Theory of Probability and Mathematical Statistics
Asymptotics for time-changed diffusions.
Raffaela Capitanelli, Mirko D'Ovidio
Download PDF
Abstract: We consider time-changed diffusions driven by generators with discontinuous coefficients. The PDE’s connections are investigated and in particular some results on the asymptotic analysis according to the behaviour of the coefficients are presented.
Keywords: Skew Brownian motion, scale function, boundary value problems
Bibliography: [1] T. Appuhamillage, V. Bokil, E. Thomann, E. Waymire, B. Wood, Occupation and local times for skew Brownian motion with applications to dispersion across an interface. Ann. Appl. Probab., 21 (2011), 1, 183–214.
[2] E. Acerbi, G. Buttazzo, Reinforcement problems in the calculus of variations, Ann. Inst. H. Poincar´e Anal. Non Lin. 3 (1986), no. 4, 273–284.
[3] H. Brezis, L.A. Caffarelli, A. Friedman, Reinforcement problems for elliptic equations and variational inequalities. Ann. Mat. Pura Appl. (4) 123 (1980), 219–246.
[4] S. Blei, On symmetric and skew Bessel processes. Stochastic Processes and their Applications, 122 (2012), 3262–3287.
[5] R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory. Academic Press, New York, 1968.
[6] R. Capitanelli, M. D’Ovidio, Skew Brownian diffusions across Koch interfaces. Potential Anal. (2016) doi: 10.1007/s11118-016-9588-4.
[7] R. Capitanelli, M.A. Vivaldi, On the Laplacean transfer across fractal mixtures. Asymptot. Anal. 83 (2013), no. 1-2, 1–33.
[8] M. Decamps , M. Goovaerts, W. Schoutens, Asymmetric skew Bessel processes and their applications to finance. Journal of Computational and Applied Mathematics, 186 (2006), 130–147.
[9] M. D’Ovidio, From Sturm-Liouville problems to fractional and anomalous diffusions. Stochastic Process. Appl. 122 (2012), no. 10, 3513–3544.
[10] J. M. Harrison, L. A. Shepp, On Skew Brownian Motion. Ann. Probab., 9 (1981), 309–313.
[11] S.Karlin, H. M. Taylor, A Second Course in Stochastic Processes. Academic Press, New York, 1981.
[12] K. Itˆo, H. P. McKean, Jr., Diffusion Processes and Their Sample Paths. Springer-Verlag, Heidelberg New York 1974.
[13] A. Lejay, On the constructions of the skew Brownian motion. Probab. Surveys, 3 (2006), 413–466.
[14] N. N. Leonenko, M. M. Meerschaert, A. Sikorskii, Fractional Pearson diffusions. J. Math. Anal. Appl., 403 (2013), no. 2, 532–546
[15] N. N. Leonenko, N. ˇSuvak, Statistical inference for reciprocal gamma diffusion process. J. Statist. Plann. Inference 140 (2010), no. 1, 30–51
[16] U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3, (1969), 510–585.
[17] U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal., 123, (1994), no. 2, 368–421.
[18] Y. Ouknine, F. Russo, G. Trutnau, On countably skewed Brownian motion with accumulation point. Electron. J. Probab. 20 (2015), no. 82, 1–27.
[19] J. M. Ramirez, Multi-skewed Brownian motion and diffusion in layered media. Proceedings of the American Mathematical Society, 139 (2011), 3739–3752.
[20] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion. 3rd edn, Springer-Verlag, Berlin, 1999.
[21] J.B. Walsh, A diffusion with a discontinuous local time. Asterisque, 52-53 (1978), 37–45.