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Stochastic Processes

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The word ``stochastic process'' is derived from the Greek noun ``stokhos'' which means ``aim''. Another related Greek word ``stokhastikos'', ``the dart game'', provides an alternative image for randomness or chance. Although the concept of Probability is often associated with dice games, the dart game seems to be more adapted to the modern approach to both Probability Theory and Stochastic Processes. Indeed, the fundamental difference between a dice game and darts is that while in the first, one cannot control the issue of the game, in the dart game, one tries to attain an objective with different degrees of success, thus, the player increases his knowledge of the game at each trial. As a result, time is crucial in the dart game, the longer you play, the better you increase your skills.

1. Definition of a Stochastic Process

The mathematical definition of a stochastic process, in the Kolmogorov model of Probability Theory, is given as follows. Let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space, that is, $\Omega$ is a non empty set called sample space, $\mathcal{F}$ is a sigma field of subsets of $\Omega$ , which represents the family of events, and $\mathbb{P}$ is a probability measure defined on $\mathcal{F}$ .

is another non empty set, and $(E,\mathcal{E})$ a measurable space to represent all possible states. Then, a stochastic process with states in is a map $X: T\times\Omega\to E$ such that for all $t\in T$ , $\omega\mapsto X(t,\omega)$ is a measurable function. In other words, a primary interpretation of a stochastic process

is as a collection of random variables, and as such, notations like $({X}_{t})_{t\in T}$ are used to refer to

, that is $X_{t}(\omega)=X(t,\omega)$ , for all $(t,\omega)\in T\times\Omega$ . If

is an ordered number set, (eg. ${\mathbb{N}}^{}$ , $\mathbb{Z}^{}$ , ${\mathbb{R}}^{+}$ , ${\mathbb{R}}^{}$ ), it is often referred as the set of time variables and taken as a subset of integers or real numbers.For each $\omega\in\Omega$ , the map $X(\cdot,\omega):t\mapsto X(t,\omega)$ is called the trajectory of the process. Thus, each trajectory is an element of

, the set of all

-valued functions defined on

. Particularly, if

is a countable set, the process is said to be indexed by discrete times (the expression Time Series is also in use in this case). Discrete time stochastic processes were the first studied in Probability Theory under the name of chains (see Markov Chains). Examples

Consider a sequence $(\xi_{n})_{n\geq 1}$ of real random variables. According to the definition, this is a stochastic process. New stochastic processes can be defined on this basis. For instance, take $(S_{n})_{n\geq 1}$ , defined as, $S_{n}=\xi_{1}+\ldots+\xi_{n}$ , for each $n\geq 1$ .
Suppose now that the random variables $(\xi_{n})_{n\geq 1}$ are independent and identically distributed on $\{-1,1\}$ with $\mathbb{P}(\xi=\pm 1)=1/2$ . Then, $(S_{n})_{n\geq 1}$ becomes a Simple Symmetric Random Walk.
Consider a real function $x:[0,\infty[\to \mathbb{R}$ , this is also a stochastic process. It suffices to consider any probability space $(\Omega,\mathcal{F},\mathbb{P})$ and define $X(\omega,t)=x(t)$ , for all $\omega\in\Omega$ , $t\geq 0$ . This is a trivial stochastic process.
Consider an initial value problem given by

$\displaystyle x^\prime=f(t,x);\;x(0)=x,$ (1)

where is a continuous function on the two variables . Newtonian Mechanics can be written within this framework, which is usually referred as a mathematical model for a closed dynamical system in Physics. That is, the system has no interaction with the environment, and time is reversible. Now define $\Omega$ as the set of all continuous functions from $[0,\infty[$ into $\mathbb{R}$ . Endow $\Omega$ with the topology of uniform convergence on compact subsets of the positive real line and call $\mathcal{F}$ the corresponding Borel $\sigma$ -field. Thus, any $\omega\in\Omega$ is a function $\omega=(\omega (t);t\geq 0)$ . Define the stochastic process $X(\omega,t)=\omega (t)$ , known as the canonical process. The initial value problem is then written as

$\displaystyle X(\omega,t)=x+\int_{0}^tf(s,X(\omega,s))ds.$ (2)

