stochastic, with the original meaning of random or chance, or random process in probability theory and related realms refers to a mathematical object that is frequently defined as a sequence of random variables within a probability space in which the index of the sequence is endowed with the interpretation of time. Stochastic processes, in other words, are known to be a collection of random variables indexed by mathematical set, which means that the stochastic process’ each random variable is associated with an element in the set in a unique manner [Parzen, E. (2015)]. An index set in this regard is the set which is used to index the random variables, and each random variable within the collection takes the values from the state space which is the same mathematical space which can be real lines, dimensional Euclidean space or integers. In this process, when a stochastic process changes between two index values, this is called an increment which can be interpreted as two points in time. It should also be noted that stochastic processes can have many outcomes owing to their randomness, with a single outcome of a stochastic process called a sample function or realization.
The basic idea concerned with process modeling is to build a model related to a process that starts from a set of sequences of events, typically generated by the process itself. Subsequently, it is also possible to use the model in order to discover properties of the process or more importantly, to predict future events based on the past history [Carfora, M. F. (2019)]. Generally speaking, a model can be employed for the purposes of describing the details of a process, classification or predicting its outcome. For example, for predicting a single variable k, the values are taken in a finite unordered set, provided with some input data 
. In the deterministic model, prediction of outcomes is with certainty besides a set of equations describing the system inputs and outputs. Deterministic models predict a single outcome from a given set of circumstances. In contrast, a situation where uncertainty is present is represented by a stochastic model, which means it is a model for a process with some sort of randomness.
As a sequence of events, a stochastic process refers to a sequence of events where the outcome at any stage is dependent on some probabilities. Stochastic models, therefore, are used for predicting possible outcomes that are weighted by their likelihoods, namely their probabilities. As known to be any process describing the evolution in time of a random phenomenon, time plays the key role in modeling stochastic processes since a stochastic model is a means to predict probability distributions of possible results by ensuring a random variation in its inputs over time. The collection of random variables is denoted as follows:
This is defined on a common probability space, taking values in a common set 
(the state space), and indexed by a set T, often either N or N or 
and considered to be discrete-time stochastic process or continuous-time stochastic process, respectively.
From a mathematical perspective, the theory of stochastic processes was settled around 1950, and since then stochastic processes have become a common means in mathematics, physics and engineering, with this theory’s application ranging from modeling of stock pricing to differential geometry [Baudoin, F. (2015)]. It is noteworthy that the state of the system cannot be precisely predicted in a stochastic process due to its current state, even with a full knowledge of all the factors affecting that process. To illustrate, detailed life history parameters of a species may be known in a population context. Yet, various unpredictable or stochastic processes are existent, like the chance nature of birth and death, namely demographic stochasticity, annual variations in climate, which is to say, environmental stochasticity. These show that accurate predicting of the precise size of a population in the future is not possible [McCarthy, M. A., & Possingham, H. P. (2006)].
As can be seen from the above examples, the probability of an event depends on various external factors. Regarding the types of stochastic processes, the Bernoulli process, as one of the simplest stochastic processes, is a sequence of independent and identically distributed random variables. Random walks, as another type of process, are typically defined as sums of random variables or random vectors in Euclidean space, which suggests that they are discrete-time processes. Simple random walk is a classical example of random walk as a stochastic process in discrete time with integers as the state space, based on Bernoulli process where each Bernoulli variable takes a positive or negative value. Finally, the Wiener process, as a stationary stochastic process, has independently distributed increments which are usually distributed being dependent on their size. The Wiener process is named after Norbert Wiener who showed its mathematical existence, also known as Brownian motion process. Numerous physical, biological and social phenomena are related to this specific stochastic process named the Brownian motion. Examples include from communication systems with noise, neuro-physical activity with disturbances, gene substitution, variations of population growth, molecular motions of particles, fluctuations in a market, to name some.
The solution or integration of differential equations require a different type of integral named the Itô stochastic integral and the ensuing methods of calculations, the Itô calculus. Stochastic process, to recall, is a part of probability theory and mathematics examining phenomena or systems, which evolve probabilistically in time, which means that they are time dependent. The Brownian motion, Poisson process and the martingales are the important aspects of stochastic calculus. The area across stochastic analysis is known to be broad. While many continuous stochastic processes are driven by Brownian motions, the basic discontinuous processes can be related to Poisson point processes. The Itô calculus is named after Kiyosi Itô, who introduced the calculus in 1942. While Itô investigated the topics of stochastic integrals, Itô’s formula, and stochastic differential equations in terms of Brownian motions. Currently, these topics are also investigated in the extended framework of martingales. Applications of this calculus is a construction of diffusion processes by stochastic differential equations and stochastic differential models. Another historical point is the Malliavin calculus, introduced by Paul Malliavin in the second half of 1970s and developed by many researchers. He named it a stochastic calculus of variation originally, and it is differential calculation on a path space. These developments opened a way for probabilistic approach to transition densities of diffusion processes that are fundamental objects in theory, having applications in many fields [Matsumoto, H., & Taniguchi, S. (2016)], [Kunita, H. (2010)] The fundamental work of Kiyosi Itô on stochastic analysis in 1942, subsequently realized an idea of Paul Lévy to describe the structure of Lévy processes, namely cadlag processes with independent increments and differential processes, currently named as Levy-Itô's canonical form of Lévy processes. These are remarkable developments in the history of stochastic analysis, which paved the way for further studies on various distributions of probability models through analytical tools like Fourier analysis, Laplace transform, differential equations and generation of function methods [Ikeda, N. et al. (2012)]. Consequently, Itô’s stochastic calculus is acknowledged to be one of the most important methods in modern probability theory for analyzing various probability models. In the recent applications, this theory enables the employment of a basic tool in the theory so that a sample functions analysis for a class of stochastic processes (semi-martingales) or generalized Itô processes. One important class of stochastic processes is Markov processes and Markov chains. A Markov process fulfills the Markov property, which is to say memoryless, which suggests it does not have any memory, while the distribution of the next state or observation exclusively depends on the current state [Karaca, Y., et al. (2022)].
Stochastic analysis as based on Itô's calculus is used to cope with questions that arise in probability theory where processes are modelled by motion along paths which typically are not differentiable. Stochastic analysis, as a basic tool in modern probability theory is used in many applied areas with stochastic processes employed as mathematical models of systems and phenomena showing variations in a random manner. Some relevant examples are electrical current fluctuating or growth of a bacterial population. The applications of stochastic processes are many in different disciplines such as neuroscience, physics, biology, chemistry, ecology, image / signal processing, information theory, computer science, and so forth.
In line with the specific stochastic processes, modeling, properties, activities, motions and fluctuations, the relevant topical areas can be put forth as follows:
- Stochastic differential equations (SDEs)
- Stochastic control
- Fractional stochastic differential equations
- The Wiener integral
- Integration and signed measures
- Forward-backward stochastic differential equations (FBSDEs)
- Finite variation processes.
- Local martingales and semimartingales
- Continuous-parameter martingale theory
- Quadratic variation and covariation
- Itô isometry
- Lévy's characterization of Brownian motion
- Solution maps
- Local solutions and Dirichlet-Poisson problem
- Cauchy problem
- Markov properties and processes
- Hidden Markov Modeling (HMM)
- Data-driven modeling of nonlinear and stochastic dynamic systems
- Identification and prediction of nonlinear and stochastic systems by data-driven methodologies
- Inverse problems with data-driven nonlinear stochastic dynamic systems
- Quantitative analysis of stochasticity and complexity
- Stochastic control
- Brownian motion
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