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Part of the book series: Applications of Mathematics ((SMAP,volume 23))

Abstract

This chapter consists of a selection of examples from the literature of applications of stochastic differential equations. These are taken from a wide variety of disciplines with the aim of stimulating the readers’ interest to apply stochastic differential equations in their own particular fields of interest and of providing an indication of how others have used models described by stochastic differential equations. Here we simply describe the equations and refer readers to the original papers for the justification and analysis of the models.

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Bibliographical Notes

  • Chandrasekhar, S. (1954). Stochastic problems in physics and astronomy. Rev. Modern Phys. 15, 2–89.

    Google Scholar 

  • Bellman, R. (1964). Stochastic processes in mathematical physics and engineering. In Proc. Sympos. Appl. Math., Volume 16. Amer. Math. Soc. Providence RI.

    Google Scholar 

  • Ricciardi, L. M. (1977). Diffusion Processes and Related Topics in Biology,Volume 14 of Lecture Notes in Biomath. Springer.

    Google Scholar 

  • Schuss, Z. (1980). Theory and Applications of Stochastic Differential Equations. Wiley Ser. Probab. Statist. Wiley, New York.

    Google Scholar 

  • van Kampen, N. G. (1981b). Stochastic Processes in Physics and Chemistry,Volume 888 of Lecture Notes in Math. North Holland.

    Google Scholar 

  • Gardiner, C. W. (1983). Handbook of Stochastic Methods. For Physics, Chemistry and the Natural Sciences,Volume 13 of Springer Ser. Synergetics. Springer.

    Google Scholar 

  • Horsthemke, W. and R. Lefever (1984). Noise Induced Transitions. Springer.

    Google Scholar 

  • Sobczyk, K. (1991). Stochastic Differential Equations: With Applications to Physics and Engineering,Volume 40 of Math. Appl. (East European Ser.). Kluwer.

    Google Scholar 

  • Gard, T. C. and D. Kannan (1976). On a stochastic differential equation modeling of prey-predator evolution. J. Appl. Probab. 13 (3), 429–443.

    Article  MathSciNet  MATH  Google Scholar 

  • Schoener, T. W. (1973). Population growth regulated by intraspecific competition for energy or time: Some simple representations. Theor. Pop. Biol. 4, 56–84.

    Article  Google Scholar 

  • Turelli, M. (1977). Random environments and stochastic calculus. Theory Pop. BioL 12, 140–178.

    Article  MathSciNet  MATH  Google Scholar 

  • Gard, T. C. (1988). Introduction to Stochastic Differential Equations. Marcel Dekker, New York.

    MATH  Google Scholar 

  • Arnold, L., W. Horsthemke, and R. Lefever (1978). White and coloured external noise and transition phenomena in nonlinear systems. Z. Phys. B29, 867.

    Google Scholar 

  • Ehrhardt, M. (1983). Invariant probabilities for systems in a random environment–with applications to the Brusselator. Bull. Math. BioL 45, 579–590.

    MathSciNet  MATH  Google Scholar 

  • Kimura, M. and T. Ohta (1971). Theoretical Aspects of Population Genetics. Princeton Univ. Press, Princeton.

    Google Scholar 

  • Levkinson, B. (1977). Diffusion apprcocimations in population genetics - how good are they? In Proc. Conf. Stochastic Differential Equations and Applications. Academic Press, New York.

    Google Scholar 

  • Shiga, T. (1985). Mathematical results on the stepping stone model of population genetics. In T. Ohta and K. Aoki (Eds.), Population Genetics and Molecular Evolution, pp. 267–279. Springer.

    Google Scholar 

  • Schöner, G., H. Haken, and J. A. S. Kelso (1986). A stochastic theory of phase transitions in human hand movement. Biol. Cybern. 53, 247–257.

    Article  MATH  Google Scholar 

  • Lansky, P. and V. Lanska (1987). Diffusion approximation of the neuronal model with synaptic reversal potentials. Biol. Cybern. 56, 19–26.

    Article  MathSciNet  MATH  Google Scholar 

  • Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Economic Theory 3 (4), 373–413.

    Article  MathSciNet  Google Scholar 

  • Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. J. Political Economy 81, 637–659.

    Article  Google Scholar 

  • Kloeden, P. E., E. Platen, and I. Wright (1992). The approximation of multiple stochastic integrals. Stochastic Anal. AppL 10 (4), 431–441.

    Article  MathSciNet  MATH  Google Scholar 

  • Karatzas, I. and S. E. Shreve (1988). Brownian Motion and Stochastic Calculus. Springer.

    Google Scholar 

  • Karatzas, I. (1989). Optimization problems in the theory of continuous trading. SIAM J. Control Optim. 27, 1221–1259.

    Article  MathSciNet  MATH  Google Scholar 

  • Musiela, M. and M. Rutkowski (1997). Martingale Methods in Financial Modelling. Theory and Applications, Volume 36 of Appt. Math. Springer.