This can be phrased as an example of a Stochastic Differential Equation, without noise term. The solution is a deterministic process which provides a description of the given closed dynamical system. Apparently, there is no great novelty and one can wonder whether the introduction of $\Omega$ is useful. However, this framework includes processes describing open dynamical systems too, embracing the interaction of the main system with the environment, and that is an important merit of the stochastic approach. Typically, the interaction of a given system with the environment is described through the action of so-called noises interfering with the main dynamics. Let us complete our example adding a noise term to the closed dynamics.
To consider the action of a noise, take a sequence $(\xi_{n})_{n\geq 1}$ of random variables defined on $\Omega$ , such that $\xi_{n}(\omega)\in\{-1,1\}$ . Let be given a probability $\mathbb{P}$ on the measurable space $(\Omega, \mathcal{F})$ such that $\mathbb{P}(\xi_{n}=\pm 1)=1/2$ . Call $S_{n}=\xi_{1}+\ldots+\xi_{n}$ and denote the greatest integer $\leq t$ . The equation

$\displaystyle X(\omega,t)=x+\int_{0}^tf(s,X(\omega,s))ds+S_{[t]}(\omega),$ (3)

is an example of a stochastic differential equation driven by a random walk. The stochastic process obtained as a solution is no longer deterministic and describes an open system dynamics.

$\bigtriangledown$

2. Distributions

The space of trajectories

is usually endowed with the product $\sigma$ -field $\mathcal{E}^{\otimes T}$ generated by all projections $\pi_{t}: E^T\to E$ , which associate to each function $x\in E^T$ its value $x(t)\in E$ , $t\in T$ . Thus, a stochastic process is, equivalently, a random variable $X:\Omega\to E^T$ , $\omega\mapsto X(\cdot,\omega)$ . The Law or Probability Distribution $P_{X}$ of a stochastic process

is the image of the probability $\mathbb{P}$ on the measurable space $(E^T,\mathcal{E}^{\otimes T})$ of all trajectories. Given a probability measure

on the space $(E^T,\mathcal{E}^{\otimes T})$ , one may construct a Canonical Process

whose distribution $P_{X}$ coincides with

. Indeed, it suffices to consider $\Omega=E^T,\;\mathcal{F}=\mathcal{E}^{\otimes T},\;\mathbb{P}=P, \;X(t,\omega)=\omega (t)$ , for each $\omega=(\omega (s);\;s\in T)\in E^T$ , $t\in T$ .

Let a finite set $I=\left\{t_{1},\ldots,t_{n}\right\}\subset T$ be given, and denote $\pi_{I}$ the canonical projection defined on with values in $E^{I}$ , such that $x\mapsto (x(t_{1}),\ldots,x(t_{n}))$ . Call $\mathcal{P}_{f}(T)$ the family of all finite subsets of . The Finite Dimensional Distributions or Marginal Probability Distributions of an -valued stochastic process is the family $(P_{X,I})_{I\in\mathcal{P}_{f}(T)}$ of distributions, where $P_{X,I}$ is defined as

$\displaystyle P_{X,I}(A)=P_{X}(\pi_{i}^{-1}(A))=\mathbb{P}\left((X(t_{1},\cdot),\ldots,X(t_{n},\cdot))\in A\right),$

(4)

for all $A\in\mathcal{E}^{\otimes I}$ . Examples

A Poisson Process $(N_{t})_{t\geq 0}$ is defined as a stochastic process with values in $\mathbb{N}$ such that

(a)

$N_{0}(\omega)=0$ and $t\mapsto N_{t}(\omega)$ is increasing, for all $\omega\in\mathbb{N}$ ,

(b)

for all $0\leq s\leq t<\infty$ , $N_{t}-N_{s}$ is independent of $(N_{u};\;u\leq s)$ ,

(c)

for all $0\leq s\leq t<\infty$ , the distribution of $N_{t}-N_{s}$ is Poisson with parameter , that is
$\displaystyle \mathbb{P}(N_{t}-N_{s}=k)=\dfrac{(t-s)^k}{k!}e^{-(t-s)}.$
A d-dimensional Brownian Motion is a stochastic process $(B_{t})_{t\geq 0}$ , taking values in $\mathbb{R}^d$ such that:

(a)

if $0\leq s<t<\infty$ , then $B_{t}-B_{s}$ is independent of $(B_{u};\;u\leq s)$ ,

(b)

if $0\leq s<t<\infty$ , then
$\displaystyle \mathbb{P}(B_{t}-B_{s}\in A)=(2\pi (t-s))^{-d/2}\int_{A}e^{-\vert x\vert^2/2(t-s)}dx,$
where represents the Lebesgue measure on $\mathbb{R}^d$ and $\vert x\vert$ is the euclidian norm in that space.
The Brownian Motion starts at if $\mathbb{P}(B_{0}=x)=1$ .

