Stochastic process

Stock market fluctuations have been modeled by stochastic processes.

In probability theory, a stochastic (/stoʊˈkæstɪk/) process, or often random process, is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.

In the simple case of discrete time, as opposed to continuous time, a stochastic process is a sequence of random variables (for example, see Markov chain, also known as discrete-time Markov chain). The random variables corresponding to various times may be completely different, the only requirement being that these different random quantities all take values in the same space (the codomain of the function). One approach may be to model these random variables as random functions of one or several deterministic arguments (in most cases, the time parameter). Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical dependence.

Familiar examples of stochastic processes include stock market and exchange rate fluctuations, signals such as speech, audio and video, medical data such as a patient's EKG, EEG, blood pressure or temperature, and random movement such as Brownian motion or random walks.

A generalization, the random field, is defined by letting the variables be parametrized by members of a topological space instead of time. Examples of random fields include static images, random terrain (landscapes), wind waves or composition variations of a heterogeneous material.

Formal definition and basic properties

Definition

Given a probability space $(\Omega, \mathcal{F}, P)$ and a measurable space $(S,\Sigma)$ , an S-valued stochastic process is a collection of S-valued random variables on $\Omega$ , indexed by a totally ordered set T ("time"). That is, a stochastic process X is a collection

\{ X_t : t \in T \}

where each $X_t$ is an S-valued random variable on $\Omega$ . The space S is then called the state space of the process.

Finite-dimensional distributions

Let X be an S-valued stochastic process. For every finite sequence $T'=( t_1, \ldots, t_k ) \in T^k$ , the k-tuple $X_{T'} = (X_{t_1}, X_{t_2},\ldots, X_{t_k})$ is a random variable taking values in $S^k$ . The distribution $\mathbb{P}_{T'}(\cdot) = \mathbb{P} (X_{T'}^{-1}(\cdot))$ of this random variable is a probability measure on $S^k$ . This is called a finite-dimensional distribution of X.

Under suitable topological restrictions, a suitably "consistent" collection of finite-dimensional distributions can be used to define a stochastic process (see Kolmogorov extension in the "Construction" section).

History of stochastic processes

Stochastic processes were first studied rigorously in the late 19th century to aid in understanding financial markets and Brownian motion. The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. Albert Einstein (in one of his 1905 papers) and Marian Smoluchowski (1906) brought the solution of the problem to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1908.

An excerpt from Einstein's paper describes the fundamentals of a stochastic model:

"It must clearly be assumed that each individual particle executes a motion which is independent of the motions of all other particles; it will also be considered that the movements of one and the same particle in different time intervals are independent processes, as long as these time intervals are not chosen too small.

We introduce a time interval $\tau$ into consideration, which is very small compared to the observable time intervals, but nevertheless so large that in two successive time intervals $\tau$ , the motions executed by the particle can be thought of as events which are independent of each other".

Construction

In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension.^[1]

Kolmogorov extension

The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions $f: X \to Y$ exists, then it can be used to specify the joint probability distribution of finite-dimensional random variables $f(x_1),\dots,f(x_n)$ . Now, from this n-dimensional probability distribution we can deduce an (n − 1)-dimensional marginal probability distribution for $f(x_1),\dots,f(x_{n-1})$ . Note that the obvious compatibility condition, namely, that this marginal probability distribution be in the same class as the one derived from the full-blown stochastic process, is not a requirement. Such a condition only holds, for example, if the stochastic process is a Wiener process (in which case the marginals are all gaussian distributions of the exponential class) but not in general for all stochastic processes. When this condition is expressed in terms of probability densities, the result is called the Chapman–Kolmogorov equation.

The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman–Kolmogorov compatibility condition.

Separability, or what the Kolmogorov extension does not provide

Recall that in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer.

The Kolmogorov extension starts by declaring to be measurable all sets of functions where finitely many coordinates $[f(x_1), \dots , f(x_n)]$ are restricted to lie in measurable subsets of $Y_n$ . In other words, if a yes/no question about f can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer.

In measure theory, if we have a countably infinite collection of measurable sets, then the union and intersection of all of them is a measurable set. For our purposes, this means that yes/no questions that depend on countably many coordinates have a probabilistic answer.

The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions. Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence. The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer. One might hope that the questions that depend on uncountably many values of a function be of little interest, but the really bad news is that virtually all concepts of calculus are of this sort. For example:

all require knowledge of uncountably many values of the function.

One solution to this problem is to require that the stochastic process be separable. In other words, that there be some countable set of coordinates $\{f(x_i)\}$ whose values determine the whole random function f.

The Kolmogorov continuity theorem guarantees that processes that satisfy certain constraints on the moments of their increments have continuous modifications and are therefore separable.

