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August  2020, 19(8): 4127-4142. doi: 10.3934/cpaa.2020184

## First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $\alpha$-stable white noise

 1 Research and Evaluation Division, Public Health Wales, Cardiff, UK 2 Department of Mathematics, Computational Foundry, Swansea University, Swansea, UK 3 Accounting and Finance Department, School of Management, Swansea University, Swansea, UK

* Corresponding author

Received  September 2019 Revised  March 2020 Published  May 2020

The aim of this paper is twofold. Firstly, we derive an explicit expression of the (theoretical) solutions of stochastic differential equations with affine coefficients driven by $\alpha$-stable white noise. This is done by means of Itô formula. Secondly, we develop a detection algorithm for the first jump time in simulation of sampling trajectories which are described by the solutions. The algorithm is carried out through a multivariate Lagrange interpolation approach. To this end, we utilise a computer simulation algorithm in MATLAB to visualise the sampling trajectories of the jump-diffusions for two combinations of parameters arising in the modelling structure of stochastic differential equations with affine coefficients.

Citation: Jiao Song, Jiang-Lun Wu, Fangzhou Huang. First jump time in simulation of sampling trajectories of affine jump-diffusions driven by $\alpha$-stable white noise. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4127-4142. doi: 10.3934/cpaa.2020184
##### References:
 [1] D. Applebaum, L$\acute{e}$vy Processes and Stochastic Calculus, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511755323. [2] C. Bardgett, E. Gourier and M. Leippold, Inferring volatility dynamics and risk premia from the S & P 500 and VIX markets, J. Financ. Econ., 131 (2019), 593-618. [3] A. Barletta, P. Magistris and D. Sloth, It only takes a few moments to hedge options, J. Econ. Dyn. Control, 100 (2019), 251-269.  doi: 10.1016/j.jedc.2018.11.008. [4] O. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24 (1997), 1-13.  doi: 10.1111/1467-9469.00045. [5] J. Campbell, S. Giglio, C. Polk and R. Turley, An intertemporal CAPM with stochastic volatility, J. Financ. Econ., 128 (2018), 207-233. [6] J. Y. Campbell, A. W. C. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997. [7] J. C. Cox, J. E. Ingersoll and S. A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [8] M. Dror, P. L$\prime$ecuyer and F. Szidarovszky, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, Springer Science & Business Media, 2002. [9] H. Du, J. Wu and W. Yang, On the mechanism of CDOs behind the current financial crisis and mathematical modeling with L$\acute{e}$vy distributions, Intel. Inform. Manag., 2 (2010), 149-158. [10] D. Duffie, D. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833. [11] D. Duffie, J. Pan and K. Singleton, Transform Analysis and Asset Pricing for Affine Jump-diffusions, Econometrica, 68 (2000), 1343-1376.  doi: 10.1111/1468-0262.00164. [12] A. Fiche, J. C. Cexus, A. Martin and A. Khenchaf, Features modeling with an $\alpha$-stable distribution: Application to pattern recognition based on continuous belief functions, Inform. Fusion, 14 (2013), 504-520. [13] R. Giacometti, M. Bertocchi, S. T. Rachev and F. J. Fabozzi, Stable distributions in the Black-Litterman approach to asset allocation, Quant. Finance, 7 (2007), 423-433.  doi: 10.1080/14697680701442731. [14] M. Hain, M. Uhrig-Homburg and N. Unger, Risk factors and their associated risk premia: An empirical analysis of the crude oil market, J. Bank. Finance, 95 (2018), 44-63. [15] A. Janicki and A. Weron, Simulation and Chaotic Behavior of $\alpha$-Stable Stochastic Processes, CRC Press, 1993. [16] S. Janson, Stable Distributions, preprint, 2011. Available from: http://www2.math.uu.se/ svante/papers/sjN12.pdf. [17] R. Jarrow, Exploring mispricing in the term structure of CDS spreads, Rev. Finance, 23 (2018), 161-198. [18] W. E. Leland, M. S. Taqqu, W. Willinger and D. W. Wilson, On the self-similar nature of Ethernet traffic, ACM SIGCOMM Comput. Commun. Rev., 23 (1993), 183-193. [19] P. Lévy, Calcul des probabilités, Gauther-Villars, 1925. [20] P.Lévy, Théorie de l'addition des variables aléatoires, Gauther-Villars, 1937. [21] B. Mandelbrot, The Pareto-L$\acute{e}$vy Law and the Distribution of Income, Int. Econ. Rev., 1 (1960), 79-106. [22] G. Samorodnitsky and M. S. Taqqu, Stable Random Processes: Stochastic Models with Infinite Variance, CRC Press, 1994. [23] M.F.Shlesinger, G.M.Zaslavsky and U.Frisch, L$\acute{e}$vy flights and related topics in physics, in Lecture notes in physics, vol. 450, (1995), Springer-Verlag. doi: 10.1007/3-540-59222-9. [24] J. Song and J. Wu, A detection algorithm for the first jump time in sample trajectories of jump-diffusions driven by $\alpha$-stable white noise, Commun. Statist. Theory Meth., 48 (2019), 4888-4902.  doi: 10.1080/03610926.2018.1500602. [25] O. Vasicek, An equilibrium characterization of the term structure, J. Financ. Econ., 52 (1977), 177-188. [26] J. Wu and W. Yang, Valuation of synthetic CDOs with affine jump-diffusion processes involving Lévy stable distributions, Math. Comput. Model., 57 (2013), 570-583.  doi: 10.1016/j.mcm.2012.06.038. [27] V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986. [28] C. Zopounidis and P. M. Pardalos, Managing in Uncertainty: Theory and Practice, Springer Science & Business Media, 2013.

