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Henri Schurz

Henri Schurz, Professor

Professor; Ph.D., Humboldt University Berlin, 1997. Stochastic analysis, probability and math. statistics, stochastic differential equations (ordinary, partial), stochastic difference equations, stability, Lyapunov functionals, numerical methods, convergence and stability, balanced methods, numerical integration, mathematical finance, random vibrations, noisy circuits, noisy beams and plates, mathematical epidemiology and biology.

Research Interests

My research focuses on stochastic analysis and its applications. In particular, I am working on the qualitative behavior of stochastic dynamical systems such as systems of stochastic differential equations (ordinary, partial, functional), their numerical methods and stochastic difference equations. The applications range from mechanical engineering (random vibrations, oscillators), laser physics (injection locking), finance (interest rates, dynamic asset pricing), marketing (innovation diffusion) to biology and ecology (random epidemics, population models).

 

Selected Publications

SIU Henri Schurz

Neckers 271
618-453-6577
hschurz@math.siu.edu
Personal Website

  1. H. Schurz: `Logistic stochastic differential equations with power-law', J. Comput. Math., 2024 (accepted).
  2. T. Chandrasena, S. Chandrasena and H. Schurz: `Analysis of stochastic SEIR(S) model with random total populations and variable diffusion rates', Electr. J. Qual. Theory Differ. Equat. {\bf 2024} (5), p. 1-26, 2024. https://doi.org/10.14232/ejqtde.2024.1.55.
  3. H. Schurz: `A brief review on stability investigations of numerical methods
    for systems of stochastic differential equations', AIMS Networks Heterogeneous Media, {\bf 19} (1), p. 355-383, 2024. https://doi.org/10.3934/nhm.2024016.
  4. B. Belinskiy and H. Schurz: `Analysis of damped rotating disk-beam system
    excited by $L^2$-regular, velocity-dependent, space-time random noise',
    Int. J. Appl. Comput. Math. {\bf 10} (12), p. 1-30, 2024. https://doi.org/10.1007/s40819-023-01653-w.
  5. H. Schurz: `Nonlinear stochastic wave Equations in 1D with fractional Laplacian,
    power-law nonlinearity and additive Q-regular noise', Res. Appl. Math. {\bf
    20}, ID 100411, p. 1-30, 2023. https://doi.org/10.1016/j.rinam.2023.100411.
  6. A. Rodkina and H. Schurz: `A martingale approach to asymptotic stability of
    discrete SDE with bounded noise in $\R^1$', Dynam. Syst. Applic. {\bf 32}, p. 129-155, 2023. https://doi.org/10.46719/dsa2023.32.08.
  7. H. Schurz and Q.S. Alharbi: `Existence, uniqueness, and
    stability of Fourier series solutions of stochastic wave equations with cubic nonlinearities in 3D',
    Nonlinear Analysis {\bf 227} (2023), ID 113169, February 2023, p. 1-37. 
    https://doi.org/10.1016/j.na.2022.113169. 
  8. H. Schurz: `Weak convergence of balanced implicit methods for It\^o-type stochastic
    differential equations with variable step sizes', Res. Math. {\bf 9}:1, p. 1-24, 2022. Article No. 2130253. https://doi.org/10.1080/27684830.2022.2130253.
  9. H. Schurz and A. Talafha: `Existence, uniqueness, and energy of
    a modified stochastic Sine-Gordon equation with power-law nonlinearity in 1D', Int. J. Appl. Comput. Math. {\bf 7}, Article 126, p. 1-22, 2021. https://doi.org/10.1007/s40819-021-01067-6.
  10. H. Schurz: `Numeric and dynamic B-stability, exact-monotone and asymptotic two-point behavior of theta methods for stochastic differential equations' , J. Stoch. Anal. {\bf 2} (2), Article 7, p. 1-25, 2021. https://doi.org/10.31390/josa.2.2.07.
  11. H. Schurz and K. Tosun: `Stability of stochastic SIS model with disease deaths and variable diffusion rates', Electron. J. Qual. Theory Differ. Equ. {\bf 2019} (14), p. 1-24, 2019. https://doi.org/10.14232/ejqtde.2019.1.14.

  12. G. Budzban, R.H. Hughes and H. Schurz (eds.): `Probability on Algebraic and Geometric Structures', Contemporary Mathematics Vol. 668, American Mathematical Society, Providence, Rhode Island, pp. 222, 2016.
    https://doi.org/10.1090/conm/668.

