On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals
Article Sidebar
Main Article Content
Xavier Bardina
Universitat Aut`onoma de Barcelona. Departament de Matemàtiques
Carles Rovira
Universitat de Barcelona. Departament de Matem`atiques i Inform`atica
Paraules clau
strong convergence, multiple Stratonovich integral, uniform transport process
Article Details
Com citar
Bardina, Xavier; Rovira, Carles. «On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals». Publicacions Matemàtiques, 2021, vol.VOL 65, núm. 2, p. 859-76, http://raco.cat/index.php/PublicacionsMatematiques/article/view/390255.
Referències
X. Bardina, G. Binotto, and C. Rovira, The complex Brownian motion as a strong limit of processes constructed from a Poisson process, J. Math. Anal. Appl. 444(1) (2016), 700–720. DOI: 10.1016/j.jmaa.2016.06.061.
X. Bardina, M. Ferrante, and C. Rovira, Strong approximations of Brownian sheet by uniform transport processes, Collect. Math. 71(2) (2020), 319–329. DOI: 10.1007/s13348-019-00263-4.
X. Bardina and M. Jolis, Weak convergence to the multiple Stratonovich integral, Stochastic Process. Appl. 90(2) (2000), 277–300. DOI: 10.1016/S0304- 4149(00)00045-4.
M. Csorgo and L. Horvath , Rate of convergence of transport processes with an application to stochastic differential equations, Probab. Theory Related Fields 78(3) (1988), 379–387. DOI: 10.1007/BF00334201.
J. Garzon, L. G. Gorostiza, and J. A. Leon, A strong uniform approximation of fractional Brownian motion by means of transport processes, Stochastic Process. Appl. 119(10) (2009), 3435–3452. DOI: 10.1016/j.spa.2009.06.003.
J. Garzon, L. G. Gorostiza, and J. A. Leo , Approximations of fractional stochastic differential equations by means of transport processes, Commun. Stoch. Anal. 5(3) (2011), 433–456. DOI: 10.31390/cosa.5.3.01.
J. Garzon, L. G. Gorostiza, and J. A. Le ´ on´ , A strong approximation of subfractional Brownian motion by means of transport processes, in: “Malliavin Calculus and Stochastic Analysis”, Springer Proc. Math. Stat. 34, Springer, New York, 2013, pp. 335–360. DOI: 10.1007/978-1-4614-5906-4_15.
J. Garzon, S. Torres, and C. A. Tudor , A strong convergence to the Rosenblatt process, J. Math. Anal. Appl. 391(2) (2012), 630–647. DOI: 10.1016/j. jmaa.2012.02.040.
L. G. Gorostiza and R. J. Griego, Strong approximation of diffusion processes by transport processes, J. Math. Kyoto Univ. 19(1) (1979), 91–103. DOI: 10. 1215/kjm/1250522470.
L. G. Gorostiza and R. J. Griego, Rate of convergence of uniform transport processes to Brownian motion and application to stochastic integrals, Stochastics 3(4) (1980), 291–303. DOI: 10.1080/17442508008833152.
R. J. Griego, D. Heath, and A. Ruiz-Moncayo, Almost sure convergence of uniform transport processes to Brownian motion, Ann. Math. Statist. 42(3) (1971), 1129–1131. DOI: 10.1214/aoms/1177693346.
Y. Hu, “Analysis on Gaussian Spaces”, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. DOI: 10.1142/10094.
P. E. Kloeden and E. Platen, Relations between multiple Itˆo and Stratonovich integrals, Stochastic Anal. Appl. 9(3) (1991), 311–321. DOI: 10.1080/073629991 08809243
D. F. Kuznetsov, “Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach”, Lambert Academic Publishing, Saarbr¨ucken, 2012.
D. F. Kuznetsov, Expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity based on generalized iterated Fourier series converging pointwise, Preprint (2018). arXiv:1801.00784v6.
E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36(5) (1965), 1560–1564. DOI: 10.1214/aoms/ 1177699916.
X. Bardina, M. Ferrante, and C. Rovira, Strong approximations of Brownian sheet by uniform transport processes, Collect. Math. 71(2) (2020), 319–329. DOI: 10.1007/s13348-019-00263-4.
X. Bardina and M. Jolis, Weak convergence to the multiple Stratonovich integral, Stochastic Process. Appl. 90(2) (2000), 277–300. DOI: 10.1016/S0304- 4149(00)00045-4.
M. Csorgo and L. Horvath , Rate of convergence of transport processes with an application to stochastic differential equations, Probab. Theory Related Fields 78(3) (1988), 379–387. DOI: 10.1007/BF00334201.
J. Garzon, L. G. Gorostiza, and J. A. Leon, A strong uniform approximation of fractional Brownian motion by means of transport processes, Stochastic Process. Appl. 119(10) (2009), 3435–3452. DOI: 10.1016/j.spa.2009.06.003.
J. Garzon, L. G. Gorostiza, and J. A. Leo , Approximations of fractional stochastic differential equations by means of transport processes, Commun. Stoch. Anal. 5(3) (2011), 433–456. DOI: 10.31390/cosa.5.3.01.
J. Garzon, L. G. Gorostiza, and J. A. Le ´ on´ , A strong approximation of subfractional Brownian motion by means of transport processes, in: “Malliavin Calculus and Stochastic Analysis”, Springer Proc. Math. Stat. 34, Springer, New York, 2013, pp. 335–360. DOI: 10.1007/978-1-4614-5906-4_15.
J. Garzon, S. Torres, and C. A. Tudor , A strong convergence to the Rosenblatt process, J. Math. Anal. Appl. 391(2) (2012), 630–647. DOI: 10.1016/j. jmaa.2012.02.040.
L. G. Gorostiza and R. J. Griego, Strong approximation of diffusion processes by transport processes, J. Math. Kyoto Univ. 19(1) (1979), 91–103. DOI: 10. 1215/kjm/1250522470.
L. G. Gorostiza and R. J. Griego, Rate of convergence of uniform transport processes to Brownian motion and application to stochastic integrals, Stochastics 3(4) (1980), 291–303. DOI: 10.1080/17442508008833152.
R. J. Griego, D. Heath, and A. Ruiz-Moncayo, Almost sure convergence of uniform transport processes to Brownian motion, Ann. Math. Statist. 42(3) (1971), 1129–1131. DOI: 10.1214/aoms/1177693346.
Y. Hu, “Analysis on Gaussian Spaces”, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. DOI: 10.1142/10094.
P. E. Kloeden and E. Platen, Relations between multiple Itˆo and Stratonovich integrals, Stochastic Anal. Appl. 9(3) (1991), 311–321. DOI: 10.1080/073629991 08809243
D. F. Kuznetsov, “Approximation of Multiple Ito and Stratonovich Stochastic Integrals: Multiple Fourier Series Approach”, Lambert Academic Publishing, Saarbr¨ucken, 2012.
D. F. Kuznetsov, Expansion of iterated Stratonovich stochastic integrals of arbitrary multiplicity based on generalized iterated Fourier series converging pointwise, Preprint (2018). arXiv:1801.00784v6.
E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 36(5) (1965), 1560–1564. DOI: 10.1214/aoms/ 1177699916.