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

Jiao Song; Jiang-Lun Wu; Fangzhou Huang

doi:10.3934/cpaa.2020184

Article Contents

2020, Volume 19, Issue 8: 4127-4142. Doi: 10.3934/cpaa.2020184

This issue Previous Article Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems Next Article Computing eigenpairs of two-parameter Sturm-Liouville systems using the bivariate sinc-Gauss formula

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
^* Corresponding author

Received: September 2019

Revised: March 2020

Published: May 2020

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Abstract

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.

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Mathematics Subject Classification: 65C99, 68U20, 60E07, 60G17.

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Figure 1. $ \alpha_1<1 $ and $ \alpha_2>1 $: Fix $ \lambda $ = 1, $ \mu_1 $ = 1 and $ \mu_2 $ = 10

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Figure 2. $ \alpha_1<1 $ and $ \alpha_2>1 $: $ \lambda $ changes when $ \alpha_1 $ = 0.5 and $ \alpha_2 $ = 1.5

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Figure 3. $ \alpha_1<1 $ and $ \alpha_2>1 $: $ \mu_2 $ changes when $ \alpha_1 $ = 0.25 and $ \alpha_2 $ = 1.75

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Figure 4. $ \alpha_1>1 $ and $ \alpha_2<1 $: Fix $ \lambda $ = 1, $ \mu_1 $ = 1 and $ \mu_2 $ = 10

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Figure 5. $ \alpha_1>1 $ and $ \alpha_2<1 $: Fix $ \lambda $ = 1, $ \mu_1 $ = 10 and $ \mu_2 $ = 1

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Figure 6. $ \alpha_1>1 $ and $ \alpha_2<1 $: $ \mu_2 $ changes when $ \alpha_1 $ = 1.5 and $ \alpha_2 $ = 0.5

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Table 1. 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

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Table 2. 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

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