  13. H. Schurz and A. Talafha: `Existence, uniqueness, and energy of a modified stochastic Sine-Gordon equation on one-dimensional domain', AMS Contemp. Math. {\bf 668}, p. 179-197, 2016. https://doi.org/10.1090/conm/668/13406.
  14. H. Schurz and K. Tosun: `Stochastic asymptotic stability of SIR model with variable diffusion rates', J. Dynamics Differ. Equat. {\bf 27} (1), p. 69-82, 2015. https://doi.org/doi:10.1007/s10884-014-9415-9.
  15. H. Schurz and H.M. Hazaimeh: `Existence, uniqueness and stability of stochastic wave equation with cubic nonlinearities in two dimensions', J. Math. Anal. Appl. {\bf 418} (2), p. 775-795, 2014. https://doi.org/10.1016/j.jmaa.2014.04.011.
  16. S. Xu, H. Schurz and Y. Chen: `Parameter optimization of relaxed ordered subsets pre-computed back projection (BP) based penalized-likelihood (OS-PPL) reconstruction in limited-angle X-ray tomography', Computerized medical imaging and graphics: the official journal of the Computerized Medical Imaging Society, Comput Med Imaging Graph. 2013 Jun; {\bf 37} (4), p. 304-312, 2013. https://doi.org/10.1016/j.compmedimag.2013.04.005.
  17. H. Schurz: `Stochastic heat equations with cubic nonlinearity and Q-regular additive noise in 2D'}, Discrete Contin. Dyn. Syst. Supplemental {\bf 2013}, p. 673-684, 2013. https://doi.org/10.3934/proc.2013.2013.673.
  18. Z. Liu, Q. Wang, N. Ansari, H. Schurz: `Nonlinear optimal control of stochastic recurrent neural
    networks with multiple time delays', Proceedings of the American Control
    Conference, IEEE COMPUTER SOC, Los Alamitos, p. 6424-6429, 2012. https://doi.org/10.1109/acc.2012.6314714.
  19. Z. Liu, H. Schurz, N. Ansari: `Theoretic design of differential minimax controllers for stochastic cellular neural networks', Neural Networks {\bf 26}, p. 110-117, 2012. https://doi.org/10.1016/j.neunet.2011.09.003.
  20. H. Schurz: `Almost sure convergence and asymptotic stability of systems of linear stochastic difference equations in $R^d$ driven by $L^2$-martingales', J. Difference Equ. Appl. {\bf 18}, No. 8, p. 1333-1343, 2012. https://doi.org/10.1080/10236198.2011.561796.
  21. B. Belinskiy and H. Schurz: `Undamped nonlinear beam excited by additive $L^2$-regular noise', J. Comput. Appl. Math. {\bf 235}, No. 17, 5284-5306, 2011. https://doi.org/10.1016/j.cam.2011.05.039.
  22. H. Schurz: `Stochastic wave equations with cubic nonlinearity and
    Q-regular additive noise in $R^2$', Discrete Contin. Dyn. Syst. Supplemental
    {\bf 2011}, Vol. 2, ed. Wei Feng et al, p. 1299-1308, 2011. https://doi.org/10.3934/proc.2011.2011.1299.
  23. J.A.D. Appleby, A. Rodkina and H. Schurz: `On an exponential martingale approach to almost sure stability of Ito SDEs in $R^1$', Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. {\bf 18}, No. 4, p. 471-484, 2011.
