The means of the solutions are estimated through Monte Carlo method.
📌 3 Key Takeaways
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Objective & Hypothesis:We consider linearizing transformations of the one-dimensional nonlinear stochastic differential equations driven by Wiener and compound Poisson processes, namely finite activity Levy processes.
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Key Findings & Data:We apply our method to a number ofstochastic differential equations including Cox-Ingersoll-Ross short-term interest rate model, log-mean reverting asset pricing model and geometric Ornstein- Uhlenbeck equation all with additional jump terms.
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Conclusion & Impact:The means of the solutions are estimated through Monte Carlo method.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
We consider linearizing transformations of the one-dimensional nonlinear stochastic differential equations driven by Wiener and compound Poisson processes, namely finite activity Levy processes. We present linearizability criteria and derive the required transformations. We use a stochastic integrating factor method to solve the linearized equations and provide closed-form solutions. We apply our method to a number ofstochastic differential equations including Cox-Ingersoll-Ross short-term interest rate model, log-mean reverting asset pricing model and geometric Ornstein- Uhlenbeck equation all with additional jump terms. We use their analytical solutions to illustrate the accuracy of the numerical approximations obtained from Euler and Maghsoodi discretization schemes. The means of the solutions are estimated through Monte Carlo method.
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Cite This Article
İYİGÜNLER, İ., ÇAĞLAR, M., ÜNAL, G. (2012). Exact solvability of stochastic differential equations driven by finite activity levy processes. Mathematical and Computational Applications, Cilt: 17 - Sayı: 1, 68–82. https://ajindex.com/articles/279832
D. Applebaum, Levy Processes and Stochastic Calculus, Cambridge University Press, 2004
F. Black and P. Karasinski, Bond and option pricing when short rates are lognormal. Financial Analysts Journal, July-August 52-59, 1991
J. C. Cox, J. E. Ingersoll and S. A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica 53 385-408 1985
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, 2004
L. Clewlow, C. Strickland and V. Kaminski, Extending Mean-Reversion Jump Diusion, Energy Power Risk Management, Risk Waters Group 5 (10), January, (2000)
A. K. Dixit and R. S. Pindyck, Investment under Uncertainty, Princeton University Press, 2000
J. Duan, J. Yan, General Matrix-Valued Inhomogeneous Linear Stochastic Dierential Equations, Statistics and Probability Letters 78 2361-2365, 2008
D. Epstein, N. Mayor, P. Schonbucher, A. E. Whalley and P. Wilmott, The Valuation of a Firm Advertising Optimally, Quarterly Review of Economics and Finance 38-2 149-166, 1999
P. V. Gapeev, Solving stochastic jump dierential equations, Preprint, Centre for Mathematical Physics and Stochastics (MaPhySto) Research Report 47, December 2002
T. C. Gard, Introduction to Stochastic Differential Equations Marcel Dekker, 1988
C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences Springer-Verlag, 1985
F. B. Hanson, Applied Stochastic Processes and Control for Jump-Diusions: Modeling, Analysis and Computation SIAM Books, 2007
D. Higham and P. E. Kloeden, Numerical methods for nonlinear stochastic dierential equations with jumps, Numerische Mathematik 101-1, 101-119, 2005
N. Ikeda and S. Watanabe, Stochastic Dierential Equations and Diusion Processes, North-Holland, 1988
J. Jacod, Equations dierentielles lineaires, la methode de variation des constantes. Seminaire de Probabilites XVI. Lecture Notes in Mathematics 920 442-446, 1982
Jacod, J. and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer- Verlag 1987
J. Jang, Jump Diusion Processes and their Applications in Insurance and Finance, Insurance: Mathematics and Economics 41-1, 62-70, 2007
M. Jeanblanc, M. Yor and M. Chesney, Mathematical Methods for Financial Markets, Springer 2009
Y. Maghsoodi, Exact solutions and doubly efficient approximations and simulation of jump-diusion Itô equations, Stochast. Anal. Appl. 16 1049-1072, 1998 20. Y. Maghsoodi, Mean Square Ef