Applications of second order stochastic integral equations to electrical networks

The theory of stochastic differential equations is used in various fields of science and engineering. This paper deals with vector-valued stochastic integral equations. We show some applications of the presented theory to the problem of modelling RLC electrical circuits by noisy parameters. From practical point of view, the second-order RLC circuits are of major importance, as they are the building blocks of more complex physical systems. The mathematical models of such circuits lead to the second order differential equations. We construct stochastic models of the RLC circuit by replacing a coefficient in the deterministic system with a noisy one. In this paper we present the analytic solution of these equations using the Itô calculus and compute confidence intervals for the stochastic solutions. Numerical simulations in the examples are performed using Matlab.

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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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Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

Mathematical Problems in Engineering ◽

10.1155/2015/794607 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Mariusz Michta

Keyword(s):

Differential Equations ◽

Stochastic Processes ◽

Fuzzy Sets ◽

Stochastic Differential Equations ◽

Integral Equations ◽

Metric Space ◽

Level Sets ◽

Stochastic Integral ◽

Stochastic Integral Equations ◽

Fuzzy Solutions

The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itô’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms.

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Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2010/603819 ◽

2010 ◽

Vol 2010 ◽

pp. 1-16 ◽

Cited By ~ 1

Author(s):

K. Balachandran ◽

J.-H. Kim

Keyword(s):

Integral Equations ◽

Existence And Uniqueness ◽

Mixed Type ◽

Fixed Point Theorems ◽

Existence Of Solutions ◽

Stochastic Integral ◽

Sufficient Conditions ◽

Fredholm Integral Equations ◽

Equations Of Mixed Type ◽

Stochastic Integral Equations

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

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