  24. S. Gan, H. Schurz and H. Zhang: `Mean square convergence of stochastic theta-methods for nonlinear neutral stochastic differential delay equations', Int. J. Numer. Anal. Model. {\bf 8}, No. 2, p. 201-213, 2011.
  25. H. Schurz: `Almost sure asymptotic stability and convergence of stochastic Theta methods applied to systems of linear SDEs in $R^d$', Random Oper. Stoch. Equ. {\bf 19} (2), p. 111-129, 2011. https://doi.org/10.1515/ROSE.2011.007.
  26. H. Schurz: `Algorithmic solution of stochastic differential equations', Algorithms {\bf 3} (3), p. 216-223, 2010. https://doi.org/10.3390/a3030216.
  27. H. Schurz: `Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in $R^1$',
    Electron. J. Differ. Equ. Conf. {\bf 19}, p. 221-233, 2010. https://ejde.math.txstate.edu/conf-proc/19/s1/schurz.pdf.
  28. J.A.D. Appleby, A. Rodkina and H. Schurz: Non-positivity and oscillations of solutions of nonlinear stochastic difference equations with state-dependent noise', J. Difference Equ. Appl. {\bf 16} (7), p. 807-830, 2010.https://doi.org/10.1080/10236190802495303.
  29. J.A.D. Appleby, A. Rodkina and H. Schurz: `On the oscillation ofsolutions of stochastic difference equations', Mat. Ensen. Univ. (N. S.) {\bf 17} (2),
    p. 1-10, 2009.
  30. A. Rodkina and H. Schurz: `On positivity and boundedness of solutions of nonlinear stochastic difference equations', Discrete Contin. Dyn. Syst. -Supplement {\bf 2009}, p. 640-649, 2009.
  31. Ziqian Liu, Qunjing Wang and H. Schurz: `Inverse optimal noise-to-state stabilization of stochastic recurrent neural networks driven by noise of unknown covariance', Optimal Control Appl. Methods {\bf 30} (2), p. 163-178, 2009.
  32. H. Schurz: `New stochastic integrals, oscillation theorems and energy identities', Commun. Appl. Anal. {\bf 13} (2), p. 181-194, 2009.
  33. H. Schurz: `Verification of Lyapunov functions for the analysis of stochastic Lienard equations', J. Sound and Vibration {\bf 325}, p. 938-949, 2009. https://doi.org/10.1016/j.jsv.2009.04.004.
  34. I. Bashkirtseva, L.B. Ryashko and H. Schurz: `Analysis of noise-induced transitions for Hopf system with additive and multiplicative random disturbances', Chaos Solitons Fractals {\bf 39} (1), p. 72-82, 2009.
  35. A. Rodkina, H. Schurz and L. Shaikhet, `Almost sure stability of some stochastic dynamical systems with memory', Discrete Contin. Dynam. Syst. A {\bf 21} (2), p. 571-593, 2008.
  36. H. Schurz: `How rates of $L^p$-convergence carry over to numerical approximations of some convex, non-smooth functionals of SDEs', Int. J. Numer. Anal. Model. {\bf 5} (1), p. 55-72, 2008.
  37. H. Schurz: `Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise', Discrete Cont. Dynam. Syst. Series S {\bf 1} (2), p. 353-363, 2008. https://10.3934/dcdss.2008.1.353.
  38. E. Moro and H. Schurz: `Boundary preserving semi-analytical numerical algorithms for stochastic differential equations', SIAM J. Sci. Comput. {\bf 29} (4), p. 1525-1549, 2007.https://doi.org/10.1137/05063725X.

