Stochastic process

A stochastic process is defined as a collection of random variables defined on a common probability space $(\Omega ,{\mathcal {F}},P)$ , where $\Omega$ is a sample space, ${\mathcal {F}}$ is a $\sigma$ -algebra, and $P$ is a probability measure; and the random variables, indexed by some set $T$ , all take values in the same mathematical space $S$ , which must be measurable with respect to some $\sigma$ -algebra $\Sigma$ .^[28]

In other words, for a given probability space $(\Omega ,{\mathcal {F}},P)$ and a measurable space $(S,\Sigma )$ , a stochastic process is a collection of $S$ -valued random variables, which can be written as:^[80]

\{X(t):t\in T\}.

Historically, in many problems from the natural sciences a point $t\in T$ had the meaning of time, so $X(t)$ is a random variable representing a value observed at time $t$ .^[133] A stochastic process can also be written as $\{X(t,\omega ):t\in T\}$ to reflect that it is actually a function of two variables, $t\in T$ and $\omega \in \Omega$ .^[28]^[134]

There are other ways to consider a stochastic process, with the above definition being considered the traditional one.^[68]^[69] For example, a stochastic process can be interpreted or defined as a $S^{T}$ -valued random variable, where $S^{T}$ is the space of all the possible functions from the set $T$ into the space $S$ .^[27]^[68] However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined.^[135]

Index set

The set $T$ is called the index set^[4]^[51] or parameter set^[28]^[136] of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set $T$ the interpretation of time.^[1] In addition to these sets, the index set $T$ can be another set with a total order or a more general set,^[1]^[54] such as the Cartesian plane $\mathbb {R} ^{2}$ or $n$ -dimensional Euclidean space, where an element $t\in T$ can represent a point in space.^[48]^[137] That said, many results and theorems are only possible for stochastic processes with a totally ordered index set.^[138]

State space

The mathematical space $S$ of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, $n$ -dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.^[1]^[5]^[28]^[51]^[56]

Sample function

A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process.^[28]^[139] More precisely, if $\{X(t,\omega ):t\in T\}$ is a stochastic process, then for any point $\omega \in \Omega$ , the mapping

X(\cdot ,\omega ):T\rightarrow S,

is called a sample function, a realization, or, particularly when $T$ is interpreted as time, a sample path of the stochastic process $\{X(t,\omega ):t\in T\}$ .^[50] This means that for a fixed $\omega \in \Omega$ , there exists a sample function that maps the index set $T$ to the state space $S$ .^[28] Other names for a sample function of a stochastic process include trajectory, path function^[140] or path.^[141]

Increment

An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if $\{X(t):t\in T\}$ is a stochastic process with state space $S$ and index set $T=[0,\infty )$ , then for any two non-negative numbers $t_{1}\in [0,\infty )$ and $t_{2}\in [0,\infty )$ such that $t_{1}\leq t_{2}$ , the difference $X_{t_{2}}-X_{t_{1}}$ is a $S$ -valued random variable known as an increment.^[48]^[49] When interested in the increments, often the state space $S$ is the real line or the natural numbers, but it can be $n$ -dimensional Euclidean space or more abstract spaces such as Banach spaces.^[49]

Further definitions

Law

For a stochastic process $X\colon \Omega \rightarrow S^{T}$ defined on the probability space $(\Omega ,{\mathcal {F}},P)$ , the law of stochastic process $X$ is defined as the image measure:

\mu =P\circ X^{-1},

where $P$ is a probability measure, the symbol $\circ$ denotes function composition and $X^{-1}$ is the pre-image of the measurable function or, equivalently, the $S^{T}$ -valued random variable $X$ , where $S^{T}$ is the space of all the possible $S$ -valued functions of $t\in T$ , so the law of a stochastic process is a probability measure.^[27]^[68]^[142]^[143]

For a measurable subset $B$ of $S^{T}$ , the pre-image of $X$ gives

X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\},

so the law of a $X$ can be written as:^[28]

\mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}).

The law of a stochastic process or a random variable is also called the probability law, probability distribution, or the distribution.^[133]^[142]^[144]^[145]^[146]

Finite-dimensional probability distributions

For a stochastic process $X$ with law $\mu$ , its finite-dimensional distribution for $t_{1},\dots ,t_{n}\in T$ is defined as:

\mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1},

This measure $\mu _{t_{1},..,t_{n}}$ is the joint distribution of the random vector $(X({t_{1}}),\dots ,X({t_{n}}))$ ; it can be viewed as a "projection" of the law $\mu$ onto a finite subset of $T$ .^[27]^[147]

For any measurable subset $C$ of the $n$ -fold Cartesian power $S^{n}=S\times \dots \times S$ , the finite-dimensional distributions of a stochastic process $X$ can be written as:^[28]

{\displaystyle \mu _{t_{1},\dots ,t_{n}}(C)=P{\Big (}{\big \{}\omega \in \Omega

The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.^[57]

Stationarity

Stationarity is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if $X$ is a stationary stochastic process, then for any $t\in T$ the random variable $X_{t}$ has the same distribution, which means that for any set of $n$ index set values $t_{1},\dots ,t_{n}$ , the corresponding $n$ random variables

X_{t_{1}},\dots X_{t_{n}},

all have the same probability distribution. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.^[148]^[149] But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.^[148]^[150]^[151]

When the index set $T$ can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations.^[148] The intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same.^[152] A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.^[148]

