The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

On Henstock-Kurzweil method to Stratonovich integral

Haifeng Yang; Tin Lam Toh

Mathematica Bohemica (2016)

  • Volume: 141, Issue: 2, page 129-142
  • ISSN: 0862-7959

Abstract

top
We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the “tail” term, that is, f ( W t ) = f ( W 0 ) + ∫ 0 t f ' ( W s ) ∘ d W s . Further, the condition on the integrands in this paper is weaker than the classical one.

How to cite

top

Yang, Haifeng, and Toh, Tin Lam. "On Henstock-Kurzweil method to Stratonovich integral." Mathematica Bohemica 141.2 (2016): 129-142. <http://eudml.org/doc/276987>.

@article{Yang2016,
abstract = {We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the “tail” term, that is, \[ f(W\_\{t\})= f(W\_\{0\})+\int \_\{0\}^\{t\}f^\{\prime \}(W\_\{s\})\circ \{\rm d\}W\_\{s\}. \] Further, the condition on the integrands in this paper is weaker than the classical one.},
author = {Yang, Haifeng, Toh, Tin Lam},
journal = {Mathematica Bohemica},
keywords = {Itô formula; Henstock-Kurzweil approach; Stratonovich integral},
language = {eng},
number = {2},
pages = {129-142},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Henstock-Kurzweil method to Stratonovich integral},
url = {http://eudml.org/doc/276987},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Yang, Haifeng
AU - Toh, Tin Lam
TI - On Henstock-Kurzweil method to Stratonovich integral
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 129
EP - 142
AB - We use the general Riemann approach to define the Stratonovich integral with respect to Brownian motion. Our new definition of Stratonovich integral encompass the classical Stratonovich integral and more importantly, satisfies the ideal Itô formula without the “tail” term, that is, \[ f(W_{t})= f(W_{0})+\int _{0}^{t}f^{\prime }(W_{s})\circ {\rm d}W_{s}. \] Further, the condition on the integrands in this paper is weaker than the classical one.
LA - eng
KW - Itô formula; Henstock-Kurzweil approach; Stratonovich integral
UR - http://eudml.org/doc/276987
ER -

References

top
  1. Chew, T.-S., Tay, J.-Y., Toh, T.-L., 10.14321/realanalexch.27.2.0495, Real Anal. Exch. 27 (2002), 495-514. (2002) Zbl1067.60025MR1922665DOI10.14321/realanalexch.27.2.0495
  2. Durrett, R., Probability: Theory and Examples, Cambridge University Press, Cambridge (2010). (2010) Zbl1202.60001MR2722836
  3. Gradinaru, M., Nourdin, I., Russo, F., Vallois, P., 10.1016/j.anihpb.2004.06.002, Ann. Inst. Henri Poincaré, Probab. Stat. 41 (2005), 781-806. (2005) Zbl1083.60045MR2144234DOI10.1016/j.anihpb.2004.06.002
  4. Karatzas, I., Shreve, S. E., Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics 113 Springer, New York (1991). (1991) Zbl0734.60060MR1121940
  5. Kurzweil, J., Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces, Series in Real Analysis 7 World Scientific, Singapore (2000). (2000) Zbl0954.28001MR1763305
  6. Lee, P. Y., Lanzhou Lectures on Henstock Integration, Series in Real Analysis 2 World Scientific, London (1989). (1989) Zbl0699.26004MR1050957
  7. Lee, T. W., 10.1017/S144678870001692X, J. Aust. Math. Soc., Ser. A 21 (1976), 64-71. (1976) Zbl0314.28009MR0435334DOI10.1017/S144678870001692X
  8. Loo, K. P., The Non-Uniform Riemann Approach to Anticipating Stochastic Integrals. Ph.D thesis, National University of Singapore Singapore (2002/2003). (2002) MR2061311
  9. McShane, E. J., Stochastic Calculus and Stochastic Models, Probability and Mathematical Statistics 25 Academic Press, New York (1974). (1974) Zbl0292.60090MR0443084
  10. Nourdin, I., Réveillac, A., Swanson, J., 10.1214/EJP.v15-843, Electron. J. Probab. (electronic only) 15 (2010), Article No. 70, 2117-2162. (2010) Zbl1225.60089MR2745728DOI10.1214/EJP.v15-843
  11. Øksendal, B., Stochastic Differential Equations. An Introduction with Applications, Universitext Springer, Berlin (2003). (2003) Zbl1025.60026MR2001996
  12. Protter, P. E., Stochastic Integration and Differential Equations, Stochastic Modelling and Applied Probability 21 Springer, Berlin (2004). (2004) Zbl1041.60005MR2020294
  13. Protter, P., 10.1214/aop/1176995088, Ann. Probab. 7 (1979), 276-289. (1979) Zbl0404.60062MR0525054DOI10.1214/aop/1176995088
  14. Toh, T.-L., Chew, T.-S., 10.1007/s10587-005-0052-7, Czech. Math. J. 55 (2005), 653-663. (2005) Zbl1081.26005MR2153089DOI10.1007/s10587-005-0052-7
  15. Toh, T.-L., Chew, T.-S., 10.1016/S0022-247X(03)00059-3, J. Math. Anal. Appl. 280 (2003), 133-147. (2003) Zbl1022.60055MR1972197DOI10.1016/S0022-247X(03)00059-3
  16. Toh, T.-L., Chew, T.-S., A variational approach to Itô's integral, Trends in Probability and Related Analysis. Proc. Symp. Analysis and Probability (SAP'98), Taipei, Taiwan, 1998 World Scientific Publishing Singapore N. Kono et al. (1999), 291-299. (1999) Zbl0981.60054MR1819215
  17. Xu, J., Lee, P. Y., Stochastic integrals of Itô and Henstock, Real Anal. Exch. 18 (1993), 352-366. (1993) Zbl0806.28009MR1228401

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.