  1. H. Bessaih and H. Schurz: `Upper bounds on the rate of convergence of
    truncated stochastic infinite-dimensional differential systems with $H$-regular
    noise', J. Comput. Appl. Math. {\bf 208} (2), p. 354-361, 2007.
  2. H. Schurz: `A numerical method for nonlinear stochastic
    wave equations in $\R^1$', Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. {\bf
    14} (S2), p. 74-78, 2007 (DCDIS A Supplement, "Advances in Dynamical Systems").
  3. H. Schurz: `Nonlinear stochastic wave equations in $\R^1$ with power-law nonlinearity and additive space-time noise',
    Contemporary Math. {\bf 440}, p. 223-242, 2007. https://doi.org/10.1090/conm/440.
  4. H. Schurz: `Modeling, analysis and discretization of stochastic logistic equations'}, Int. J. Numer. Anal. Model. {\bf 4} (2), p. 178-197, 2007.
  5. H. Schurz: Applications of numerical methods and its analysis for systems of stochastic differential equations, Bull. Karela Math. Soc. 4 (1) (2007), 1-85.
  6. H. Schurz: Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise, J. Math. Anal. Appl. 332 (1) (2007) 334-345.
  7. J. Appleby, A. Rodkina and H. Schurz: `Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic
    differential equations', Discrete Cont. Dynam. Syst. {\bf B 6} (4), p. 667-696, 2006. https://doi.org/10.3934/dcdsb.2006.6.667.
  8. C. Kahl and H. Schurz: `Balanced implicit Milstein methods for ordinary SDEs', Monte Carlo Methods Appl. {\bf 12} (2), p. 143-170, 2006. https://doi.org/10.1515/156939606777488842.
  9. H. Schurz: An axiomatic approach to numerical approximations of stochastic processes, Int. J. Numer. Anal. Model. 3 (4) (2006) 459-480.
  10. H. Schurz: Stochastic $\alpha$-calculus, a fundamental theorem and Burkholder-Davis-Gundy-type estimates, Dynam. Syst. Applic. 15 (2) (2006) 241-268.
  11. H. Schurz: `On estimation of $L^p$-errors of It\^o-Riemann-type numerical
    quadratures for stochastic integrals along Wiener paths', Proceedings of Neural, Parallel and Scientific Computations {\bf 3}, Dynamic Publishers, Atlanta, p. 221-225, 2006.
  12. H. Schurz: Stability of numerical methods for ordinary SDEs along Lyapunov-type and other functions with variable step sizes, Electr. Trans. Numer. Anal. 20 (2005), p. 27-49.
  13. H. Schurz: Dissipation of mean energy of discretized linear oscillators under random perturbations, Discrete Contin. Dyn. Syst. (Suppl. Vol. 2005), p. 778-783, 2005.
  14. H. Schurz: `Convergence and stability of balanced implicit methods for SDEs with variable step sizes', Int. J. Numer. Anal. Model. {\bf 2}(2), p. 197-220, 2005.
  15. A. Rodkina and H. Schurz: `Almost sure asymptotic stability of drift-implicit$\theta$-methods for bilinear ordinary stochastic differential equations in $R^1$',Comput. Appl. Math. {\bf 180} (1), p. 13-31, 2005.https://doi.org/https://doi.org/10.1016/j.cam.2004.09.060
  16. A. Rodkina and H. Schurz: `On global asymptotic stability of solutions to some in-arithmetic-mean-sense drift-monotone stochastic difference equations', Int. J. Numer. Anal. Model. {\bf 2} (3), p. 355-366, 2005.
  17. A. Rodkina and H. Schurz: `A theorem on asymptotic stability of solutions to nonlinear stochastic difference equations with Volterra type noises', SACTA {\bf 6} (1), p. 23-34, 2004.
  18. A. Rodkina and H. Schurz: `Global asymptotic stability of solutions to cubic stochastic difference equations'}, Adv. Diff. Equat. {\bf 1} (3),p. 249-260, 2004.
  19. H. Schurz: `General theorems on numerical approximation of stochastic processes on the Hilbert space $H_2([0,T],\mu,R^d)$}', Electr. Trans. Numer. Anal.{\bf 16}, p. 50-69, 2003.
  20. H. Schurz: `Remarks on Taylor series expansions and conditional expectations for Stratonovich SDEs with complete $V$-commutativity', Stoch. Anal. Appl. {\bf 21} (4), p. 865-894, 2003. https://doi.org/10.1081/sap-120022867.
  21. H. Schurz: `Numerical analysis of stochastic differential equations', in Handbook of Stochastic Analysis and Applications, ed. by D. Kannan and V. Lakshmikantham, Marcel Dekker, Basel, p. 237-359, 2002. https://doi.org/10.1201/9781482294705.
  22. H. Schurz: Moment attractivity, stability and contractivity exponents of stochastic dynamical systems, Discrete Contin. Dyn. Syst. 7 (3), p. 487-515, 2001.
  23.  H. Schurz: `Numerical analysis of stochastic differential equations', in Handbook of Stochastic Analysis and Applications, ed. by D. Kannan and V. Lakshmikantham, Marcel Dekker, Basel, p. 237-359, 2002. https://doi.org/10.1201/9781482294705.