$\bigtriangledown$

3. Construction of canonical processes

An important problem in the construction of a canonical stochastic process given the family of its finite dimensional distributions was solved by Kolmogorov in the case of a countable set

and extended to continuous time later by several authors. At present, a particular case, general enough for applications, is the following version of the Daniell-Kolmogorov Theorem. Suppose that

is a Polish space (complete separable metric space) and let $\mathcal{E}$ be its Borel $\sigma$ -field. Let

be a subset of ${\mathbb{R}}^{+}$ . Suppose that for each $I\in\mathcal{P}_{f}(T)$ a probability $P_{I}$ is given on the space $(E^{I},\mathcal{E}^{\otimes I})$ . Then, there exists a probability

on $(E^T,\mathcal{E}^{\otimes T})$ such that for all $I\in\mathcal{P}_{f}(T)$ ,

$\displaystyle P_{I}(A)=P\circ\pi_{I}^{-1}(A)=P(\pi_{I}^{-1}(A)),$

(5)

for all $A\in\mathcal{E}^{\otimes I}$ , if and only if the following Consistency Condition is satisfied:

$\displaystyle P_{I}=P_{J}\circ\pi_{J,I}^{-1},$

(6)

for all $I,J\in\mathcal{P}_{f}(T)$ such that $I\subset J$ , where $\pi_{J,I}$ denotes the canonical projection from the space $E^{J}$ onto $E^{I}$ . Example Consider $J=\{t_{1},\ldots,t_{n}\}$ and let $\Phi_{t}$ be the normal distribution of mean zero and variance $t\geq 0$ , that is,

$\displaystyle \Phi_{t}(A)=(2\pi t)^{-1/2}\int_{A}e^{-x^2/2t}dx.$

Let $P_{J}=\Phi_{t_{1}}\otimes\Phi_{t_{2}-t_{1}}\otimes\ldots\Phi_{t_{n}-t_{n-1}}$ , that is for all Borel sets $A_{1},\ldots,A_{n}$ ,

$\displaystyle P_{J}(A_{1}\times A_{2}\times\ldots\times A_{n})=\Phi_{t_{1}}(A_{1})\Phi_{t_{2}-t_{1}}(A_{2})\ldots\Phi_{t_{n}-t_{n-1}}(A_{n}).$

This is a probability on $\mathbb{R}^n$ . Take $I=\{t_{1},\ldots,t_{n-1}\}$ . Notice that $\pi_{J,I}^{-1}(A_{1}\times\ldots\times A_{n-1})=A_{1}\times\ldots\times A_{n-1}\times \mathbb{R}$ , thus

$\displaystyle P_{I}(A_{1}\times A_{2}\times\ldots\times A_{n-1})=\Phi_{t_{1}}(A_{1})\Phi_{t_{2}-t_{1}}(A_{2})\ldots\Phi_{t_{n-1}-t_{n-2}}(A_{n-1})$

$\displaystyle =P_{J}(A_{1}\times A_{2}\times\ldots\times \mathbb{R}).$

$\bigtriangledown$

4. Regularity of trajectories

Another interpretation of a stochastic process is based on regularity properties of trajectories. Indeed, if one knows that each trajectory belongs almost surely to a function space $S\subset E^T$ , endowed with a $\sigma$ -field $\mathcal{S}$ , one may provide another characterization of the stochastic process

as an

-valued random variable, $\omega\mapsto X(\cdot,\omega)$ defined on $\Omega$ .

Regarding the regularity, Kolmogorov first proved one of the most useful criteria on continuity of trajectories. Suppose that $X=(X(t,\omega);\;t\in [0,1], \;\omega\in \Omega)$ is a real-valued stochastic process and assume that there exist $\alpha,\delta>0$ and $0<C<\infty$ such that

$\displaystyle \mathbb{E}\left (\left\vert X(t+h)-X(t)\right\vert ^\alpha\right )<C\left\vert h\right\vert ^{1+\delta},$

(7)

for all $t\in [0,1]$ and all sufficiently small

, then

has continuous trajectories with probability 1. Therefore, if

satisfies (7), then there exists a random variable $\widetilde{X}:\Omega\to C[0,1]$ , where

is the metric space of real continuous functions defined on

, endowed with the metric of uniform distance, such that $\mathbb{P}\left(\left\{\omega\in\Omega:\; X(\cdot,\omega)=\widetilde{X}(\omega)\right\}\right)=1$ .

