A simulation of integral and derivative of the solution of a stochastici integral equation

Nguyen Quy Hy; Nguyen Thi Minh

Annales Polonici Mathematici (1992)

  • Volume: 57, Issue: 1, page 1-12
  • ISSN: 0066-2216

Abstract

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A stochastic integral equation corresponding to a probability space ( Ω , Σ ω , P ω ) is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable u ( · ) : ℝ 1 → ℝ m . One constructs stochastic processes η ( 1 ) ( t ) , η ( 2 ) ( t ) connected with a Markov chain and with the space ( Ω , Σ ω , P ω ) . The expected values of η ( i ) ( t ) (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative x u ' ( t ) .

How to cite

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Nguyen Quy Hy, and Nguyen Thi Minh. "A simulation of integral and derivative of the solution of a stochastici integral equation." Annales Polonici Mathematici 57.1 (1992): 1-12. <http://eudml.org/doc/262278>.

@article{NguyenQuyHy1992,
abstract = {A stochastic integral equation corresponding to a probability space $(Ω,Σ_ω,P_ω)$ is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable $u(·):ℝ^1 → ℝ^m$. One constructs stochastic processes $η^\{(1)\}(t)$, $η^\{(2)\}(t)$ connected with a Markov chain and with the space $(Ω,Σ_ω,P_ω)$. The expected values of $η^\{(i)\}(t)$ (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative $x^\{\prime \}_u(t)$.},
author = {Nguyen Quy Hy, Nguyen Thi Minh},
journal = {Annales Polonici Mathematici},
keywords = {stochastic integral equation; Markov chain; integral representation of a solution},
language = {eng},
number = {1},
pages = {1-12},
title = {A simulation of integral and derivative of the solution of a stochastici integral equation},
url = {http://eudml.org/doc/262278},
volume = {57},
year = {1992},
}

TY - JOUR
AU - Nguyen Quy Hy
AU - Nguyen Thi Minh
TI - A simulation of integral and derivative of the solution of a stochastici integral equation
JO - Annales Polonici Mathematici
PY - 1992
VL - 57
IS - 1
SP - 1
EP - 12
AB - A stochastic integral equation corresponding to a probability space $(Ω,Σ_ω,P_ω)$ is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable $u(·):ℝ^1 → ℝ^m$. One constructs stochastic processes $η^{(1)}(t)$, $η^{(2)}(t)$ connected with a Markov chain and with the space $(Ω,Σ_ω,P_ω)$. The expected values of $η^{(i)}(t)$ (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative $x^{\prime }_u(t)$.
LA - eng
KW - stochastic integral equation; Markov chain; integral representation of a solution
UR - http://eudml.org/doc/262278
ER -

References

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  1. [1] Nguyen Ngoc Cuong, On a solution of a class of random integral equations relating to the renewal theory by the Monte-Carlo method, Ph. D. Thesis, University of Hanoi, 1983. 
  2. [2] N. Dunford and J. Schwartz, Linear Operators I, Interscience Publ., New York 1958. Zbl0084.10402
  3. [3] S. M. Ermakov and V. S. Nefedov, On the estimates of the sum of the Neumann series by the Monte-Carlo method, Dokl. Akad. Nauk SSSR 202 (1) (1972), 27-29 (in Russian). Zbl0251.65002
  4. [4] S. M. Ermakov, Monte Carlo Method and Related Problems, Nauka, Moscow 1975 (in Russian). 
  5. [5] Yu. M. Ermolev, Methods of Stochastic Programming, Nauka, Moscow 1976 (in Russian). 
  6. [6] Yu. M. Ermolev, Finite Difference Method in Optimal Control Problems, Naukova Dumka, Kiev 1978 (in Russian). 
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  8. [8] I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes, Saunders, Philadelphia 1965. Zbl0132.37902
  9. [9] Nguyen Quy Hy, On a probabilistic model for derivation and integration of the solution of some stochastic linear equations, Proc. Conf. Appl. Prob. Stat. Vietnam 1 (10-11), Nhatrang 7-1983. 
  10. [10] Nguyen Quy Hy and Nguyen Ngoc Cuong, On probabilistic properties of a solution of a class of random integral equations, Acta Univ. Lodz. Folia Math. 2 (1985). 
  11. [11] Nguyen Quy Hy and Nguyen Van Huu, A probabilistic model to solve a problem of stochastic control, Proc. Conf. Math. Vietnam III, Hanoi 7-1985. 
  12. [12] Nguyen Quy Hy and Bui Huy Quynh, Solution of a random integral using Monte-Carlo method, Bull. Univ. Hanoi Vol. Math. Mech. 10 (1980). 
  13. [13] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Gos. Izdat. Fiz.-Mat. Liter., Moscow 1959 (in Russian). Zbl0127.06102
  14. [14] A. I. Khisamutdinov, A unit class of estimates for calculating functionals of solutions to II kind integral equations by the Monte Carlo method, Zh. Vychisl. Mat. i Mat. Fiz. 10 (5) (1970), 1269-1280 (in Russian). 
  15. [15] Bui Huy Quynh, On a randomised model for solving stochastic integral equation of the renewal theory, Bull. Univ. Hanoi Vol. Math. Mech. 11 (1980). 

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