The Kurzweil-Henstock theory of stochastic integration

Tin-Lam Toh; Tuan-Seng Chew

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 829-848
  • ISSN: 0011-4642

Abstract

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The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock's Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.

How to cite

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Toh, Tin-Lam, and Chew, Tuan-Seng. "The Kurzweil-Henstock theory of stochastic integration." Czechoslovak Mathematical Journal 62.3 (2012): 829-848. <http://eudml.org/doc/246743>.

@article{Toh2012,
abstract = {The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock's Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.},
author = {Toh, Tin-Lam, Chew, Tuan-Seng},
journal = {Czechoslovak Mathematical Journal},
keywords = {stochastic integral; Kurzweil-Henstock; convergence theorem; stochastic integral; Kurzweil-Henstock; convergence theorem},
language = {eng},
number = {3},
pages = {829-848},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Kurzweil-Henstock theory of stochastic integration},
url = {http://eudml.org/doc/246743},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Toh, Tin-Lam
AU - Chew, Tuan-Seng
TI - The Kurzweil-Henstock theory of stochastic integration
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 829
EP - 848
AB - The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock's Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.
LA - eng
KW - stochastic integral; Kurzweil-Henstock; convergence theorem; stochastic integral; Kurzweil-Henstock; convergence theorem
UR - http://eudml.org/doc/246743
ER -

References

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