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Introduction To Stochastic Processes By Stochastic Processes Assignment Expert

The theory of stochastic processes focuses on the study and modelling of systems that evolve, or space, according to non-specific laws deterministic, that is, of a random nature.

Our statistics assignment help experts explain that the usual way to describe the evolution of the system is by utilizing sequences or collections of random variables (V). In this way, one can study how a V evolves. For example, the number of people waiting at a bank window at an instant (t) of time; the price of a company's shares at over a year; the number of unemployed in the Hospitality sector over a year.

The first basic idea is to identify a stochastic process with a sequence of V {Xn, n ∈ N} where the subscript indicates the corresponding instant of time (or space).

This initial idea can be easily generalized, allowing the moments in which V be continuous. Thus, one can speak of a collection or family of V {X_t, t ∈ R}, which gives a more accurate idea of what a stochastic process is it had to be a V. X (s) is a function that goes from a sample space S to the real line, to each point s ∈ S of the sample space can be associated with a number of the real line.

In this way, the probability of each event in S can be translated into the probability that a value of X (Va) falls in a certain interval or set of real numbers. Yes, to all this a time dimension is added, a stochastic process is obtained.

The formal definition is as follows:

Given the probability space (Ω, a, P) such that for all fixed t ∈ T ⊂ R

Xt: (Ω, a, P) - → (R, B)

W: → Xt (w) ∈ R that is, Xt is a random variable and ∀w ∈ Ω fixed, X "¢ (w) is a function of time.

Examples:

Xt: number of people waiting for a bus at time t where t ∈ [9, 10]

Xt: the price of a share of a company on a day t of the month (t = 1, 2... 30).

Xt: number of unemployed in month t (t = 1, 2... 12).

For a stochastic process to be fully defined, Va must be completely determined, that is, determine and identify the probability distribution associated with each of them and the joint distribution of all of them.

1^st Definition

The set T ⊂ R of subscripts is called a parametric set and can be continuous or countable. In this way, a first classification appears in stochastic processes of parameter continuous or discrete parameter.

2^nd Definition

The set of states E is called the set of possible values that can take the v.a {Xt} t∈R

Classification of Stochastic Processes

They can be classified according to

- The structure of the parametric set T and the set of states E.

- The probabilistic characteristics of the v.a.

Classification according to the structure of Time and State.

Stochastic processes can be classified into four types, depending on whether Time is a countable or continuous set, and whether the state is another discrete or continuous set. So:

Examples:

(i) Chain - Suppose a hydraulic system with a safety valve that is checked daily. This valve has three possible states: correct, incorrect and deteriorated. In this way, the process can be defined as Xn ≡ State in which the find the valve on day n.

(ii) Specific Process - A school bus has capacity for k people. If in a stop k or less people are waiting, then all will get on the bus. If it's not thus, the first k will go up, leaving the rest of the people on hold. It is defined, then, Xt ≡ number of people waiting at the stop in an instant of time t. Issues such as minimizing the mean waiting time, or reducing the number of people waiting.

(iii) Succession of V.a. - An oil company must decide how much oil to extract per month to maximize profits. We define, then, Xn ≡ Amount of oil at the extract in month n. This is a particular example of the so-called Inventory Theory, where the distribution of demand, for example.

(iv) Continuous Process - The waiting time is recorded at a bank window of a customer arriving in an instant t. You can define Xt ≡ Wait time for a customer arriving at time t. This is related to the Theory of Queues. For example, it is interesting to minimize time waiting or studying the distribution of Xt.

Utilities Of Stochastic Processes

Our experts of Statistics Assignment sample have deep knowledge about the applications of stochastic processes and their usefulness in the world of work. The applications are diverse and very different from the use applied to engineering, sociology, finance, business, and medicine, to meteorology or the environment, passing through the simulations of physical or chemical processes.

Stochastic processes applied to finance, by comprising an indiscriminate component, can help us to study the behaviour of several variables. Thus allowing to estimate the probable future behaviour. Of all of them, here we are sharing one of the solved Stochastic Processes Assignment Sample Online based on Measure Theory and Stochastic Integration.

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