American Institute of Mathematical Sciences

August  2013, 18(6): 1683-1696. doi: 10.3934/dcdsb.2013.18.1683

Exponential growth rate for a singular linear stochastic delay differential equation

 1 Institut für Mathematik, Technische Universität Berlin, Str. des 17. Juni 136, 10623 Berlin

Received  January 2012 Revised  March 2012 Published  March 2013

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space) of the solution of the linear scalar stochastic delay equation $d X(t) = X(t-1) d W(t)$ which does not depend on the initial condition as long as it is not identically zero. Due to the singular nature of the equation this property does not follow from available results on stochastic delay differential equations. The key technique is to establish existence and uniqueness of an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).
Citation: Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683
References:
 [1] L. Arnold, W. Kliemann and E. Oeljeklaus, Lyapunov exponents for linear stochastic systems,, in, 1186 (1986), 85. doi: 10.1007/BFb0076836. Google Scholar [2] G. Da Prato and J. Zabczyk, "Ergodicity for Infinite Dimensional Systems,", London Mathematical Society Lecture Note Series, 229 (1996). doi: 10.1017/CBO9780511662829. Google Scholar [3] H. Furstenberg, Noncommuting random products,, Trans. Amer. Math. Soc., 108 (1963), 377. Google Scholar [4] M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations,, Prob. Theory Rel. Fields, 149 (2011), 223. doi: 10.1007/s00440-009-0250-6. Google Scholar [5] P. Hall and C. Heyde, "Martingale Limit Theory and its Application,", Probability and Mathematical Statistics, (1980). Google Scholar [6] R. Z. Has'minskiĭ, Necessary and sufficient conditions for asymptotic stability of linear stochastic systems,, Theory Probability Appl., 12 (1967), 144. Google Scholar [7] R. S. Liptser and A. N. Shiryayev, "Statistics of Random Processes. I. General Theory,", Translated by A. B. Aries, (1977). Google Scholar [8] S. Mohammed, Nonlinear flows of stochastic linear delay equations,, Stochastics, 17 (1986), 207. doi: 10.1080/17442508608833390. Google Scholar [9] S. Mohammed and M. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. I. The multiplicative ergodic theory,, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 69. Google Scholar [10] S. Mohammed and M. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. II. Examples and case studies,, Ann. Probab., 25 (1997), 1210. doi: 10.1214/aop/1024404511. Google Scholar [11] M. Scheutzow, Exponential growth rates for stochastic delay differential equations,, Stoch. Dyn., 5 (2005), 163. doi: 10.1142/S0219493705001468. Google Scholar

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References:
 [1] L. Arnold, W. Kliemann and E. Oeljeklaus, Lyapunov exponents for linear stochastic systems,, in, 1186 (1986), 85. doi: 10.1007/BFb0076836. Google Scholar [2] G. Da Prato and J. Zabczyk, "Ergodicity for Infinite Dimensional Systems,", London Mathematical Society Lecture Note Series, 229 (1996). doi: 10.1017/CBO9780511662829. Google Scholar [3] H. Furstenberg, Noncommuting random products,, Trans. Amer. Math. Soc., 108 (1963), 377. Google Scholar [4] M. Hairer, J. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations,, Prob. Theory Rel. Fields, 149 (2011), 223. doi: 10.1007/s00440-009-0250-6. Google Scholar [5] P. Hall and C. Heyde, "Martingale Limit Theory and its Application,", Probability and Mathematical Statistics, (1980). Google Scholar [6] R. Z. Has'minskiĭ, Necessary and sufficient conditions for asymptotic stability of linear stochastic systems,, Theory Probability Appl., 12 (1967), 144. Google Scholar [7] R. S. Liptser and A. N. Shiryayev, "Statistics of Random Processes. I. General Theory,", Translated by A. B. Aries, (1977). Google Scholar [8] S. Mohammed, Nonlinear flows of stochastic linear delay equations,, Stochastics, 17 (1986), 207. doi: 10.1080/17442508608833390. Google Scholar [9] S. Mohammed and M. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. I. The multiplicative ergodic theory,, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 69. Google Scholar [10] S. Mohammed and M. Scheutzow, Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. II. Examples and case studies,, Ann. Probab., 25 (1997), 1210. doi: 10.1214/aop/1024404511. Google Scholar [11] M. Scheutzow, Exponential growth rates for stochastic delay differential equations,, Stoch. Dyn., 5 (2005), 163. doi: 10.1142/S0219493705001468. Google Scholar
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