Markov process

Markov process are an important class of the stochastic processes. The Markov property means that evolution of the Markov process in the future depends only on the present state and does not depend on past history. The Markov process does not remember the past if the present state is given. Hence, the Markov process is called the process with memoryless property.

Basic types of Markov process

Markov processes are classified according to the nature of the time parameter and the nature of the state space. With respect to state space, a Markov process can be either a discrete-state Markov process or a continuous-state Markov process. A discrete state Markov process is called a Markov chain. Similarly, with respect to time, a Markov process can be either a discrete-time. Markov process or a continuous-time Markov process. Thus, there are four basic types of Markov processes:

1. Discrete-time Markov chain
2. Continuous-time Markov chain
3. Discrete-time Markov process
4. Continuous Markov process

Structure of Markov processes

A jump process is a stochastic process that makes transitions between discrete states at times that can be fixed or random. In such a process, the system enters a state, spends an amount of time called the holding time (or sojourn time), and then jumps to another state where it spends another holding time, and so on. If the jump times are t= 0<t1<t2<..., then the sample path of the process inconstant between t1 and t1+1. If the jump times are discrete, the jump process is called a jump chain.

There are two types of jump processes:

1. Pure (or nonexplosive)
2. Explosive

If the holding times of a continuous-time jump process are exponentially distributed, the process is called a Markov jump process. A Markov jump process is a continuous-time Markov chain if the holding time depends only on the current state. If the holding times of a discrete-time jump process are geometrically distributed, the process is called a Markov jump chain. However, not all discrete-time Markov chains are Markov jump chains. For many discrete-time Markov chains, transactions occur in equally spaced intervals, such as every day, every week, and every year .

Examples of Markov process

• Markov Chains: A Markov chain is a stochastic process in which the future state of the system depends only on the present state and not on the past history. It is often used to model systems such as queues, computer networks, and biological systems. A real-life example of a Markov chain is the random movement of a particle in a gas, where the probability of the particle moving from one state to another depends only on the current state, and not on the previous movements of the particle.
• Hidden Markov Models: Hidden Markov models are probabilistic models used to describe sequence data. They are used to model the underlying process that generates the observed data. A real-life example of a hidden Markov model is the stock market, where the underlying states of the market are unknown and the observed data is the price of a stock at a given time.
• Continuous Time Markov Chain: A continuous time Markov chain is a type of Markov process where the state of the system evolves over continuous time. A real-life example of a continuous time Markov chain is the movement of a particle in a fluid, where the velocity of the particle is constantly changing and depends only on the current state of the system.

Advantages of Markov process

One of the advantages of Markov process is that it is simple to understand and use.

• It can be used to model complex systems that cannot be described using deterministic approaches.
• It allows one to predict the probability of future states given the present state.
• It can be used to model random events such as stock market fluctuations and coin tosses.
• It can be used to model the behavior of complex systems such as chemical reactions and biological processes.
• It can be used to make decisions in uncertain environments.
• It can be used to optimize processes such as scheduling and routing.
• It is computationally efficient and can be used to solve large problems in a relatively short amount of time.
• It can be used to model the behavior of non-Markovian systems.

Limitations of Markov process

Markov process has several limitations:

• It only works with discrete time intervals. The Markov process assumes that the system moves from one state to another in a fixed but unknown amount of time. This means that it cannot be used to model continuous processes.
• Markov process also assumes that transition from one state to another is independent of its previous states. It does not take into account the effects of past events on the transition rates.
• The Markov process also assumes that transition rates are constant over time. This is not always true in real-world applications, where transition rates can change due to external factors.
• Markov process also assumes that all the states are equally likely to occur. This is not necessarily true in real-world applications, where some states may be more likely than others.

Other approaches related to Markov process

Markov processes are related to other approaches in stochastic processes, including:

• Hidden Markov Models (HMMs): HMMs are a class of stochastic processes that are used in various fields, such as natural language processing, speech recognition, and bioinformatics. They are composed of a set of states and a set of probabilities that determine the evolution of the system from one state to another.
• Markov Decision Processes (MDPs): MDPs are a class of stochastic processes that are used to model decision-making problems. They are composed of a set of states, a set of actions, and a set of rewards that determine the evolution of the system based on the action taken.
• Markov Chain Monte Carlo (MCMC): MCMC is a class of stochastic processes used to generate samples from a probability distribution. It uses a Markov chain to sample from a probability distribution given an initial state.

In summary, Markov processes are related to several other approaches in stochastic processes, including HMMs, MDPs, and MCMC. These approaches are used in various fields to solve different types of problems.

Footnotes

Markov process — recommended articles

Markov model — Markov Analysis — Probability theory — Mollier chart — Taylor rule — Continuation Pattern — Black box model — Stochastic volatility — Types of forecasts

References

• Freidlin M.I., (2012), Processes and Differential Equations, Birkhäuser, Basel.
• Grabski F., (2014), Semi-Markov Processes, Elsevier, Amsterdam.
• Ibe O.C., (2013), Processes for Stochastic Modeling, Elsevier, Boston.
• Kijima M., (2012), Markov Processes for Stochastic Modeling, Springer, London.
• Taira K., (2010), Boundary Value Problems and Markov Processes, Springer Science & Business Media, New York.

Author: Natalia Węgrzyn