Article Contents
Article Contents

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

• * Corresponding author
• 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.

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

 Citation:

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

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

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

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

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

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

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

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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