5. Wiener Measure, Brownian Motion

The above result is crucial to construct the Wiener Measure on the space

or, more generally, on $C({\mathbb{R}}^{+})$ , which is the law of the Brownian Motion. Indeed, by means of Kolmogorv's Consistency Theorem, one first constructs a probability measure

on the product space $({\mathbb{R}}^{{\mathbb{R}}^{+}},\mathcal{B}({\mathbb{R}}^{})^{\otimes {\mathbb{R}}^{+}})$ , where $\mathcal{B}({\mathbb{R}}^{})$ is the Borel $\sigma$ -field of ${\mathbb{R}}^{}$ , considering the consistent family of probability distributions

$\displaystyle P_{I}=\Phi_{t_{1}}\otimes \Phi_{t_{2}-t_{1}}\otimes\ldots\otimes\Phi_{t_{n}-t_{n-1}},$

(8)

where $I=\left\{t_{1},\ldots,t_{n}\right\}$ , and $\Phi_{t}$ denotes the normal distribution with mean 0 and variance

. Since the family $(P_{I})_{I\in\mathcal{P}_{f}({\mathbb{R}}^{+})}$ is consistent, there exists a unique

probability measure on $({\mathbb{R}}^{{\mathbb{R}}^{+}},\mathcal{B}({\mathbb{R}}^{})^{\otimes {\mathbb{R}}^{+}})$ such that $P_{I}=P\circ\pi_{I}^{-1}$ . One can construct the canonical process with law

which should correspond to the Brownian Motion. Unfortunately, the set of real-valued continuous functions defined on ${\mathbb{R}}^{+}$ is not an element of $\mathcal{B}({\mathbb{R}}^{})^{\otimes {\mathbb{R}}^{+}}$ . However, thanks to (7) one proves that the exterior probability measure

defined by

is concentrated on the subset $C({\mathbb{R}}^{+})$ of ${\mathbb{R}}^{{\mathbb{R}}^{+}}$ thus, the restriction $P_{W}$ of

to $C({\mathbb{R}}^{+})$ gives the good definition of Wiener Measure. Thus, a canonical version of the Brownian Motion is given by the canonical process on the space $C({\mathbb{R}}^{+})$ .

6. Series expansion in

In the early years of the Theory of Stochastic Processes, a number of authors, among them Karhunen and Loève, explored other regularity properties of trajectories, deriving some useful representations by means of series expansions in an

space. More precisely, let $T\in \mathcal{B}({\mathbb{R}}^{+})$ be given and call ${\mathfrak{h}}=L^2(T)$ the Hilbert space of all real-valued Lebesgue-square integrable functions defined on

. Suppose that all trajectories $X(\cdot, \omega)$ belong to ${\mathfrak{h}}$ for all $\omega\in\Omega$ , and denote $({e}_{n})_{n\in {\mathbb{N}}^{}}$ an orthonormal basis of ${\mathfrak{h}}$ . Therefore, $x_{n}(\omega)=\langle X(\cdot,\omega),e_{n}\rangle$ satisfies $\sum_{n\in{\mathbb{N}}^{}}\left\vert x_{n}(\omega)\right\vert ^2<\infty$ , for all $\omega\in\Omega$ . And the series

$\displaystyle \sum_{n\in{\mathbb{N}}^{}}x_{n}(\omega)e_{n},$

(9)

converges in ${\mathfrak{h}}$ , providing a representation of $X(\cdot,\omega)$ . So that, by an abuse of language one can represent $X(t,\omega)$ by $\sum_{n\in{\mathbb{N}}^{}}x_{n}(\omega)e_{n}(t)$ .