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##### References:
 [1] D. Applebaum, L$\acute{e}$vy Processes and Stochastic Calculus, Cambridge University Press, 1994.  doi: 10.1017/CBO9780511755323. [2] C. Bardgett, E. Gourier and M. Leippold, Inferring volatility dynamics and risk premia from the S & P 500 and VIX markets, J. Financ. Econ., 131 (2019), 593-618. [3] A. Barletta, P. Magistris and D. Sloth, It only takes a few moments to hedge options, J. Econ. Dyn. Control, 100 (2019), 251-269.  doi: 10.1016/j.jedc.2018.11.008. [4] O. Barndorff-Nielsen, Normal inverse Gaussian distributions and stochastic volatility modelling, Scand. J. Statist., 24 (1997), 1-13.  doi: 10.1111/1467-9469.00045. [5] J. Campbell, S. Giglio, C. Polk and R. Turley, An intertemporal CAPM with stochastic volatility, J. Financ. Econ., 128 (2018), 207-233. [6] J. Y. Campbell, A. W. C. Lo and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, 1997. [7] J. C. Cox, J. E. Ingersoll and S. A. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242. [8] M. Dror, P. L$\prime$ecuyer and F. Szidarovszky, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, Springer Science & Business Media, 2002. [9] H. Du, J. Wu and W. Yang, On the mechanism of CDOs behind the current financial crisis and mathematical modeling with L$\acute{e}$vy distributions, Intel. Inform. Manag., 2 (2010), 149-158. [10] D. Duffie, D. Filipović and W. Schachermayer, Affine processes and applications in finance, Ann. Appl. Probab., 13 (2003), 984-1053.  doi: 10.1214/aoap/1060202833. [11] D. Duffie, J. Pan and K. Singleton, Transform Analysis and Asset Pricing for Affine Jump-diffusions, Econometrica, 68 (2000), 1343-1376.  doi: 10.1111/1468-0262.00164. [12] A. Fiche, J. C. Cexus, A. Martin and A. Khenchaf, Features modeling with an $\alpha$-stable distribution: Application to pattern recognition based on continuous belief functions, Inform. Fusion, 14 (2013), 504-520. [13] R. Giacometti, M. Bertocchi, S. T. Rachev and F. J. Fabozzi, Stable distributions in the Black-Litterman approach to asset allocation, Quant. Finance, 7 (2007), 423-433.  doi: 10.1080/14697680701442731. [14] M. Hain, M. Uhrig-Homburg and N. Unger, Risk factors and their associated risk premia: An empirical analysis of the crude oil market, J. Bank. Finance, 95 (2018), 44-63. [15] A. Janicki and A. Weron, Simulation and Chaotic Behavior of $\alpha$-Stable Stochastic Processes, CRC Press, 1993. [16] S. Janson, Stable Distributions, preprint, 2011. Available from: http://www2.math.uu.se/ svante/papers/sjN12.pdf. [17] R. Jarrow, Exploring mispricing in the term structure of CDS spreads, Rev. Finance, 23 (2018), 161-198. [18] W. E. Leland, M. S. Taqqu, W. Willinger and D. W. Wilson, On the self-similar nature of Ethernet traffic, ACM SIGCOMM Comput. Commun. Rev., 23 (1993), 183-193. [19] P. Lévy, Calcul des probabilités, Gauther-Villars, 1925. [20] P.Lévy, Théorie de l'addition des variables aléatoires, Gauther-Villars, 1937. [21] B. Mandelbrot, The Pareto-L$\acute{e}$vy Law and