Stochastic Model / Process: Definition and Examples

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A stochastic process is a system of countable events, where the events occur according to some well-defined random process. Strictly speaking, a stochastic process is also concerned with the sequence in which the events occur in time, but we shall take the more usual broader definition to include counting systems where the order is of no importance.

This section describes four very fundamental stochastic processes: Clicking on any of the text of the diagram will take you directly to an explanation of that component. Having an understanding of stochastic processes is really quite critical to good risk analysis modeling, so we encourage you to put some effort into this section.

In school physics lessons we learn about different types of motion simple harmonic, rolling, sliding, parabolic flight, etc. Attached to each type of motion we learn some equations that completely describe whatever we wish to know. Then we got out into the world and see any number of examples of these types of motion. Learn to identify the types of motion in the real world, and know the associated equations, and you can describe a huge number of problems, from planetary motion, electrons around an atom, missile and rocket trajectories, to rolling a snowball down a hill.

The same applies to these stochastic processes: The diagram above illustrates how these stochastic processes are related. For example, the binomial process has three parameters: If we know what is stochastic combined with real life examples two, we can estimate the third, as depicted by the arrows joining these three components together.

Thus, for example, the Beta, Binomial and Negative Binomial distributions are describing different aspects of the same process and are therefore intimately related. More interestingly still, the Poisson process can be viewed very similarly and since the Poisson mathematics are derived from the Binomial process where n is made large and p is small, there is a strong relationship between the corresponding Poisson and Binomial distributions.

The Hypergeometric process is similarly linked. Central Limit Theorem applies to many of these distributions and explains why they look very similar to Normal distributions for certain parameter values. A very great deal of risk analysis problems can be tackled with a good knowledge of these four processes. We look at the theory and assumptions behind each process, and the distributions that are used in their modeling. This approach provides us with an excellent opportunity to become very familiar with a number of important distributions, and to see the relationships between them, even between the distributions of the different stochastic processes.

This section also discusses a generalization of the Poisson process where the times between events are independent what is stochastic combined with real life examples identically distributed with an arbitrary distribution, a type of randomness known as a renewal process which is often used in modeling equipment reliability, for example. Finally, some examples are given of mixture processes. These are random processes where one or more of the defining parameters like a binomial probability, for example may itself be a random variable.

There are some very useful theoretical results that come out of mixture processes, and in Monte Carlo simulation this is something that do we quite naturally anyway what is stochastic combined with real life examples simply nesting distributions.

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Stochastic Processes and Applications August - July Stochastic Analysis deals with models which involve uncertainties or randomness. Uncertainty, complexity and dynamism have been continuing challenges to our understanding and control of our physical environment. Everyday we encounter signals which cannot be modeled exactly by an analytic expression or in a deterministic way. Examples of such signals are ordinary speech waveforms, seismological signals, biological signals, temperature histories, communication signals etc.

In manufacturing domain no machine is totally reliable. Every machine fails at some random time. Thus in a typical manufacturing system which involves a large number of machines, the total number of machines at any time cannot be determined in a deterministic way.

In a market driven economy, the stock market is volatile, the interest rates fluctuate in a random fashion. One can give any number of examples from our daily life events where uncertainty prevails in an essential way. This gives us the realization that many real life phenomena require the analysis of a system in a probabilistic setting rather than in a deterministic setting.

Thus stochastic models are becoming increasingly important for understanding or making performance evaluation of complex systems in a broad spectrum of fields. Under the current thematic programme we first plan to carry out training and research in mathematical finance.

Mathematical finance is an interdisciplinary area. It lies at the crossroad of probability theory, partial differential equations, numerical methods, statistical analysis and economics. We plan to organize a series of lectures on Mathematical Finance throughout the year. We also plan to organize a workshop for researchers from the academic institutions and practitioners from the finance industries followed by a conference.

In this manner, we wish to create a big group in this area across the country which would grow to meet the challenges thrust on us by the changing socio-economic-political scenario in our country. Stochastic processes have played a significant role in various engineering disciplines like power systems, robotics, automotive technology, signal processing, manufacturing systems, semiconductor manufacturing, communication networks, wireless networks etc.

This will form the second part of the thematic programme. In communication networks, unpredictability and randomness arise for several reasons. For one, connections come in and leave randomly. At the time of connection establishment, circuit switched networks typically need to find sufficient resources for the connection in the absence of which, a connection is refused. For packet switched networks such as the internet, even though the above constraint is not there, however, each packet is routed individually by the switches based on current information available and can take a different path.

Also, packets can be dropped if sufficient resources are not available. In most networks in particular, packet switched , the packets that individual connections pump into the network are of varying sizes one packet may have a different size than the other in each connection and thus each packet holds the network for a varying random amount of time. Moreover, the links can fail randomly and so reliability of the medium is also an issue.

From the perspective of the user, the total time needed to transmit, say a file, should be the least possible. Whereas processing, transmission and propagation delays of packets are not significant in general, queuing delays are.

The latter depend on the size of packets and also the number of connections operational at a given time. Stochastic processes are thus crucially used in the design, analysis and control of networks. Control in networks can be broadly classified under four heads - admission control, routing, flow and congestion control, and resource allocation. Designing good control strategies requires a good knowledge of the above mentioned topics such as stochastic control in particular under partial information , parameter estimation, simulation based optimization, queuing theory, learning theory etc.

Another area where stochastic processes have important applications is in the area of neuroscience. This will comprise the third part of the thematic programme. The firing of neurons can be modeled as a first passage time problem from stochastic processes. The output spike train is a point stochastic process.

Analysis of EEG signals from the brain makes intensive use of stochastic processes, in particular, vector autoregressive processes. Among the awesome repertoire of tasks that the human brain can accomplish, one of the more fascinating ones is how the electrical activity of millions of brain cells neurons is translated into precise sequences of movements. One of the greatest challenges in applied neuroscience is to build prosthetic limbs controlled by neural signals from the brain.

The ultimate goal is to provide paralytic patients and amputees with the means to move and communicate by controlling the prosthetic device using brain activity. Scientists and engineers are slowly getting closer to building such devices thanks to studies revealing a strong connection between the activity of neurons in the brain's cerebral cortex and the movements of limbs. To realize the above goal of building prosthetic limbs, one tool which plays a critical role is the theory of stochastic processes.

As part of the thematic programme, we plan to organize seminars , compact courses , workshops and conferences, etc.