On the Stability of Stochastic Jump Kinetics

Stefan  Engblom

doi:10.4236/am.2014.519300

Stefan Engblom

Abstract: Motivated by the lack of a suitable constructive framework for analyzing popular stochastic models of Systems Biology, we devise conditions for existence and uniqueness of solutions to certain jump stochastic differential equations (SDEs). Working from simple examples we find reasonable and explicit assumptions on the driving coefficients for the SDE representation to make sense. By “reasonable” we mean that stronger assumptions generally do not hold for systems of practical interest. In particular, we argue against the traditional use of global Lipschitz conditions and certain common growth restrictions. By “explicit”, finally, we like to highlight the fact that the various constants occurring among our assumptions all can be determined once the model is fixed. We show how basic long time estimates and some limit results for perturbations can be derived in this setting such that these can be contrasted with the corresponding estimates from deterministic dynamics. The main complication is that the natural path-wise representation is generated by a counting measure with an intensity that depends nonlinearly on the state.

Keywords: Nonlinear Stability, Perturbation, Continuous-Time Markov Chain, Jump Process, Uncertainty, Rate Equation

Cite this paper: Engblom, S. (2014) On the Stability of Stochastic Jump Kinetics. Applied Mathematics, 5, 3217-3239. doi: 10.4236/am.2014.519300.

References

[1] Lestas, I., Vinnicombe, G. and Paulsson, J. (2010) Fundamental Limits on the Suppression of Molecular Fluctuations. Nature, 467, 174-178.
http://dx.doi.org/10.1038/nature09333

[2] Raj, A. and van Oudenaarden, A. (2008) Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences. Cell, 135, 216-226.
http://dx.doi.org/10.1016/j.cell.2008.09.050

[3] Samoilov, M.S. and Arkin, A.P. (2006) Deviant Effects in Molecular Reaction Pathways. Nature Biotechnology, 24, 1235-1240.
http://dx.doi.org/10.1038/nbt1253

[4] Gillespie, D.T. (1976) A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics, 22, 403-434.
http://dx.doi.org/10.1016/0021-9991(76)90041-3

[5] Gillespie, D.T. (1992) A Rigorous Derivation of the Chemical Master Equation. Physica A, 188, 404-425.
http://dx.doi.org/10.1016/0378-4371(92)90283-V

[6] Protter, P.E. (2005) Stochastic Integration and Differential Equations. Number 21 in Stochastic Modelling and Applied Probability. 2nd Edition, Version 2.1, Springer, Berlin.

[7] Applebaum, D. (2004) Lévy Processes and Stochastic Calculus, Volume 93 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge.

[8] Situ, R. (2005) Theory of Stochastic Differential Equations with Jumps and Applications. Mathematical and Analytical Techniques with Applications to Engineering. Springer, New York.

[9] Øksendal, B. and Zhang, T. (2007) The Itô-Ventzell Formula and Forward Stochastic Differential Equations Driven by Poisson Random Measures. Osaka Journal of Mathematics, 44, 207-230.

[10] Hausenblas, E. (2007) SPDEs Driven by Poisson Random Measure with Non Lipschitz Coefficients: Existence Results. Probability Theory and Related Fields, 137, 161-200.

[11] Marinelli, C., Prévôt, C. and R?ckner, M. (2010) Regular Dependence on Initial Data for Stochastic Evolution Equations with Multiplicative Poisson Noise. Journal of Functional Analysis, 258, 616-649.
http://dx.doi.org/10.1016/j.jfa.2009.04.015

[12] Marinelli, C. and R?ckner, M. (2010) Well-Posedness and Asymptotic Behavior for Stochastic Reaction-Diffusion Equations with Multiplicative Poisson Noise. Electronic Journal of Probability, 15, 1529-1555.
http://dx.doi.org/10.1214/EJP.v15-818

[13] Filipovic, D., Tappe, S. and Teichmann, J. (2010) Jump-Diffusions in Hilbert Spaces: Existence, Stability and Numerics. Stochastics, 82, 475-520.
http://dx.doi.org/10.1080/17442501003624407

[14] van Kampen, N.G. (2004) Stochastic Processes in Physics and Chemistry. 2nd Edition, Elsevier, Amsterdam.

