Studies in the Theory of Random Processes

By A. V. Skorokhod

This text is devoted to the development of certain probabilistic methods in the specific field of stochastic differential equations and limit theorems for Markov processes. Specialists, researchers, and students in the field of probability will find it a source of important theorems as well as a remarkable amount of advanced material in compact form.
The treatment begins by introducing the basic facts of the theory of random processes and constructing the auxiliary apparatus of stochastic integrals. All proofs are presented in full. Succeeding chapters explore the theory of stochastic differential equations, permitting the construction of a broad class of Markov processes on the basis of simple processes. The final chapters are devoted to various limit theorems connected with the convergence of a sequence of Markov chains to a Markov process with continuous time. Topics include the probability method of estimating how fast the sequence converges in the limit theorems and the precision of the limit theorems.

• Language: English
• Publisher: Dover Publications
• Release date: Jul 28, 2014
• ISBN: 9780486781464

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INDEX

CHAPTER 1

CERTAIN FACTS FROM THE THEORY OF RANDOM PROCESSES

1. Basic definitions associated with the concept of a random process. We shall say that a probability field {Ω, F, P} is given if a σ-algebra F of subsets of the set Ω – F is defined and a measure P is defined on the σ-algebra such that P{Ω} = 1. We shall call the elements ω of the set Ω elementary events and the elements A, B, ... of the σ-algebra F random events.

Let X be a metric space. If the function ξ(ω) is defined for almost all points relative to the measure P, if ω assumes value from X, and if for every Borel set A in X, the set {ξ(ω) ∈ A} of those ω for which ξ(ω) ∈ A belongs to F, then ξ(ω) is called a random variable with values in X, or simply a random variable in X. If X is the m-dimensional Euclidean space R(m), the variable ξ(ω) is called an m-dimensional random vector. In this case, if ξ(ω) is a random variable in R(m), and if

exists, then the vector ∫ξ(ω)P (dω) is called the mathematical expectation of the vector ξ(ω) and is denoted by Μξ(ω).

A sequence of random variables ξn(ω) with values in X converges to the random variable ξ0(ω)

(a) almost everywhere (with probability 1), if ξn(ω) → ξ0(ω) with respect to the measure P for almost all ω,

(b) in probability, if for every ∊ > 0,

where ρ(x, y) is the distance in X between x and y.

The function ξ(t, ω), defined for t ∈ [t0, T] and for every t ∈ [t0, T] for almost all ω relative to the measure P, is called a random process defined on [t0, T] with values in X if for every t ∈ [t0, T], ξ(t, ω) is a random variable in X. A process ξ(t, ω) is said to be stochastically continuous at a point t1 if for every ∊ > 0,

a process is said to be stochastically continuous from the right at the point t1 if for every ∊ > 0,

Stochastic continuity from the left is defined in an analogous manner.

A process is stochastically continuous (from the right, from the left) if it is stochastically continuous at every point where it is defined.

Suppose that Β1 is a σ-algebra of Borel sets on the interval [t0, T], that [t0, T] × Ω is the set of pairs (t, ω) where t ∈ [t0, T], ω ∈ Ω, that Β1 × F is the minimal σ-algebra of subsets of [t0, T] × Ω containing all subsets of the form Δ × A, and that Δ × A is the set of pairs of points (t, ω) for which t ∈ Δ, ω ∈ A, and Δ ∈ Β1, A ∈ F. If the function ξ(t, ω) is measurable relative to the σ-algebra Β1 × F, then the process ξ(t, ω) is said to be measurable.

Let us examine the process ξ(t, ω) in R(m). Let Λ be a set that is everywhere dense on [t0, T]. A process ξ(t, ω) is said to be Λ-separable (separable relative to the set Λ) if there exists an event A1 with probability 0 (P{A1} = 0), such that for every closed set F in R(m) and (open) interval Δ, the event

implies the event

If the process ξ(t, ω) is separable relative to some set, we shall call it separable.

The probabilities

where t1, t2, ..., tk ∈ [t0, T], and where A1, A2, ..., Ak are Borel sets of the space on which the process is defined, are called k-dimensional distributions of the process ξ(t, ω). The collection of k-dimensional distributions with all possible values of k is called the set of finite-dimensional distributions of the process ξ(t, ω).

