Stochastic asymptotic exponential stability of stochastic integral equations

1972 ◽  
Vol 9 (1) ◽  
pp. 169-177 ◽  
Author(s):  
Chris P. Tsokos ◽  
M. A. Hamdan
Keyword(s):  
Integral Equation ◽  
Exponential Stability ◽  
Stochastic Integral ◽  
Stability Theorem ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Classical Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R+.The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

Stochastic asymptotic exponential stability of stochastic integral equations

1972 ◽  
Vol 9 (01) ◽  
pp. 169-177
Author(s):  
Chris P. Tsokos ◽  
M. A. Hamdan
Keyword(s):  
Integral Equation ◽  
Exponential Stability ◽  
Stochastic Integral ◽  
Stability Theorem ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Classical Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R +. The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

scholarly journals Mean-square stability of a class of stochastic integral equations

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.

scholarly journals On a random solution of a nonlinear perturbed stochastic integral equation of the Volterra type

1973 ◽  
Vol 9 (2) ◽  
pp. 227-237 ◽  
Author(s):  
J. Susan Milton ◽  
Chris P. Tsokos
Keyword(s):  
Integral Equation ◽  
Stochastic Integral ◽  
Random Variable ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Almost Everywhere ◽  
Image Position ◽  
Probability Measure Space ◽  
Complete Probability ◽  
Vector Random Variable

The object of this present paper is to study a nonlinear perturbed stochastic integral equation of the formwhere ω ∈ Ω, the supporting set of the complete probability measure space (Ω A, μ). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; ω), of the above equation is defined to be a vector random variable which satisfies the equation μ almost everywhere.

On a stochastic integral equation of the Volterra type in telephone traffic theory

1971 ◽  
Vol 8 (02) ◽  
pp. 269-275 ◽  
Author(s):  
W. J. Padgett ◽  
C. P. Tsokos
Keyword(s):  
Integral Equation ◽  
Stochastic Integral ◽  
Random Variable ◽  
Scalar Function ◽  
List Type ◽  
Volterra Type ◽  
Stochastic Integral Equation ◽  
Image Position ◽  
Traffic Theory ◽  
Telephone Traffic

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where: (i) ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P); (ii) x(t; ω) is the unknown random variable for t ∊ R +, where R + = [0, ∞); (iii) y(t; ω) is the stochastic free term or free random variable for t ∊ R +; (iv) k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and (v) f(t, x) is a scalar function defined for t ∊ R + and x ∊ R, where R is the real line.

The method of V. M. Popov for differential systems with random parameters

1971 ◽  
Vol 8 (2) ◽  
pp. 298-310 ◽  
Author(s):  
Chris P. Tsokos
Keyword(s):  
Integral Equation ◽  
Frequency Response ◽  
Absolute Stability ◽  
Stochastic Integral ◽  
Random Parameters ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Differential Systems ◽  
Frequency Response Method ◽  
Response Method

The aim of this paper is to investigate the existence of a random solution and the stochastic absolute stability of the differential systems (1.0)–(1.1) and (1.2)–(1.3) with random parameters. These objectives are accomplished by reducing the differential systems into a stochastic integral equation of the convolution type of the form (1.4) and utilizing a generalized version of V. M. Popov's frequency response method.

scholarly journals On estimation of the Hurst index of solutions of stochastic integral equations

2008 ◽  
Vol 48 ◽  
Author(s):  
Kęstutis Kubilius ◽  
Dmitrij Melichov
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Hurst Index ◽  
Stochastic Integral Equation ◽  
Stochastic Integral Equations ◽  
Convergence Almost Surely ◽  
Strongly Consistent

Let X be a solution of a stochasti Let X be a solution of a stochastic integral equation driven by a fractional Brownian motion BH and let Vn(X, 2) = \sumn k=1(\DeltakX)2, where \DeltakX = X( k+1/n ) - X(k/n ). We study the ditions n2H-1Vn(X, 2) convergence almost surely as n → ∞ holds. This fact is used to obtain a strongly consistent estimator of the Hurst index H, 1/2 < H < 1.  

On a stochastic integral equation of the Volterra type in telephone traffic theory

1971 ◽  
Vol 8 (2) ◽  
pp. 269-275 ◽  
Author(s):  
W. J. Padgett ◽  
C. P. Tsokos
Keyword(s):  
Integral Equation ◽  
Stochastic Integral ◽  
Random Variable ◽  
Scalar Function ◽  
List Type ◽  
Volterra Type ◽  
Stochastic Integral Equation ◽  
Image Position ◽  
Traffic Theory ◽  
Telephone Traffic

In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where: (i)ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P);(ii)x(t; ω) is the unknown random variable for t ∊ R+, where R+ = [0, ∞);(iii)y(t; ω) is the stochastic free term or free random variable for t ∊ R+;(iv)k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and(v)f(t, x) is a scalar function defined for t ∊ R+ and x ∊ R, where R is the real line.

On the Uryson type of stochastic integral equations

1974 ◽  
Vol 76 (1) ◽  
pp. 297-305 ◽  
Author(s):  
S. T. Hardiman ◽  
Chris P. Tsokos
Keyword(s):  
Integral Equations ◽  
Measure Space ◽  
Fixed Point Theorems ◽  
Stochastic Integral ◽  
Sufficient Conditions ◽  
Random Solution ◽  
Stochastic Approximations ◽  
Stochastic Integral Equations ◽  
Image Position ◽  
Probability Measure Space

AbstractAn investigation of the random or stochastic integral equations of the formandis presented, where ω ∈ Ω, the supporting set of the probability measure space (Ω, A, P). The existence and uniqueness of a random solution, a second-order stochastic process, of the equations is considered. Several theorems utilizing fixed point theorems and successive stochastic approximations give sufficient conditions for the existence of a random solution.

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