Filtrations

Given a probability space $(\Omega,\mathcal{F},P)$ , a filtration is a weakly increasing collection of sigma-algebras on $\Omega$ , $\{\mathcal{F}_t, t\in T\}$ , indexed by some totally ordered set $T$ , and bounded above by $\mathcal{F}$ , i.e. for s,t $\in T$ with s < t,

\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}

A stochastic process $X$ on the same time set $T$ is said to be adapted to the filtration if, for every t $\in T$ , $X_t$ is $\mathcal{F}_t$ -measurable.^[2]

Natural filtration

Given a stochastic process $X = \{X_t : t\in T\}$ , the natural filtration for (or induced by) this process is the filtration where $\mathcal{F}_t$ is generated by all values of $X_s$ up to time s = t, i.e. $\mathcal{F}_t = \sigma(\{X_s^{-1}(A) : s\leq t,A \in \Sigma\})$ .

A stochastic process is always adapted to its natural filtration.

Classification

Stochastic processes can be classified according to the cardinality of its index set (usually interpreted as time) and state space.

Discrete time and discrete state space

If both $t$ and $X_t$ belong to $\mathbb{N}$ , the set of natural numbers, then we have models which lead to Markov chains. For example:

(a) If $X_t$ means the bit (0 or 1) in position $t$ of a sequence of transmitted bits, then $X_t$ can be modelled as a Markov chain with two states. This leads to the error-correcting Viterbi algorithm in data transmission.

(b) If $X_t$ represents the combined genotype of a breeding couple in the $t$ ^th generation in an inbreeding model, it can be shown that the proportion of heterozygous individuals in the population approaches zero as $t$ goes to ∞.^[3]

Continuous time and continuous state space

The paradigm of continuous stochastic process is that of the Wiener process. In its original form the problem was concerned with a particle floating on a liquid surface, receiving "kicks" from the molecules of the liquid. The particle is then viewed as being subject to a random force which, since the molecules are very small and very close together, is treated as being continuous and since the particle is constrained to the surface of the liquid by surface tension, is at each point in time a vector parallel to the surface. Thus, the random force is described by a two-component stochastic process; two real-valued random variables are associated to each point in the index set, time, (note that since the liquid is viewed as being homogeneous the force is independent of the spatial coordinates) with the domain of the two random variables being R, giving the x and y components of the force. A treatment of Brownian motion generally also includes the effect of viscosity, resulting in an equation of motion known as the Langevin equation.^[4]

Discrete time and continuous state space

If the index set of the process is N (the natural numbers), and the range is R (the real numbers), there are some natural questions to ask about the sample sequences of a process {X_i}_{i ∈ N}, where a sample sequence is {X_i(ω)}_{i ∈ N}.

What is the probability that each sample sequence is bounded?
What is the probability that each sample sequence is monotonic?
What is the probability that each sample sequence has a limit as the index approaches ∞?
What is the probability that the series obtained from a sample sequence from $f(i)$ converges?
What is the probability distribution of the sum?

Main applications of discrete time continuous state stochastic models include Markov chain Monte Carlo (MCMC) and the analysis of Time Series.

Continuous time and discrete state space

Similarly, if the index space I is a finite or infinite interval, we can ask about the sample paths {X_t(ω)}_{t ∈ I}

What is the probability that it is bounded/integrable...?
What is the probability that it has a limit at ∞
What is the probability distribution of the integral?

References

↑ Karlin, Samuel & Taylor, Howard M. (1998). An Introduction to Stochastic Modeling, Academic Press. ISBN 0-12-684887-4.
↑ Durrett, Rick (2010). Probability: Theory and Examples (Fourth ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-76539-8.
↑ Allen, Linda J. S., An Introduction to Stochastic Processes with Applications to Biology, 2nd Edition, Chapman and Hall, 2010, ISBN 1-4398-1882-7
↑ Gardiner, C. Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences, 3rd ed., Springer, 2004, ISBN 3540208828

Wio, S. Horacio, Deza, R. Roberto & Lopez, M. Juan (2012). An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics. World Scientific Publishing. ISBN 978-981-4374-78-1.
Papoulis, Athanasios & Pillai, S. Unnikrishna (2001). Probability, Random Variables and Stochastic Processes. McGraw-Hill Science/Engineering/Math. ISBN 0-07-281725-9.
Boris Tsirelson. "Lecture notes in Advanced probability theory".
Doob, J. L. (1953). Stochastic Processes. Wiley.
Klebaner, Fima C. (2011). Introduction to Stochastic Calculus With Applications. Imperial College Press. ISBN 1-84816-831-4.
Bruce Hajek (July 2006). "An Exploration of Random Processes for Engineers".
"An 8 foot tall Probability Machine (named Sir Francis) comparing stock market returns to the randomness of the beans dropping through the quincunx pattern". Index Funds Advisors.
"Popular Stochastic Processes used in Quantitative Finance". sitmo.com.
"Interactive Web Application: Stochastic Processes used in Quantitative Finance". TuringFinance.com.
"Addressing Risk and Uncertainty".

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

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