    Google Scholar 

  • Dothan, M. U. (1990). Prices in Financial Markets. Oxford Univ. Press.

    MATH  Google Scholar 

  • Johnson, H. and D. Shanno (1987). Option pricing when the variance is changing. J. Financial and Quantitative Analysis 22, 143–151.

    Article  Google Scholar 

  • Hofmann, N., E. Platen, and M. Schweizer (1992). Option pricing under incomplete, ness and stochastic volatility. Math. Finance 2 (3), 153–187.

    Article  MATH  Google Scholar 

  • Obukhov, A. M. (1959). Description of turbulence in terms of Lagrangian variables. Adv. Geophys. 6, 113–115.

    Article  Google Scholar 

  • Bywater, R. J. and P. M. Chung (1973). Turbulent flow fields with two dynamically significant scales. AIAA papers, 73–646, 991–10.

    Google Scholar 

  • Yaglom, A. M. (1980). Applications of stochastic differential equations to the description of turbulent equations. In Stochastic Differential Systems,Volume 25 of Lecture Notes in Control and Inform. Sci.,pp. 13–27. Springer.

    Google Scholar 

  • Durbin, P. A. (1983). Stochastic differential equations and turbulent dispersion. NASA RP-1103.

    Google Scholar 

  • Drummond, I. T., S. Duane, and R. R. Horgan (1984). Scalar diffusion in simulated helical turbulence with molecular diffusivity. J. Fluid Mech. 138, 75–91.

    Article  MATH  Google Scholar 

  • Pope, S. B. (1985). PDF methods in turbulent reactive flows. Prog. Energy Comb. Sc. 11, 119–192.

    Article  MathSciNet  Google Scholar 

  • Le Gland, F. (1981). Estimation de paramétres dans les processus stochastiques en observation incompléte: Application d un probléme de radio-astronomie. Ph. D. thesis, Dr. Ing. Thesis, Univ. Paris IX (Dauphine).

    Google Scholar 

  • Pardoux, E. and M. Pignol (1984). Study of the stability of the solution of a bilinear sde with periodic coefficients. application to the motion of helicopter blades. In Analysis and Optimization of Systems, Part 2,Volume 63 of Lecture Notes in Control and Inform. Sci.,pp. 92–103. Springer. (in French).

    Google Scholar 

  • Pardoux, E. and D. Talay (1988). Stability of nonlinear differential systems with parametric excitation. In G. I. Schueller and F. Ziegler (Eds.), Proc. IUTAM Sympos., Innsbruck 1987, Nonlinear Stochastic Dynamic Engineering Systems. Springer.

    Google Scholar 

  • Pardoux, E. and D. Talay (1988). Stability of nonlinear differential systems with parametric excitation. In G. I. Schueller and F. Ziegler (Eds.), Proc. IUTAM Sympos., Innsbruck 1987, Nonlinear Stochastic Dynamic Engineering Systems. Springer.

    Google Scholar 

  • Sagirow, P. (1970). Stochastic Methods in the Dynamics of Satellites,Volume 57 of ICMS Lecture Notes. Springer.

    Google Scholar 

  • Balakrishnan, A. V. (1986). On a class of stochastic differential equations which do not satisfy Lipschitz conditions. In Stochastic Differential Systems,Volume 78 of Lecture Notes in Control and Inform. Sci.,pp. 27–35. Springer.

    Google Scholar 

  • Harris, C. J. (1976). Simulation of nonlinear stochastic equations with applications in modelling water pollution. In C. A. Brebbi (Ed.), Mathematical Models for Environmental Problems, pp. 269–282. Pentech Press, London.

    Google Scholar 

  • Harris, C. J. (1977). Modelling, simulation and control of stochastic systems with application in wastewater treatment. Internat. J. Systems. Sci. 8 (4), 393–411.

    Article  Google Scholar 

  • Harris, C. J. (1979). Simulation of multivariate nonlinear stochastic systems. Internat. J. Numer. Methods Engrg. 14, 37–50.

    Article  MATH  Google Scholar 

  • Miwail, R., T. Ognean, and S. Straja (1987). Stochastic modelling of a biochemical reactor. Hungar. J. Indus. Chem. i5, 55–62.

    Google Scholar 

  • Finney, B. A., D. S. Bowles, and M. P. Windham (1983). Random differential equations in river quality modelling. Water Resources Res. 18, 122–134.

    Article  Google Scholar 

  • Unny, T. E. and K. Karmeshu (1983). Stochastic nature of outputs from conceptual reservoir model cascades. J. Hydrology 68, 161–180.

    Google Scholar 

  • Unny, T. E. (1984). Numerical integration of stochastic differential equations in catchment modelling. Water Res. 20, 360–368.