Two processes ξ1(t, ω) and ξ2(t, ω) defined on [t0, T] are said to be stochastically equivalent if for every t ∈ [t0, T],

Stochastically equivalent processes have identical finite-dimensional distributions.

THEOREM 1. For every process ξ(t, ω) in X, there is a separable process stochastically equivalent to it. If a process ξ(t, ω) is defined on [t0, T] and is stochastically continuous at all points t ∈ [t0, T] with the possible exception of a finite number of points, then there exists a separable measurable process that is stochastically equivalent to the process ξ(t, ω).

A process ξ(t, ω) defined on [t0, T] is said to be continuous with probability 1 if for almost all ω ∈ Ω, ξ(t, ω) as a function of t is defined and continuous on [t0, T].

THEOREM 2 (KOLMOGOROV). Let ξ(t, ω) be a separable process defined on [t0, T] and assuming values in R(m). If there exists α > 0, β > 0, С > 0 such that for t1, t2 ∈ [t0, T],

then the process ξ(t, ω) is continuous with probability 1.

Let us examine a process ξ(t, ω) defined on [t0, T] with values in R(m). Let us designate by Φ(m)[t0, T] the space of all functions x(t) defined on [t0, T] and assuming values in R(m). Let Ct1 (A), where t1 ∈ (t0, T] and A is a Borel set in R(m), be the set of all x(t) for which x(t1) ∈ A. A set formed by the intersection of a finite number of sets of the form Ct1 (A) is called a cylindrical set. Let us indicate by F(m)[t0, T] the minimal σ-algebra of subsets of Ф(m)[t0, T] containing all cylindrical sets. The measure μ(A), uniquely determined on F(m)[t0, T] by the relations

for all k, t1, t2, …, tk in [t0, T] and for all Borel sets A1, A2, …, Ak in R(m), will be called the measure in the function space corresponding to the process ξ(t, ω), or simply the measure corresponding to the process ξ(t, ω). Suppose that to every set t1, t2, ..., tk in [t0, T] and to all natural numbers k there corresponds a k-dimensional distribution

then for all j ≤ k and for the Borel sets A1, …, Aj–1, Aj+1, …, Ak in R(m), the relations

hold. We shall call such a set of distributions consistent or joint.

THEOREM 3 (KOLMOGOROV). If Рt1,t2,…,tk (А1, A2, …, Ak) is a collection of joint distributions, then there exists a random process ξ(t, ω) for which these distributions will be finite-dimensional. To every joint collection of distributions, there corresponds a unique measure on F(m)[t0, T].

Suppose that ξ(t, ω) is a random process defined on [t0, T] and that [t1, t2] ⊂ [t0, T]. Let us examine the minimal σ-algebra of events that contains all events of the form {ξ(s, ω) ∈ A} for s ∈ [t1, t2] and for an arbitrary Borel set A in the space of values of the process. We shall henceforth designate this σ-algebra by F([t1, t2], ξ(t, ω)).

2. Conditional probabilities and mathematical expectations. Suppose that a probability field {Ω, F, P} is given and that F1 is a σ-algebra of events in F, F1 ⊂ F. If ξ(ω) is a certain random variable in R(m), then the random variable M(ξ(ω)/F1) measured relative to F1 and satisfying the relation

for all A ∈ F1 for which

is called the conditional mathematical expectation of the variable ξ(ω) relative to the σ-algebra F1. If M|ξ(ω)| < ∞, then the conditional mathematical expectation exists and is uniquely determined to within a set of measure zero in the variable ω. In the case in which F1 is the minimal σ-algebra containing all events of the form

where ξ1(ω), ξ2(ω), …, ξn(ω) are particular random variables and A1, A2, …, An are Borel sets in the space of their values, we write

instead of M(ξ(ω)/F1). We shall call this quantity the conditional mathematical expectation of the quantity ξ(ω) for fixed ξ1(ω), ξ2(ω), …, ξn(ω).

