Abstract
Consider a multidimensional stochastic differential equation of the form \(X_{t}=x+\int_{0}^{t}b(X_{s-})\,ds+\int_{0}^{t}f(X_{s-})\,dZ_{s}\), where (Z s )s≥0 is a symmetric stable process. Under suitable assumptions on the coefficients, the unique strong solution of the above equation admits a density with respect to Lebesgue measure, and so does its Euler scheme. Using a parametrix approach, we derive an error expansion with respect to the time step for the difference of these densities.
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References
Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probab. Theory Relat. Fields 104(1), 43–60 (1996)
Bally, V., Talay, D.: The law of the Euler scheme for stochastic differential equations, II. Convergence rate of the density. Monte Carlo Methods Appl. 2, 93–128 (1996)
Bichteler, K., Gravereaux, J.B., Jacod, J.: Malliavin Calculus for Processes with Jumps. Stochastics Monographs, vol. 2 (1987)
Breiman, L.: Probability. Addison-Wesley, Reading (1968)
Dynkin, E.B.: Markov Processes. Springer, Berlin (1963)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. Wiley, New York (1966)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, New York (1964)
Guyon, J.: Euler scheme and tempered distributions. Stoch. Process. Appl. 116(6), 877–904 (2006)
Hausenblas, E.: Error analysis for approximation of stochastic differential equations driven by Poisson random measures. SIAM J. Numer. Anal. 40(1), 87–113 (2002)
Hausenblas, E., Marchis, J.: A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure. BIT Numer. Math. 46, 773–811 (2006)
Imkeller, P., Pavlyukevich, I.: First exit times of SDEs driven by stable Lévy processes. Stoch. Process. Appl. 116(4), 611–642 (2006)
Jacod, J.: The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Probab. 5(32), 1830–1872 (2004)
Jacod, J., Kurtz, T.G., Méléard, S., Protter, P.: The approximate Euler method for Lévy driven stochastic differential equations. Ann. Inst. H. Poincaré Probab. Stat. 41(3), 523–558 (2005)
Janicki, A., Michna, Z., Weron, A.: Approximation of stochastic differential equations driven by α-stable Lévy motion. Appl. Math. (Warsaw) 24(2), 149–168 (1996)
Kolokoltsov, V.: Symmetric stable laws and stable-like jump diffusions. Proc. Lond. Math. Soc. 80, 725–768 (2000)
Konakov, V., Mammen, E.: Local limit theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Relat. Fields 117, 551–587 (2000)
Konakov, V., Mammen, E.: Edgeworth type expansions for Euler schemes for stochastic differential equations. Monte Carlo Methods Appl. 8(3), 271–285 (2002)
Konakov, V., Menozzi, S.: Weak error for stable driven SDEs: expansion of the densities. Tech. Report LPMA (2010). http://www.hal.fr
Konakov, V., Menozzi, S., Molchanov, S.: Explicit parametrix and local limit theorems for some degenerate diffusion processes. Tech. Report LPMA (2008)
McKean, H.P., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. J. Differ. Geom. 1, 43–69 (1967)
Picard, J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996)
Protter, P.: Stochastic Integration and Differential Equations. Application of Mathematics, vol. 21. Springer, Berlin (2004)
Protter, P., Talay, D.: The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 1(25), 393–423 (1997)
Samorodnittsky, G., Taqqu, M.: Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance. Chapman and Hall, New York (1994)
Stuck, B.W., Kleiner, B.Z.: A statistical analysis of telephone noise. Bell Syst. Tech. J. 53, 1263–1320 (1974)
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 94–120 (1990)
Weron, A., Weron, R.: Computer simulation of Lévy α-stable variables and processes. In: Lecture Notes in Physics, vol. 457, pp. 379–392. Springer, Berlin (1995)
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Konakov, V., Menozzi, S. Weak Error for Stable Driven Stochastic Differential Equations: Expansion of the Densities. J Theor Probab 24, 454–478 (2011). https://doi.org/10.1007/s10959-010-0291-x
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DOI: https://doi.org/10.1007/s10959-010-0291-x