    Article  Google Scholar 

  • Bodo, B. A., M. E. Thompson, and T. E. Unny (1987). A review on stochastic differential equations for applications in hydrology. J. Stochastic Hydrology and Hydraulics 1, 81–100.

    Article  MATH  Google Scholar 

  • Haghighat, F., M. Chandrashekar, and T. E. Unny (1987). Thermal behaviour in buildings under random conditions. Appl. Math. Modelling 11, 349–356.

    Article  Google Scholar 

  • Haghighat, F., P. Fazio, and T. E. Unny (1988). A predictive stochastic model for indoor air quality. Building and Environment 23, 195–201.

    Article  Google Scholar 

  • Bolotin, V. V. (1960). Statistical theory of the seismic design of structures. In Proc. 2nd WEEE Japan, pp. 1365.

    Google Scholar 

  • Shinozuka, M. (1972). Monte-Carlo solution of structural dynamics. J. Computer Structures 2, 855–874.

    Article  Google Scholar 

  • Kozin, F. (1977). An approach to characterizing, modelling and analyzing earthquake excitation records. Volume 225 of CISM Lecture Notes, pp. 77–109. Springer.

    Google Scholar 

  • Fischer, U. and M. Engelke (1983). Polar cranes for nuclear power stations in earthquake areas. In K. Hennig (Ed.), Random Vibrations and Reliability, pp. 45–54. Akademie Verlag, Berlin.

    Google Scholar 

  • Shinozuka, M. (1971). Simulation of multivariate and multidimensional random differential processes. J. Acoust. Soc. Amer. 49, 357–367.

    Article  Google Scholar 

  • Shinozuka, M. (1972). Monte-Carlo solution of structural dynamics. J. Computer Structures 2, 855–874.

    Article  Google Scholar 

  • Shinozuka, M. and C. M. Jan (1972). Digital simulation of random processes and its applications. J. Sound Vibrat. 25, 111–128.

    Article  Google Scholar 

  • Shinozuka, M. and Y. K. Wen (1972). Monte-Carlo solution of nonlinear vibrations. AIAA J. 10, 37–40.

    Article  MATH  Google Scholar 

  • Hennig, K. and G. Grunwald (1984). Treatment of flow induced pendulum oscillations. Kernenergie 27, 286–292.

    Google Scholar 

  • Friedrich, H., C. Lange, and E. Lindner (1987). Simulation of stochastic processes. Akad. der Wiss. der DDR, Institut für Mechanik, FMC Series no. 35, Karl-Marx-Stadt (Chemnitz), 154 pp.

    Google Scholar 

  • Wedig, W. (1988). Pitchfork and Hopf bifurcations in stochastic systems - effective methods to calculate Lyapunov exponents. In P. Krée and W. Wedig (Eds.), Effective Stochastic Analysis. Springer.

    Google Scholar 

  • Sobczyk, K. (1986). Modelling of random fatigue crack growth. Eng. Fracture Mech. 24, 609–623.

    Article  Google Scholar 

  • Gragg, R. F. (1981). Stochastic switching in absorptive optical bistability. Ph. D. thesis, Univ. Texas at Austin.

    Google Scholar 

  • Horsthemke, W. and R. Lefever (1984). Noise Induced Transitions. Springer.

    Google Scholar 

  • Smith, A. M. and C. W. Gardiner (1988). Simulation of nonlinear quantum damping using the positive representation. Univ. Waikato Research Report.

    Google Scholar 

  • Horsthemke, W. and R. Lefever (1984). Noise Induced Transitions. Springer.

    Google Scholar 

  • Ivanov, M. F. and V. F. Shvec (1979). Numerical solution of stochastic differential equations for modeling collisions in a plasma. Chisl. Metody Mekh. Sploshn. Sredy 10, 64–70. (in Russian).

    Google Scholar 

  • Fogelson, A. L. (1984). A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting. J. Comput. Phys. 56 (1), 111–134.

    Article  MathSciNet  MATH  Google Scholar 

  • Veuthey, A. L. and J. Stucki (1987). The adenylate kinase reaction acts as a frequency filter towards fluctuations of ATP utilization in the cell. Biophys. Chem. 26, 19–28.

    Article  Google Scholar 

  • Horsthemke, W. and R. Lefever (1984). Noise Induced Transitions. Springer.

    Google Scholar 

  • Geman, S. and C. Hwang (1986). Diffusions for global optimization. SIAM J. Control Optim. 24 (5), 1031–1043.

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein, L. (1988). Mean-square rates of convergence in the continuous time simulation annealing algorithm in Vd. Adv. in AppL Probab. 9 35–39.

    Google Scholar 

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Kloeden, P.E., Platen, E. (1992). Applications of Stochastic Differential Equations. In: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12616-5_7

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  • DOI: https://doi.org/10.1007/978-3-662-12616-5_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08107-1

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