If the σ-algebra F1 coincides with the σ-algebra F([t1, t2], ξ(t, ω)), where ξ(t, ω) is a particular random process, then instead of M(ξ(ω)/F1), we sometimes write Μ(ξ(ω)/ξ(t, ω), t ∈ [t1, t2]). We shall call this quantity the conditional mathematical expectation of the variable ξ(ω) for a fixed value of ξ(t, ω) on [t1, t2].

Let us denote by χA(x) the characteristic function of the Borel set A in R(m). Then the quantity

is called the conditional probability of the event {ξ(ω) ∈ A} relative to F1. The quantities

and

are defined analogously to the conditional mathematical expectations.

The collection of conditional probabilities P{ξ(ω) ∈ A/F1} for all possible Borel sets A is called the conditional distribution of the variable ξ(ω) relative to F1.

We note that the relation

follows from (2.1) for all A in F1.

From Formula (2.2), it is easy to deduce the following lemma:

LEMMA. If for a sequence ξn(ω) of variables in R(m) and for some F1,

in probability, then ξn(ω) → 0 in probability.

3. Processes with independent increments. A process ξ(t) defined on [t0, T] and assuming values in R(m) is called a process with independent increments if, for any t1 < t2 < · · · < tk in [t0, T], the variables

are independent of each other, that is, if for all Borel sets A0, A1, …, Ak in R(m) the relation

holds. It is obvious that finite-dimensional distributions of the process ξ(t), and consequently the measure corresponding to the process ξ(t), are completely determined in this case by the distribution of the quantity ξ(t0) and by the distributions ξ(t2) – ξ(t1) for all possible values of t1 and t2 in [t0, T]. The distribution of the quantity ξ(t0) can be arbitrary. For a stochastically continuous process with independent increments (and henceforth we shall consider only such processes with independent increments), the distribution of the variable ξ(t2) – ξ(t1) for t1 < t2 is determined by its characteristic function :

where a(t) is a continuous function on [t0, T] with values in R(m), A(t) is a continuous function on [t0, T] whose values are nonnegative linear symmetrical operators in R(m) A(t2) – A(t1) is also a nonnegative operator if t1 < t2, and, finally, G is the measure defined on the Borel sets of the space [t0, T] × R(m) i.e., of the space of pairs of points (t; u), t ∈ [t0, T], и ∈ R(m), such that

[Here, (x, y) denotes the scalar product of the vectors x and y in R(m).]

A process with independent increments ξ(t) that is defined f or t ≥ 0 and that satisfies the condition ξ(0) = 0 will be called a homogeneous process with independent increments if the distribution of the variable ξ(t + h) – ξ(t) depends only on h. Finite-dimensional distributions of such a process are completely determined by the distributions of the variables ξ(t) for all possible t > 0. For stochastically continuous homogeneous processes with independent increments, we can write the logarithm of the characteristic function in the form

where γ ∈ R(m) and A is a linear symmetric nonnegative operator in R(m). M is a measure defined on the Borel sets of the space R(m) such that

Processes whose measure G (for homogeneous M) is identically equal to zero are called normal processes. If a process with independent increments and with probability 1 is continuous, it must necessarily be normal; the converse is true for stochastically continuous separable processes. A one-dimensional normal process (that is, one with numerical values) with independent increments w(t) for which

is called a process of Brownian motion. We shall be encountering this process repeatedly in what follows.

A one-dimensional homogeneous process with independent increments υ(t) that assumes nonnegative integral values is called a Poisson process if for some a > 0,

4. Markov processes. A process ξ(t) defined on [t0, T] and assuming values in R(m) is called a Markov process if for every s < t in [t0, T], with probability 1 the relation

is fulfilled.

From (4.1), it follows that for every t1 < t2 · · · < tk and for every Borel set A, with probability 1 the relation

is fulfilled. If a function P(t1, x, t2, A) defined for all t1 < t2, t1, t2 ∈ [t0, T], x ∈ R(m) and all Borel sets A in R(m) is measurable with respect to x, and if it is a measure with respect to A such that with probability 1

then such a function is called a transition probability function of the Markov process ξ(t). If a Markov process ξ(t) has a transition probability function