A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process $X$ is said to be stationary in the wide sense, then the process $X$ has a finite second moment for all $t\in T$ and the covariance of the two random variables $X_{t}$ and $X_{t+h}$ depends only on the number $h$ for all $t\in T$ .^[152]^[153] Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.^[153]^[154]

Filtration

A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration $\{{\mathcal {F}}_{t}\}_{t\in T}$ , on a probability space $(\Omega ,{\mathcal {F}},P)$ is a family of sigma-algebras such that ${\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}}$ for all $s\leq t$ , where $t,s\in T$ and $\leq$ denotes the total order of the index set $T$ .^[51] With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process $X_{t}$ at $t\in T$ , which can be interpreted as time $t$ .^[51]^[155] The intuition behind a filtration ${\mathcal {F}}_{t}$ is that as time $t$ passes, more and more information on $X_{t}$ is known or available, which is captured in ${\mathcal {F}}_{t}$ , resulting in finer and finer partitions of $\Omega$ .^[156]^[157]

Modification

A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process $X$ that has the same index set $T$ , state space $S$ , and probability space $(\Omega ,{\cal {F}},P)$ as another stochastic process $Y$ is said to be a modification of $X$ if for all $t\in T$ the following

P(X_{t}=Y_{t})=1,

holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law^[158] and they are said to be stochastically equivalent or equivalent.^[159]

Instead of modification, the term version is also used,^[150]^[160]^[161]^[162] however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse.^[163]^[142]

If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.^[161]^[162]^[164] The theorem can also be generalized to random fields so the index set is $n$ -dimensional Euclidean space^[165] as well as to stochastic processes with metric spaces as their state spaces.^[166]

Indistinguishable

Two stochastic processes $X$ and $Y$ defined on the same probability space $(\Omega ,{\mathcal {F}},P)$ with the same index set $T$ and set space $S$ are said be indistinguishable if the following

P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1,

holds.^[142]^[158] If two $X$ and $Y$ are modifications of each other and are almost surely continuous, then $X$ and $Y$ are indistinguishable.^[167]

Separability

Separability is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to other conditions, its index set must be a separable space,^[b] which means that the index set has a dense countable subset.^[150]^[168]

More precisely, a real-valued continuous-time stochastic process $X$ with a probability space $(\Omega ,{\cal {F}},P)$ is separable if its index set $T$ has a dense countable subset $U\subset T$ and there is a set $\Omega _{0}\subset \Omega$ of probability zero, so $P(\Omega _{0})=0$ , such that for every open set $G\subset T$ and every closed set $F\subset \textstyle R=(-\infty ,\infty )$ , the two events $\{X_{t}\in F{\text{ for all }}t\in G\cap U\}$ and $\{X_{t}\in F{\text{ for all }}t\in G\}$ differ from each other at most on a subset of $\Omega _{0}$ .^[169]^[170]^[171] The definition of separability^[c] can also be stated for other index sets and state spaces,^[174] such as in the case of random fields, where the index set as well as the state space can be $n$ -dimensional Euclidean space.^[30]^[150]

The concept of separability of a stochastic process was introduced by Joseph Doob,.^[168] The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.^[172] Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable.^[175] A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification.^[168]^[170]^[176] Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.^[136]

Independence

Two stochastic processes $X$ and $Y$ defined on the same probability space $(\Omega ,{\mathcal {F}},P)$ with the same index set $T$ are said be independent if for all $n\in \mathbb {N}$ and for every choice of epochs $t_{1},\ldots ,t_{n}\in T$ , the random vectors $\left(X(t_{1}),\ldots ,X(t_{n})\right)$ and $\left(Y(t_{1}),\ldots ,Y(t_{n})\right)$ are independent.^[177]^{: p. 515}

Uncorrelatedness

Two stochastic processes $\left\{X_{t}\right\}$ and $\left\{Y_{t}\right\}$ are called uncorrelated if their cross-covariance $\operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} \left[\left(X(t_{1})-\mu _{X}(t_{1})\right)\left(Y(t_{2})-\mu _{Y}(t_{2})\right)\right]$ is zero for all times.^[178]^{: p. 142} Formally:

\left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ uncorrelated}}\quad \iff \quad \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}

.

Independence implies uncorrelatedness

If two stochastic processes $X$ and $Y$ are independent, then they are also uncorrelated.^[178]^{: p. 151}

Orthogonality

Two stochastic processes $\left\{X_{t}\right\}$ and $\left\{Y_{t}\right\}$ are called orthogonal if their cross-correlation $\operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} [X(t_{1}){\overline {Y(t_{2})}}]$ is zero for all times.^[178]^{: p. 142} Formally:

\left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ orthogonal}}\quad \iff \quad \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}

.

Skorokhod space

A Skorokhod space, also written as Skorohod space, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as $[0,1]$ or $[0,\infty )$ , and take values on the real line or on some metric space.^[179]^[180]^[181] Such functions are known as càdlàg or cadlag functions, based on the acronym of the French phrase continue à droite, limite à gauche.^[179]^[182] A Skorokhod function space, introduced by Anatoliy Skorokhod,^[181] is often denoted with the letter $D$ ,^[179]^[180]^[181]^[182] so the function space is also referred to as space $D$ .^[179]^[183]^[184] The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, $D[0,1]$ denotes the space of càdlàg functions defined on the unit interval $[0,1]$ .^[182]^[184]^[185]

Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space.^[181]^[183] Such spaces contain continuous functions, which correspond to sample functions of the Wiener process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.^[184]^[186]

Regularity

In the context of mathematical construction of stochastic processes, the term regularity is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues.^[187]^[188] For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample functions being continuous.^[189]^[190]