Example Consider and the Haar orthonormal basis on the space ${\mathfrak{h}}=L^2([0,1])$ constructed by induction as follows: $e_{1}(t)=1$ for all $t\in [0,1]$ ;

$\displaystyle e_{2^m+1}=\begin{cases} &2^{m/2},\; \text{ if $0\leq t<2^{-m-1}$\... .../2},\;\text{ if $2^{-m-1}\leq t<2^{-m}$},\ &0,\;\text{otherwise}. \end{cases}$

And finally, define $e_{2^m+j}(t)=e_{2^m+1}\left(t-2^{-m}(j-1)\right)$ , for $j=1,\ldots,2^m$ , $m=0,1,\ldots$ . Consider the Schauder functions $(s_n)_{n\in\mathbb{N}}$ as $s_n (t)=\int_0^te_n(s)ds$ , for each

. Given a sequence $(b_{n})_{n\geq 1}$ of independent standard normal random variables (that is, with distribution $\mathcal{N}(0,1)$ ), the $L^2(\Omega\times [0,1])$ -convergent series $\sum_{n\geq 1}b_{n}(\omega)s_{n}(t)$ provides a representation of the Brownian Motion $(B_{t})_{t\in [0,1]}$ . $\bigtriangledown$

7. The General Theory of Processes

The General Theory of Processes emerged in the seventies as a contribution of the Strasbourg School initiated by Paul André Meyer. This Theory uses the concept of a History or Filtration, which consists of an increasing family of $\sigma$ -fields $\mathbb{F}=({F}_{t})_{t\in T}$ , where

is an ordered set, $\mathcal{F}_{s}\subset\mathcal{F}_{t}\subset\mathcal{F}$ for all $s\leq t$ . Thus, a stochastic process

is adapted to $\mathbb{F}$ if for all $t\in T$ , the variable $X(t,\cdot)$ is $\mathcal{F}_{t}/\mathcal{E}$ -measurable. Stronger measurability conditions mixing regularity conditions have been introduced motivated by the construction of stochastic integrals and the modern theory of Stochastic Differential Equations. Let $T={\mathbb{R}}^{+}$ and assume

to be a Polish space endowed with the $\sigma$ -field of its Borel sets. Denote $C_{E}=C({\mathbb{R}}^{+},E)$ (respectively $D_{E}=D({\mathbb{R}}^{+},E)$ ) the space of all

-valued continuous functions defined on ${\mathbb{R}}^{+}$ to

(resp. the space of all

-valued functions which have left hand limit at each point

and are right-continuous at

, endowed with the Skorokhod's topology). Consider now the family $\mathcal{C}_{E}$ (resp. $\mathcal{D}_{E}$ ) of all $\mathbb{F}$ -adapted stochastic processes $X:{\mathbb{R}}^{+}\times\Omega\to E$ such that their trajectories belong to $C_{E}$ (resp. to $D_{E}$ ). The Predictable (resp. Optional) $\sigma$ -field on the product set ${\mathbb{R}}^{+}\times \Omega$ is the one generated by $\mathcal{C}_{E}$ (resp. $\mathcal{D}_{E}$ ), that is $\mathcal{P}=\sigma (\mathcal{C}_{E})$ , (resp. $\mathcal{O}=\sigma (\mathcal{D}_{E})$ ). Then, a process

is predictable (resp. optional) if $(t,\omega)\mapsto X(t,\omega)$ is measurable with respect to $\mathcal{P}$ , (resp. $\mathcal{O}$ ). A crucial notion in the development of this theory is that of Stopping Time: a function $\tau :\Omega\to [0,\infty]$ is a stopping time if for all

, $\left\{\omega\in\Omega:\;\tau (\omega)\leq t\right\}$ is an element of the $\sigma$ -field $\mathcal{F}_{t}$ . This definition is equivalent to say that $\tau$ is a stopping time if and only if $(t,\omega)\mapsto1_{[0,\tau (\omega)[}(t)$ is an optional process, where the notation $1_{A}$ is used for the indicator or characteristic function of a set

