Representation of the characteristic function of a stochastic integral

Let X(t) be a continuous, homogeneous stochastic process with independent increments characterized by a, σ, M, N in the Lévy representation formula. In this note we obtain the Lévy canonical representation of the characteristic function of a stochastic integral (in the sense of convergence in probability) of the form (where υ(t) is a non-decreasing, non-negative and left-continuous function) in terms of υ(t), a, σ, M, N.

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Characterization of stable processes by identically distributed stochastic integrals

Advances in Applied Probability ◽

10.2307/1426427 ◽

1980 ◽

Vol 12 (3) ◽

pp. 689-709 ◽

Cited By ~ 8

Author(s):

M. Riedel

Keyword(s):

Stochastic Process ◽

Stable Process ◽

Convergence In Probability ◽

Stochastic Integrals ◽

Stable Processes ◽

Process With Independent Increments ◽

Independent Increments ◽

The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

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Fourier series associated with the sample functions of a stochastic process

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100057157 ◽

1970 ◽

Vol 67 (1) ◽

pp. 101-106

Author(s):

G. Samal

Keyword(s):

Stochastic Process ◽

Continuous Process ◽

Finite Measure ◽

Unit Interval ◽

Finite Mass ◽

Process With Independent Increments ◽

Independent Increments ◽

Image Position ◽

Almost All ◽

Essential Restriction

We consider a stochastically continuous process ω(t, α) with independent increments, whose sample functions are bounded in the unit interval 0 ≤ t ≤ 1 for almost all α. If ω(t, α) is a process with independent increments, the characteristic function of ω(t, α) is of the form exp {tψ(u)} where where F is a σ-finite measure with finite mass outside every neighbourhood of o, and a and σ are constants. There is no essential restriction in supposing ω(0, α) = 0.

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Characterization of stable processes by identically distributed stochastic integrals

Advances in Applied Probability ◽

10.1017/s000186780003545x ◽

1980 ◽

Vol 12 (03) ◽

pp. 689-709 ◽

Cited By ~ 1

Author(s):

M. Riedel

Keyword(s):

Stochastic Process ◽

Stable Process ◽

Convergence In Probability ◽

Stochastic Integrals ◽

Stable Processes ◽

Process With Independent Increments ◽

Independent Increments ◽

The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

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A characterization of stable processes

Journal of Applied Probability ◽

10.1017/s0021900200032915 ◽

1969 ◽

Vol 6 (02) ◽

pp. 409-418 ◽

Cited By ~ 5

Author(s):

Eugene Lukacs

Keyword(s):

Stochastic Process ◽

Distribution Function ◽

Interior Point ◽

Random Variable ◽

Stable Processes ◽

Process With Independent Increments ◽

Independent Increments

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t 1 , t 2 ɛ T, t 1 〈 t2; the random variable X(t 2) – X(t 1) is called the increment of the process X(t) over the interval [t 1, t 2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0 P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

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Mean-square stability of a class of stochastic integral equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700045184 ◽

1972 ◽

Vol 7 (3) ◽

pp. 337-352

Author(s):

W.J. Padgett

Keyword(s):

Stochastic Process ◽

Integral Equations ◽

Stochastic Integral ◽

Second Order ◽

Mean Square ◽

Random Solution ◽

Mean Square Stability ◽

Stochastic Integral Equations ◽

Exponentially Stable ◽

Image Position

The object of this paper is to investigate under very general conditions the existence and mean-square stability of a random solution of a class of stochastic integral equations in the formfor t ≥ 0, where a random solution is a second order stochastic process {x(t; w) t ≥ 0} which satisfies the equation almost certainly. A random solution x(t; w) is defined to be stable in mean-square if E[|x(t; w)|2] ≤ p for all t ≥ 0 and some p > 0 or exponentially stable in mean-square if E[|x(t; w)|2] ≤ pe-at, t ≥ 0, for some constants ρ > 0 and α > 0.

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Note on the Asymptotic Normality of a Stochastic Process with Independent Increments

Proceedings of the Japan Academy ◽

10.2183/pjab1945.47.supplementii_989 ◽

1971 ◽

Vol 47 (SupplementII) ◽

pp. 989-993

Author(s):

Motoo HORI ◽

Tetsuo FUJIMAGARI

Keyword(s):

Stochastic Process ◽

Asymptotic Normality ◽

Process With Independent Increments ◽

Independent Increments

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Characterization of stochastic processes by stochastic integrals

Advances in Applied Probability ◽

10.2307/1426983 ◽

1983 ◽

Vol 15 (1) ◽

pp. 81-98 ◽

Cited By ~ 6

Author(s):

B. L. S. Prakasa Rao

Keyword(s):

Stochastic Process ◽

Stochastic Processes ◽

Recent Work ◽

Stochastic Integrals ◽

Stable Processes ◽

Process With Independent Increments ◽

Independent Increments

Let be a continuous homogeneous stochastic process with independent increments. A review of the recent work on the characterization of Wiener and stable processes and connected results through stochastic integrals is presented. No proofs are given but appropriate references are mentioned.

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