[1] Al-Mohy, A. H. & Higham, N. J. (2009) A new scaling and squaring algorithm for the matrix exponential. SIAM J. Matrix Anal. Appl. 31 (3), 970–989.

[2] Al-Mohy, A. H. & Higham, N. J. (2011) Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comp. 33, 488–511.

[3] Anderson, D. (2012) An efficient finite difference method for parameter sensitivities of continuous time Markov chains. SIAM J. Numer. Anal. 50, 2237–2258.

[4] Anderson, D. & Kurtz, T. (2011) Continuous time Markov chain models for chemical reaction networks. In: Koeppl, H., Setti, G., di Bernardo, M., Densmore, D. (editors), Design and Analysis of Biomolecular Circuits, New York: Springer.

[5] Barker, J. R., Nguyen, T. L., Stanton, J. F., Aieta, M. C. C., Gabas, F., Kumar, T. J. D., Li, C. G. L., Lohr, L. L., Maranzana, A., Ortiz, N. F., Preses, J. M. & Stimac, P. J. (2016) *Multiwell-2016 Software Suite*, Technical Report, University of Michigan, Ann Arbor, Michigan, USA.

[6] Bolley, C. & Crouzeix, M. (1978) Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques. RAIRO Anal. Numér. 12, 237–245.

[7] Celledoni, E. & Iserles, A. (2001) Methods for the approximation of the matrix exponential in a Lie-algebraic setting. IMA J. Numer. Anal. 21, 463–488.

[8] Corwin, I. (2014) Macdonald processes, quantum integrable systems and the Kardar–Parisi–Zhang universality class. In: *Proceedings of the International Congress of Mathematicians*.

[9] Corwin, I. (2016) Kardar–Parisi–Zhang Universality. Notices of the American Mathematical Society, March 2016 Not. AMS 63.

[10] Drawert, B., Trogdon, M., Toor, S., Petzold, L. & Hellander, A. (2016) Molns: A cloud platform for interactive, reproducible, and scalable spatial stochastic computational experiments in systems biology using pyurdme. SIAM J. Sci. Comput. 38, C179–C202.

[11] Edelman, A. & Kostlan, E. (1994) *The Road from Kac's Matrix to Kac's Random Polynomials*, Technical Report, University of California, Berkeley.

[12] Edelman, A. & Rao, N. R. (2005) Random matrix theory. Acta Numer. 14, 233–297.

[13] Evans, S. N., Sturmfels, B. & Uhler, C. (2010) Commuting birth-and-death processes. Ann. Appl. Probab. 20, 238–266.

[14] Giles, M. & Glasserman, P. (2006) Smoking adjoints: Fast Monte Carlo Greeks. Risk 19 (1), 88–92.

[15] Gillespie, D. T. (2002) The chemical Langevin and Fokker–Planck equations for the reversible isomerization reaction. J. Phys. Chem. A 106, 5063–5071.

[16] Gorenflo, R., Kilbas, A., Mainardi, F. & Rogosin, S. (2014) Mittag-Leffler Functions, Related Topics and Applications, New York: Springer.

[17] Gunawardena, J. (2012) A linear framework for time-scale separation in nonlinear biochemical systems. PLoS One 7, e36321.

[18] Gunawardena, J. (2014) Time-scale separation: Michaelis and Menten's old idea, still bearing fruit. FEBS J. 281, 473–488.

[19] Hairer, M. (2014) Singular stochastic PDEs. In: *Proceedings of the International Congress of Mathematicians*.

[20] Hellander, A., Klosa, J., Lötstedt, P. & MacNamara, S. (2017) Robustness analysis of spatiotemporal models in the presence of extrinsic fluctuations. SIAM J. Appl. Math. 77 (4), 1157–1183.

[21] Higham, D. J. (2008) Modeling and simulating chemical reactions. SIAM Rev. 50, 347–368.

[22] Hilfinger, A. & Paulsson, J. (2011) Separating intrinsic from extrinsic fluctuations in dynamic biological systems. Proc. Acad. Natl. Sci. 109, 12167–72.

[23] Hilgers, P. V. & Langville, A. N. (2006) The five greatest applications of Markov chains. In: *Proceedings of the Markov Anniversary Meeting*, Boston Press, Boston, MA.

[24] Hochbruck, M. & Lubich, C. (2003) On Magnus integrators for time-dependent Schrödinger equations. SIAM J. Numer. Anal. 41, 945–963.

[25] Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. & Zanna, A. (2000) Lie-group methods. Acta Numer. 9, 215–365.

[26] Jahnke, T. & Huisinga, W. (2007) Solving the chemical master equation for monomolecular reaction systems analytically. J. Math. Biol. 54, 1–26. cited By 97.

[27] Kac, M. (1957) Probability and Related Topics in Physical Sciences, Summer Seminar in Applied Mathematics, American Mathematical Society, Boulder, Colorado.

[28] Kormann, K. & MacNamara, S. (2016) Error control for exponential integration of the master equation, Technical Report.

[29] Kurtz, T. (1980) Representations of Markov processes as multiparameter time changes. Ann. Probab. 8, 682–715.

[31] Macnamara, S. (2015) Cauchy integrals for computational solutions of master equations. ANZIAM J. 56, 32–51.

[32] MacNamara, S., Burrage, K. & Sidje, R. (2008) Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. Sim. 6, 1146–1168.

[33] MacNamara, S., Henry, B. I. & McLean, W. (2016) Fractional Euler limits and their applications. *SIAM J. Appl. Math.*

[35] Magnus, W. (1954) On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7, 649–673.

[36] Moler, C. & Loan, C. V. (2003) Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49.

[37] Munthe-Kaas, H. Z., Quispel, G. R. W. & Zanna, A. (2001) Generalized polar decompositions on Lie groups with involutive automorphisms. Found. Comput. Math. 1, 297–324.

[38] Pavliotis, G. A. & Stuart, A. (2008) Multiscale Methods: Averaging and Homogenization, New York: Springer.

[39] Reddy, S. C. & Trefethen, L. N. (1994) Pseudospectra of the convection-diffusion operator. *SIAM J. Appl. Math.*

[40] Strang, G. & MacNamara, S. (2014) Functions of difference matrices are Toeplitz plus Hankel. SIAM Rev. 56, 525–546.

[41] Timm, C. (2009) Random transition-rate matrices for the master equation. Phys. Rev. E 80, 021140, New Jersey.

[42] Trefethen, L. N. & Chapman, S. J. (2004) Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57, 1233–1264.

[43] Trefethen, L. N. & Embree, M. (2005) Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press.

[44] Weber, M. F. & Frey, E. (2016) Master equations and the theory of stochastic path integrals, Rep Prog Phys. 2017 Apr; 80 (4):046601. doi: 10.1088/1361-6633/aa5ae2.

[45] Wei, J. & Norman, E. (1964) On global representations of the solutions of linear differential equations as a product of exponentials. Proc. Amer. Math. Soc. 15, 327–334.