  24. H. Schurz: `On moment-dissipative stochastic dynamical
    systems', Dynam. Syst. Appl. {\bf 10} (1), p. 11-44, 2001.
  25. H. Schurz: `Preservation of probabilistic laws through Euler methods for Ornstein--Uhlenbeck process', Stoch. Anal. Appl. {\bf 17} (3), p. 463-486, 1999.https://doi.org/10.1080/07362999908809613.
  26. H. Schurz: `The invariance of asymptotic laws of linear stochastic systems under discretization', Z. Angew. Math. Mech. {\bf 79} (6), p. 375-382, 1999. https://doi.org/10.1002/(SICI)1521-4001(199906)79:6.
  27. G.N. Mil'shtein, E. Platen, H. Schurz: `Balanced implicit methods for stiff stochastic systems', SIAM J. Numer. Anal. {\bf 35} (3), p. 1010-1019, 1998. https://doi.org/10.1137/s0036142994273525.
  28. L.B. Ryashko and H. Schurz: `Mean square stability analysis of some stochastic systems', Dynam. Syst. Appl. {\bf 6} (2), p. 165-190, 1997.
  29. H. Schurz: Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications, Logos-Verlag, Berlin, pp. 288, 1997 (Research Monograph, ISBN 3-931216-94-2).
  30. Karmeshu, H. Schurz: `Moment evolution of the outflow-rate from nonlinear conceptual reservoirs', in Surface Water-Hydrology {\bf 1}, ed. by V. P. Singh and B. Kumar, Kluwer Academic Publishers, Dordrecht, p. 403-413, 1996.
  31. H. Schurz: `Modelling and analysis of stochastic innovation diffusion', Z. Angew. Math. Mech. {\bf 76} (Suppl. 3, I-XV), p. 366-369, 1996.
  32. H. Schurz: `Numerical regularization for SDEs: Construction of nonnegative solutions', Dynam. Syst. Appl. {\bf 5} (1), p. 323-352, 1996.
  33. H. Schurz: `Asymptotical mean square stability of an equilibrium point of some linear numerical solutions', Stoch. Anal. Appl. {\bf 14} (3), p. 313-354, 1996. https://doi.org/10.1080/07362999608809442.
  34. P.E. Kloeden, E. Platen, H. Schurz, M. S{\o}rensen: `On effects of discretization on estimators of drift parameters for diffusion processes', J. Appl. Prob. {\bf 33}, p. 1061-1076, 1996. http://dx.doi.org/10.1017/S0021900200100488.
  35. Karmeshu, H. Schurz: `Effects of distributed delay on the stability of structures under seismic excitation and multiplicative noise', SADHANA {\bf 20} (2-4), p. 451-474, 1995.
  36. Karmeshu, H. Schurz: `Stochastic stability of structures under active control with distributed time delays', Applications of Statistics and Probability: Civil Engineering Reliability and Risk Analysis, Proc. ICASP 7 (Paris, July 1995), ed. M. Lemaire, J.-L. Favre, A. Mebarki, A.A. BALKEMA Publishers, Rotterdam, p. 1111-1119, 1995.
  37. S.S. Artemiev, H. Schurz: `Stiff systems of stochastic differential equations with small noise and their numerical solution' (in Russian), Report No. 1039, Vychisl. Tsentr Ross. Akad. Nauk Sib. Otd., Novosibirsk, p. 1-24, 1995.
  38. T.A. Averina, S.S. Artem'ev, H. Schurz: `Numerical analysis of stochastic auto-oscilla\-ting systems', Bull. Novosib. Comput. Cent., Ser. Numer. Anal. {\bf 6}, p. 9-27, 1995.
  39. H. Schurz: `Remarks on preservation of asymptotical mean square and pathwise properties of bilinear SDE by numerical methods', Proceedings AMCA-95, Internat. Conf. Advanced Mathematics, Computations and Applications (Novosibirsk, June 1995), NCC Publisher, Novosibirsk, p. 291-292, 1995.
  40. P.E. Kloeden, E. Platen, H. Schurz: Numerical solution of stochastic differential equations through computer experiments', Universitext, Springer, New York, pp. 292, 1994 (2nd edition 1997, 3rd printing 2003). (Advanced textbook, ISBN 3-540-57074-8). https://doi.org/10.1007/978-3-642-57913-4.
  41. P.E. Kloeden, E. Platen, H. Schurz: `Higher order approximate Markov chain filters', in Stochastic Processes: A Festschrift in Honour of Gopinath Kallianpur, ed. by S. Cambanis et al., Springer, New York, p. 181-190, 1993. https://doi.org/10.1007/978-1-4615-7909-0_21.
  42. P.E. Kloeden, E. Platen, H. Schurz: `Effective simulation of optimal
    trajectories in stochastic control', in Optimization: Techniques and
    Applications, ed. by K.H. Phua et al., Vol. {\bf 2}, World Scientific,
    Singapore, p. 633-644, 1992.
  43. H. Schurz: `Approximate Markov chain filtering', in Proc. 1st Workshop on Stochastic Numerics (Ed. by E. Platen, Gosen, September 1992), in Report No. {\bf 21}, IAAS, Berlin, p. 83-84, 1992.
  44. P.E. Kloeden, E. Platen, H. Schurz: `The numerical solution of nonlinear stochastic dynamical systems: a brief introduction', J. Bifur. Chaos {\bf 1} (2), p. 277-286, 1991. https://doi.org/10.1142/s021812749100021x.