Abstract
In this final chapter we apply the strong and weak schemes introduced in the preceding chapters to typical, illustrative situations for which the numerical solution of a stochastic differential equation can provide useful information and insights. The direct simulation of trajectories by a strong scheme allows the behaviour of a stochastic dynamical system to be visualized. Theoretically derived parametric estimators and finite-state Markov chain filters can be tested using simulated data. In addition, weak schemes will be used to calculate frequency histograms of invariant measures, moments and functional integrals, such as Lyapunov exponents, of solutions of stochastic differential equations. Both strong and weak schemes are applied to investigate different aspects of a common problem in the stability and bifurcation of stochastic systems. Finally, a finance model involving the computation of option prices and hedging strategies is simulated. In all cases, the computations will be left as exercises for the reader with only the results given here.
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Literature for Chapter 6
Treutlein & Schulten (1985) used Monte-Carlo methods to simulate the noisy Bonhoeffer-Van der Pol equations, see also Kloeden, Platen & Schurz (1991). The t heory of stochastic flows is presented in Kunita (1990), while the examples of stochas t ic flows on the circle and th e torus were taken from Carverhill , Chappel & Elworthy (1986), from Baxendale (1986) and from Baxendale & Harris (1986); resp ectively. For sur veys of parametric estimation with cont inuous sample of a diffusion see Basawa & Prakasa Rao (1980) and Kutoyants (1984) . The discret e sampling case is treated in Le Breton (1976) , Dacunha-Castelle & Florens Zmirou (1986) , Florens-Zmirou (1989), Campillo & Le Gland (1989) and Genon-Catalot (1990). The mod el considered in Subsection 6.4.A is taken from Kloeden, Platen, Schurz & Ser ensen (1992) . Exponential famili es of processes are treated in Kuchler & Ser ensen (1989) and quasi -likelihood est imators in Hutton & Nelson (1986), Godambe & Heyde (1987) and Ser ensen (1990). See Shiga (1985) for population mod els. Filtering of Markov chains goes back to Wonham (1965), Zakai (1969) and Fujisaki, Kallianpur & Kunita (1972) . Discrete time approximations of optimal filters have been considered by Clark & Cameron (1980) , Newton (1986) and Kloeden, Platen & Schurz (1993). Functional integrals like those in Subse ction 6.4.A have been investigated by Blankenship & Baras (1981) , amongst others. The ergodic convergence criterion is du e to Talay (1987). See also Talay (1990) for his ord er f3 = 2.0 weak scheme. The explicit formul a for the top Lyapunov exponent in Subsection 6.5.A is from Baxendale (1986). See Talay (1989) for the num erical approximation of Lyapunov exponent s. The theory of stochastic stability is develop ed in Hasminski (1980) and, from the perspective of Lyapunov exponents , in Arnold & Wihstutz (1986). See Ehrhardt (1983) for the noisy Bru sselator equat ions. For finan ce models see Black & Scholes (1973), Hull & White (1987), Johnson & Shanno (1987), Scott (1987), Wiggins (1987) and Follmer & Schweizer (1991). The model in Subsection 6.6.B is taken from Hofmann, Plat en & Schweizer (1992).
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© 1994 Springer-Verlag Berlin Heidelberg
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Kloeden, P.E., Platen, E., Schurz, H. (1994). Applications. In: Numerical Solution of SDE Through Computer Experiments. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57913-4_6
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DOI: https://doi.org/10.1007/978-3-642-57913-4_6
Publisher Name: Springer, Berlin, Heidelberg
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