The development of the General Theory of Processes encountered at least two serious difficulties which could not be solved in the framework of Measure Theory and required a use of Capacity Theory. They are the Section Theorem and the Projection Theorem. The Section Theorem asserts that if the probability space $(\Omega,\mathcal{F},\mathbb{P})$ is complete (that is $\mathcal{F}$ contains all $\mathbb{P}$ -null sets) and $A\in\mathcal{O}$ , then there exists a stopping time $\tau$ such that its graph is included in . And the Projection Theorem states that given an optional set $A\subset {\mathbb{R}}^{+}\times \Omega$ , the projection $\pi (A)$ on $\Omega$ belongs to the complete $\sigma$ -field $\mathcal{F}$ . For instance, this result allows to prove that given a Borel set of the real line, the random variable $\tau_{B}(\omega)=\inf\left\{t\geq 0:\;X(t,\omega)\in B\right\}$ ( $\inf\emptyset=\infty$ ), defines a stopping time for an $\mathbb{F}$ -adapted process with trajectories in almost surely, provided the filtration $\mathbb{F}$ is right-continuous, that is, for all $t\geq 0$ , $\mathcal{F}_{t}=\mathcal{F}_{t+}:=\bigcap_{s>t}\mathcal{F}_{s}$ , and in addition each $\sigma$ -field contains all $\mathbb{P}$ -null sets. Within this theory, the system $(\Omega, \mathcal{F},({\mathcal{F}}_{t})_{t\in {\mathbb{R}}^{+}},\mathbb{P})$ is usually called a Stochastic Basis and a system $(\Omega, \mathcal{F},({\mathcal{F}}_{t})_{t\in T}, E,\mathcal{E}, \mathbb{P},({X}_{t})_{t\in T})$ provides the whole structure needed to define an -valued adapted stochastic process.

Attending to measurability properties only, stochastic processes may be classified as optional or predictable, as mentioned before, for which no probability is needed. However, richer properties of processes strongly depend on the probability considered in the stochastic basis. For instance, the definitions of martingales, submartingales, supermartingales, semimartingales depend on a specific probability measure, through the concept of conditional expectation. Let us mention that semimartingales form the most general class of possible integrands to give a rigorous meaning to Stochastic Integrals and Stochastic Differential Equations.

Probability is moreover fundamental for introducing concepts as Markov Process, Gaussian Process, Stationary Sequence and Stationary Process.

8. Extensions of the Theory

Extensions to the theory have included changing either the nature of

to consider Random Fields, where $t\in T$ may have the meaning of a space label (

is no more a subset of the real line), or the state space

, to deal for instance with measure-valued processes, or random distributions. Example. Let $(T,\mathcal{T},\nu)$ be a $\sigma$ -finite measure space, and $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. Call $\mathcal{T}_{\nu}$ the family of all sets $A\in\mathcal{T}$ such that $\nu(A)<\infty$ . A Gaussian white noise based on $\nu$ is a random set function

defined on $\mathcal{T}_{\nu}$ and values in ${\mathbb{R}}^{}$ such that

(a): is centered Gaussian and $\mathbb{E}\left (W(A)^2\right )=\nu (A)$ , for all $A\in\mathcal{T}_{\nu}$ ;
(b): If $A\cap B=\emptyset$ , then and are independent.

In particular, if $T={{\mathbb{R}}^{+}}^2$ , $\mathcal{T}$ the corresponding Borel $\sigma$ -field, and $\nu=\lambda$ the product Lebesgue measure, define $B_{t_{1},t_{2}}=W(]0,t_{1}]\times ]0,t_{2}])$ , for all $(t_{1},t_{2})\in T$ . The process $(B_{t_{1},t_{2}})_{(t_{1},t_{2})\in T}$ is called the Brownian sheet. $\bigtriangledown$

Going further, on the state space consider the algebra $\mathfrak{E}$ of all bounded $\mathcal{E}$ -measurable complex-valued functions. Then, to each -valued stochastic process one associates a family of maps $j_{t}:\mathfrak{E}\to L^\infty_{{\mathbb{C}}^{}}(\Omega,\mathcal{F},\mathbb{P})$ , where $j_{t}(f)(\omega)=f(X(t,\omega))$ , for all $t\geq 0$ , $\omega\in\Omega$ . The family $({j}_{t})_{t\in {\mathbb{R}}^{+}}$ , known as the Algebraic Flow can be viewed as a family of complex random measures (each $j_{t}$ is a Dirac measure supported by $X(t,\omega)$ ) or, better, as a -homomorphism between the two -algebras $\mathfrak{E},\;L^\infty_{{\mathbb{C}}^{}}(\Omega,\mathcal{F},\mathbb{P})$ , the operation being here the complex conjugation. The stochastic process is completely determined by the algebraic flow $({j}_{t})_{t\in {\mathbb{R}}^{+}}$ .