[15] Higham, D.J. and Kloeden, P.E. (2005) Numerical Methods for Nonlinear Stochastic Differential Equations with Jumps. Numerische Mathematik, 101, 101-119.
http://dx.doi.org/10.1007/s00211-005-0611-8

[16] Li, T.J. (2007) Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems. Multiscale Modeling & Simulation, 6, 417-436.
http://dx.doi.org/10.1137/06066792X

[17] Engblom, S. (2009) Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics. Multiscale Modeling & Simulation, 8, 46-68.
http://dx.doi.org/10.1137/080733723

[18] Anderson, D.F. (2012) An Efficient Finite Difference Method for Parameter Sensitivities of Continuous Time Markov Chains. SIAM Journal on Numerical Analysis, 50, 2237-2258.
http://dx.doi.org/10.1137/110849079

[19] Kawamura, A. (2009) Lipschitz Continuous Ordinary Differential Equations Are Polynomial-Space Complete. 24th Annual IEEE Conference on Computational Complexity, Paris, 15-18 July 2009, 149-160.

[20] Hutzenthaler, M., Jentzen, A. and Kloeden, P.E. (2011) Strong and Weak Divergence in Finite Time of Euler’s Method for Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients. Proceedings of the Royal Society A, 467, 1563-1576.

[21] Chen, W.Y. and Bokka, S. (2005) Stochastic Modeling of Nonlinear Epidemiology. Journal of Theoretical Biology, 234, 455-470.
http://dx.doi.org/10.1016/j.jtbi.2004.11.033

[22] Ewens, W.J. (2004) Mathematical Population Genetics I. Theoretical Introduction, Volume 27 of Interdisciplinary Applied Mathematics. 2nd Edition, Springer, New York.

[23] Escudero, C., Buceta, J., de la Rubia, F.J. and Lindenberg, K. (2004) Extinction in Population Dynamics. Physical Review E, 69, 021908.

[24] Ethier, S.N. and Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York.
http://dx.doi.org/10.1002/9780470316658

[25] Brémaud, P. (1999) Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Number 31 in Texts in Applied Mathematics. Springer, New York.
http://dx.doi.org/10.1007/978-1-4757-3124-8

[26] Plyasunov, S. (2005) On Hybrid Simulation Schemes for Stochastic Reaction Dynamics.
http://arxiv.org/abs/math/0504477

[27] Brémaud, P. (1981) Point Processes and Queues: Martingale Dynamics. Springer Series in Statistics. Springer, New York.

[28] Daley, D.J. and Vere-Jones, D. (2003) An Introduction to the Theory of Point Processes, Volume I: Elementary Theory and Methods. 2nd Edition, Springer, New York.

[29] Kurtz, T.G. (1978) Strong Approximation Theorems for Density Dependent Markov Chains. Stochastic Processes and Their Applications, 6, 223-240.
http://dx.doi.org/10.1016/0304-4149(78)90020-0

[30] Rathinam, M., Sheppard, P.W. and Khammash, M. (2010) Efficient Computation of Parameter Sensitivities of Discrete Stochastic Chemical Reaction Networks. Journal of Chemical Physics, 132, 034103.
http://dx.doi.org/10.1063/1.3280166

[31] Kurtz, T.G. (1982) Representation and Approximation of Counting Processes. In: Fleming, W.H. and Gorostiza, L.G., Eds., Advances in Filtering and Optimal Stochastic Control, Vol. 42, Lecture Notes in Control and Information Sciences, Springer, Berlin, 177-191.

[32] Str?m, T. (1975) On Logarithmic Norms. SIAM Journal on Numerical Analysis, 12, 741-753.
http://dx.doi.org/10.1137/0712055

[33] Briat, C., Gupta, A. and Khammash, M. (2014) A Scalable Computational Framework for Establishing Long-Term Behavior of Stochastic Reaction Networks. PLoS Computational Biology, 10, e1003669.

[34] Rathinam, M. (2014) Moment Growth Bounds on Continuous Time Markov Processes on Non-Negative Integer Lattices. To appear in Quart. Appl. Math.
http://arxiv.org/abs/1304.5169

[35] Meyn, S.P. and Tweedie, R.L. (1993) Stability of Markovian Processes III: Foster-Lyapunov Criteria for ContinuousTime Processes. Advances in Applied Probability, 25, 518-548.
http://dx.doi.org/10.2307/1427522