Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall's lemma and Barkholder-Davis-Gundy's inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.

[1]  X. Mao,et al.  Numerical solutions of stochastic differential delay equations under local Lipschitz condition , 2003 .

[2]  X. Mao,et al.  Exponential Stability of Stochastic Di erential Equations , 1994 .

[3]  Li Ronghua,et al.  Exponential stability of numerical solutions to SDDEs with Markovian switching , 2006 .

[4]  Peter M. Kort,et al.  Capital accumulation under technological progress and learning: A vintage capital approach , 2006, Eur. J. Oper. Res..

[5]  Shaobo Zhou,et al.  Convergence of numerical solutions to neutral stochastic delay differential equations with Markovian switching , 2009 .

[6]  Natali Hritonenko,et al.  The optimal economic lifetime of vintage capital in the presence of operating costs, technological progress, and learning , 2008 .

[7]  Zhanping Wang,et al.  Stability of solution to a class of investment system , 2009, Appl. Math. Comput..

[8]  Chongzhao Han,et al.  Numerical Analysis for Stochastic Age-Dependent Population Equations , 2005, 2006 International Conference on Machine Learning and Cybernetics.

[9]  Qimin Zhang,et al.  Exponential stability of numerical solutions to a stochastic age-structured population system with diffusion , 2008 .

[10]  Chongzhao Han,et al.  Convergence of Numerical Solutions to Stochastic Age-Structured Population System , 2007, 2009 WRI Global Congress on Intelligent Systems.

[11]  Qimin Zhang,et al.  Existence, uniqueness and exponential stability for stochastic age-dependent population , 2004, Appl. Math. Comput..

[12]  Peter M. Kort,et al.  Anticipation effects of technological progress on capital accumulation: a vintage capital approach , 2006, J. Econ. Theory.

[13]  Pauli Murto,et al.  Timing of investment under technological and revenue-related uncertainties , 2007 .