the Distribution of Income, Int. Econ. Rev., 1 (1960), 79-106. [22] G. Samorodnitsky and M. S. Taqqu, Stable Random Processes: Stochastic Models with Infinite Variance, CRC Press, 1994. [23] M.F.Shlesinger, G.M.Zaslavsky and U.Frisch, L$\acute{e}$vy flights and related topics in physics, in Lecture notes in physics, vol. 450, (1995), Springer-Verlag. doi: 10.1007/3-540-59222-9. [24] J. Song and J. Wu, A detection algorithm for the first jump time in sample trajectories of jump-diffusions driven by $\alpha$-stable white noise, Commun. Statist. Theory Meth., 48 (2019), 4888-4902.  doi: 10.1080/03610926.2018.1500602. [25] O. Vasicek, An equilibrium characterization of the term structure, J. Financ. Econ., 52 (1977), 177-188. [26] J. Wu and W. Yang, Valuation of synthetic CDOs with affine jump-diffusion processes involving Lévy stable distributions, Math. Comput. Model., 57 (2013), 570-583.  doi: 10.1016/j.mcm.2012.06.038. [27] V. M. Zolotarev, One-dimensional Stable Distributions, American Mathematical Society, 1986. [28] C. Zopounidis and P. M. Pardalos, Managing in Uncertainty: Theory and Practice, Springer Science & Business Media, 2013.
$\alpha_1<1$ and $\alpha_2>1$: Fix $\lambda$ = 1, $\mu_1$ = 1 and $\mu_2$ = 10
$\alpha_1<1$ and $\alpha_2>1$: $\lambda$ changes when $\alpha_1$ = 0.5 and $\alpha_2$ = 1.5
$\alpha_1<1$ and $\alpha_2>1$: $\mu_2$ changes when $\alpha_1$ = 0.25 and $\alpha_2$ = 1.75
$\alpha_1>1$ and $\alpha_2<1$: Fix $\lambda$ = 1, $\mu_1$ = 1 and $\mu_2$ = 10
$\alpha_1>1$ and $\alpha_2<1$: Fix $\lambda$ = 1, $\mu_1$ = 10 and $\mu_2$ = 1
$\alpha_1>1$ and $\alpha_2<1$: $\mu_2$ changes when $\alpha_1$ = 1.5 and $\alpha_2$ = 0.5
Data processed for sample trajectories when $\alpha_1<1$ and $\alpha_2>1$
 $\lambda$ $\mu_1$ $\mu_2$ $\alpha_1$ $\alpha_2$ t $X^\alpha_t$ 10 1 100 0.5 1.5 0.08203 -63.86 1 0.25 100 0.75 1.25 0.1855 -303.2 1 100 1 0.75 1.75 0.1035 122.5 1 100 0.25 0.75 1.5 0.207 -896.1 10 100 0.25 0.5 1.25 0.3301 252.1 100 100 1 0.25 1.75 0.1934 -3028000 10 1 0.25 0.25 1.25 0.5762 533.7
 $\lambda$ $\mu_1$ $\mu_2$ $\alpha_1$ $\alpha_2$ t $X^\alpha_t$ 10 1 100 0.5 1.5 0.08203 -63.86 1 0.25 100 0.75 1.25 0.1855 -303.2 1 100 1 0.75 1.75 0.1035 122.5 1 100 0.25 0.75 1.5 0.207 -896.1 10 100 0.25 0.5 1.25 0.3301 252.1 100 100 1 0.25 1.75 0.1934 -3028000 10 1 0.25 0.25 1.25 0.5762 533.7
Data processed for sample trajectories when $\alpha_1>1$ and $\alpha_2<1$
 $\lambda$ $\mu_1$ $\mu_2$ $\alpha_1$ $\alpha_2$ t $X^\alpha_t$ 10 0.25 1 1.5 0.5 0.1973 -91.87 10 1 100 1.5 0.25 0.01953 95.72 1 100 1 1.25 0.5 0.04492 305.9 100 1 0.25 1.5 0.25 0.1211 346 1 1 100 1.75 0.5 0.09766 311.1 100 1 100 1.5 0.5 0.05273 -242.9 10 1 0.25 1.75 0.75 0.5742 -105.1
 $\lambda$ $\mu_1$ $\mu_2$ $\alpha_1$ $\alpha_2$ t $X^\alpha_t$ 10 0.25 1 1.5 0.5 0.1973 -91.87 10 1 100 1.5 0.25 0.01953 95.72 1 100 1 1.25 0.5 0.04492 305.9 100 1 0.25 1.5 0.25 0.1211 346 1 1 100 1.75 0.5 0.09766 311.1 100 1 100 1.5 0.5 0.05273 -242.9 10 1 0.25 1.75 0.75 0.5742 -105.1
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