Accuracy of discrete approximation for integral functionals of Markov processes
Volume 2, Issue 4 (2015), pp. 401–420
Pub. online: 30 December 2015
Type: Research Article
Open Access
Open Access
Received
4 December 2015
4 December 2015
Revised
15 December 2015
15 December 2015
Accepted
15 December 2015
15 December 2015
Published
30 December 2015
30 December 2015
Abstract
The article is devoted to the estimation of the convergence rate of integral functionals of a Markov process. Under the assumption that the given Markov process admits a transition probability density differentiable in t and the derivative has an integrable upper bound of a certain type, we derive the accuracy rates for strong and weak approximations of the functionals by Riemannian sums. We also develop a version of the parametrix method, which provides the required upper bound for the derivative of the transition probability density for a solution of an SDE driven by a locally stable process. As an application, we give accuracy bounds for an approximation of the price of an occupation time option.
References
Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. Anal. 266(6), 3543–3571 (2014). MR3165234. doi:10.1016/j.jfa.2014.01.007
Ganychenko, I., Kulik, A.: Rates of approximation of nonsmooth integral-type functionals of Markov processes. Mod. Stoch., Theory Appl. 2, 117–126 (2014). MR3316480. doi:10.15559/vmsta-2014.12
Gobet, E., Labart, C.: Sharp estimates for the convergence of the density of the Euler scheme in small time. Electron. Commun. Probab. 13, 352–363 (2008). MR2415143. doi:10.1214/ECP.v13-1393
Guerin, H., Renaud, J.-F.: Joint distribution of a spectrally negative Lévy process and its occupation time, with step option pricing in view. arXiv:1406.3130
Knopova, V.: Compound kernel estimates for the transition probability density of a Lévy process in ${\mathbb{R}}^{n}$. Theory Probab. Math. Stat. 89, 57–70 (2014). MR3235175. doi:10.1090/s0094-9000-2015-00935-2
Knopova, V., Kulik, A.: Parametrix construction of the transition probability density of the solution to an SDE driven by α-stable noise. arXiv:1412.8732
Knopova, V., Kulik, A.: Intrinsic small time estimates for distribution densities of Lévy processes. Random Oper. Stoch. Equ. 21(4), 321–344 (2013). MR3139314. doi:10.1515/rose-2013-0015
Kohatsu-Higa, A., Makhlouf, A., Ngo, H.L.: Approximations of non-smooth integral type functionals of one dimensional diffusion precesses. Stoch. Process. Appl. 124, 1881–1909 (2014). MR3170228. doi:10.1016/j.spa.2014.01.003
Kulczycki, T., Ryznar, M.: Gradient estimates of harmonic functions and transition densities for Lévy processes. arXiv:1307.7158. MR3413864. doi:10.1090/tran/6333
Mimica, A.: Heat kernel upper estimates for symmetric jump processes with small jumps of high intensity. Potential Anal. 36(2), 203–222 (2012). MR2886459. doi:10.1007/s11118-011-9225-1
Pruitt, W.E., Taylor, S.J.: The potential kernel and hitting probabilities for the general stable process in ${\mathbb{R}}^{n}$. Trans. Am. Math. Soc. 146, 299–321 (1969). MR0250372
Rosinski, J., Singlair, J.: Generalized tempered stable processes. Stab. Probab. 90, 153–170 (2010). MR2798857. doi:10.4064/bc90-0-10
Sztonyk, P.: Estimates of tempered stable densities. J. Theor. Probab. 23(1), 127–147 (2010). MR2591907. doi:10.1007/s10959-009-0208-8
Sztonyk, P.: Transition density estimates for jump Lévy processes. Stoch. Process. Appl. 121, 1245–1265 (2011). MR2794975. doi:10.1016/j.spa.2011.03.002
Watanabe, T.: Asymptotic estimates of multi-dimensional stable densities and their applications. Trans. Am. Math. Soc. 359(6), 2851–2879 (2007). MR2286060. doi:10.1090/S0002-9947-07-04152-9