Example. Consider a Brownian motion defined on a stochastic basis $(\Omega,\mathcal{F},({\mathcal{F}}_{t})_{t\in {\mathbb{R}}^{+}},\mathbb{P})$ , with states in ${\mathbb{R}}^{}$ , and call $\mathfrak{B}$ the algebra of bounded complex valued Borel function defined on the real line. $\mathfrak{B}$ is a -algebra of functions, that is, there exists an involution (the conjugation), such that $f\mapsto f^*$ is antilinear and , for all $f,g\in\mathfrak{B}$ . The algebraic flow associated to is given by $j_{t}(f)=f(B_{t})$ , for all $t\geq 0$ , and any $f\in\mathfrak{B}$ , that is $j_{t}:\mathfrak{B}\to L^\infty_{{\mathbb{C}}^{}}(\Omega,\mathcal{F},\mathbb{P})$ . If $\mathbb{P}(B_{0}=x)=1$ , then $j_{0}(f)=f(x)$ almost surely. Moreover, notice that Itô's formula implies that for all bounded of class , it holds

$\displaystyle j_{t}(f)=f(x)+\int_{0}^tj_{s}\left(\dfrac{d}{dx}f\right)dB_{s}+\int_{0}^tj_{s}\left(\dfrac{1}{2}\dfrac{d^2}{dx^2}f\right)ds.$

$\bigtriangledown$

Algebraic flows provide a suitable framework to deal with more generalized evolutions, like those arising in the description of Open Quantum System Dynamics, where the algebras are non commutative. Thus, given two unital -algebras (possibly non commutative) $\mathfrak{A}, \mathfrak{B}$ , a notion of Algebraic Stochastic Process is given by a flow $({j}_{t})_{t\in {\mathbb{R}}^{+}}$ , where $j_{t}:\mathfrak{B}\to\mathfrak{A}$ is a -homomorphisms, for all $t\geq 0$ . That is, each $j_{t}$ is a linear map, which satisfies $(j_{t}(b))^*=j_{t}(b^*)$ , $j_{t}(a^*b)=j_{t}(a)^*j_{t}(b)$ , for all $a,b\in\mathfrak{B}$ , and $j_{t}(\mathbf{1}_{\mathfrak{B}})=\mathbf{1}_{\mathfrak{A}}$ , where $\mathbf{1}_{\mathfrak{A}}$ (resp. $\mathbf{1}_{\mathfrak{B}}$ ) is the unit of $\mathfrak{A}$ (resp. $\mathfrak{B}$ ).

9. The dawning of Stochastic Analysis as a pillar of Modern Mathematics

These days, Stochastic Processes provide the better description of complex evolutionary phenomena in Nature. Coming from our understanding of the macro world, through our everyday life, exploring matter at its smallest component, stochastic modeling has become fundamental. In other words, stochastic processes have become influential in all sciences, namely, in biology (population dynamics, ecology, neurosciences), computer science, engineering (especially electric and operation research), economics (via finance), physics, among others. The new branch of Mathematics, known as Stochastic Analysis, is founded on stochastic processes. Stochastics is invading all branches of Mathematics: Combinatorics, Graph Theory, Partial and Ordinary Differential Equations, Group Theory, Dynamical Systems, Geometry, Functional Analysis, among many other specific subjects. The dawning of Stochastic Analysis era is a fundamental step in the evolution of human understanding of Chance as a natural interconnection and interaction of matter in Nature. This has been a long historical process which started centuries ago with the dart game.

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Article written by Rolando Rebolledo, reprinted with permission from Lovric, Miodrag (2011), International encyclopaedia of Statistical Science. Heidelberg: Springer Science +Business Media, LLC.

Stochastic Processes is owned by Rolando Rebolledo.

How to Cite This Entry

Rolando Rebolledo. "Stochastic Processes" (version 6). StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Freely available at http://statprob.com/encyclopedia/StochasticProcesses.html

Classification

AMS MSC:	60G05 (Probability theory and stochastic processes :: Stochastic processes :: Foundations of stochastic processes)
	60G07 (Probability theory and stochastic processes :: Stochastic processes :: General theory of processes)
	60G12 (Probability theory and stochastic processes :: Stochastic processes :: General second-order processes)
	60G15 (Probability theory and stochastic processes :: Stochastic processes :: Gaussian processes)
	60G20 (Probability theory and stochastic processes :: Stochastic processes :: Generalized stochastic processes)
	60G60 (Probability theory and stochastic processes :: Stochastic processes :: Random fields)
	60K40 (Probability theory and stochastic processes :: Special processes :: Other physical applications of random processes)

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Stochastic Processes

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Classification

Pending Errata and Addenda

Discussion