Modelling Nonlinear Dynamical Systems Using Itô Stochastic Differential Equations With Jump Process
DOI:
https://doi.org/10.60787/tnamp.v21.508Keywords:
Jumps, Levy process, Exchange rate, Diffusion process, Intensity functionAbstract
Diffusion processes have been used for describing nonlinear dynamical systems and the geometric Brownian motion has long served as a foundational model for capturing stochastic nature of processes characterized by the continuous random fluctuations. This study developed a modification to Lévy process for the Nigeria exchange rate to the US dollar using Ito stochastic differential equation by considering the case where the price movement involves sudden jumps, while capturing statistical features present in the time series. The developed extended xMJNID model, assumes that the market model has no arbitrage opportunities and the exchange rate follows a Merton model. The extension was a jump composed of Poisson process with nonconstant intensity function. Through simulation study and application to real data, the xMJNID model was shown to perform better than existing diffusion models including the Merton model and ARIMA. Comparisons were made using accuracy measures and Akaike and Bayesian information criteria.
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References
Karatas, I. and Shreve, E.S. (1994). Graduate Text in Mathematics, Brownian Motion and Stochastic Calculus, 2nd ed., Springer-Verlag, New York
Grebenkov, D.S., Sposini, V., Metzler, R., Oshanin, G. and Seno, F. (2021). Exact distributions of the maximum and range of random diffusivity processes. New Journal of Physics 23, 023014.
Stojkoski, V. Utkovski, Z. Basnarkov, and L. Kocarev, L. (2019). Cooperation dynamics in networked geometric Brownian motion. Physical Review E 99, 062312.
Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637.
Stojkoski, V. Sandev, T., Basnarkov, L., Kocarev, L. and Metzler, R. (2020). Generalised geometric Brownian motion: theory and applications to option pricing. Entropy 22, 1432 (2020).
Gupta R., Drzazga-Szczesniak, E.A., Kais, S. and Szczesniak, D. (2024). The entropy corrected geometric Brownian motion. arXiv:2403.06253v1 [physics.data-an] 10 Mar 2024.
Intarasit A. and Sattayatham, P. (2010). A Geometric Brownian motion Model with Compound Poisson Process and Fractional Stochastic Volatility. Advances and Applications in Statistics, Volume 16, Number 1, 2010, Pages 25-47
Hawkes, A. G. (2020). Hawkes jump-diffusions and finance: a brief history and review. European Journal of Finance. doi.org/10.1080/1351847X.2020.1755712.
Lefebvre, M. (2020). Exit Problems for Jump-Diffusion Processes with Uniform Jumps. Journal of Stochastic Analysis: Vol. 1: No. 1, Article 5. DOI: 10.31390/josa.1.1.05
Ma, H. (2021). Taylor Based Jump Diffusion Model of Fractional Brownian motion of Stock Price. Journal of Physics: Conference Series 1744 (2021) 032093. doi:10.1088/1742-6596/1744/3/032093
Ohnishi, M. (2003). An optimal stopping problem for a Geometric Brownian motion with Poissonian jumps. Mathematical and Computer modelling 38 (2003) 1381-1390.
Mulambula, A., Oduor, D. B. and Kwach, B. (2019). Derivation of Black-Scholes-Merton Logistic Brownian motion Differential Equation with Jump Diffusion Process. Int. J. Math. And Appl., 7(3) (2019), 85-93
Ma, H. and Li, Y. (2019). Stock Price Jump-diffusion Process Model Based on Fractional Brownian Motion Theory. 3rd International Conference on Education, Economics and Management Research (ICEEMR 2019). Advances in Social Science, Education and Humanities Research, volume 385
Mulambula, A. (2021). Logistic Brownian motion with Jumps. IRE Journals, Volume 5 Issue 3. ISSN: 2456-8880
Mulambula, A., Oduor, D. B. and Kwach, B. (2019). Derivation of Black-Scholes-Merton Logistic Brownian motion Differential Equation with Jump Diffusion Process. Int. J. Math. And Appl., 7(3) (2019), 85-93
Oduor, D. B. (2022). Mean: Reverting logistic Brownian motion with jump diffusion process on energy commodity prices. International Journal of Statistics and Applied Mathematics 2022; 7(4): 69-74
Merton, R. C. (1976). Option Pricing When Underlying Stock Returns Are Continuous. Journal of Financial Economics, 3, 125-144.
Nunno, G.D., Øksendal B. and Proske F. (2009). Malliavin Calculus for Lévy Processes with Applications to Finance. Springer Berlin, Heidelberg. ISBN 978-3-540-78571-2
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series.
Øksendal, B. and Sulem A. (2019). Applied Stochastic Control of Jump Diffusions. Springer Nature, ed 3.
Matsuda, K. (2004). Introduction to Option Pricing with Fourier Transform: Option Pricing with Exponential Lévy Models, Working Paper, Graduate School and University Center of the City University of New York.
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