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Marcus canonical integral for non-Gaussian processes and its computation: Pathwise simulation and tau-leaping algorithm
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10.1063/1.4794780
/content/aip/journal/jcp/138/10/10.1063/1.4794780
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/10/10.1063/1.4794780

Figures

Image of FIG. 1.
FIG. 1.

The smoothed function L δ(t) of the single-jump realization L(t).

Image of FIG. 2.
FIG. 2.

Shown in the figure is one specific realization of the solution. The blue solid line is the theoretical solution. The star symbol shows the numerical result by pathwise simulation algorithm and the red circle corresponds to the Ito integral. The second order Runge-Kutta methods are used for both drift and jump ODEs and the time stepsize δt = 0.01. Some other parameters are λ = 20 and σ = 0.2.

Image of FIG. 3.
FIG. 3.

For ODEs (33) and (34) , we use the Runge-Kutta methods with p = 3 and q = 3. The time stepsize is chosen to be Δt = 0.006, 0.005, 0.004, 0.003, and 0.002. Some parameters are λ = 20, σ = 0.2, T = 1. Three thousand samples are simulated. Shown in the left panel is the log-log plot of the error of mean versus time stepsizes, which gives numerical order 1.0157. The right panel shows the linear fitting of error against stepsize curve, which gives slope 4.1905.

Image of FIG. 4.
FIG. 4.

The red circle corresponds to the heat based on Marcus integral and the blue solid line corresponds to the total energy U. Some parameters are set as λ = 5, λI 2 = 1, ε = 0.1 and the time stepsize δs = 0.01 for solving both drift and jump ODEs.

Image of FIG. 5.
FIG. 5.

The red circle corresponds to the numerical heat Q and the blue solid line corresponds to the theoretical energy E which equals to Q. L(t) is a compound Poisson process with rate λ = 10 and P(xx ± I) = λ/2, where I satisfies λI 2 = 1. The stepsizes δs = 0.01 for both drift and jump ODEs.

Image of FIG. 6.
FIG. 6.

The distribution of X at time T = 10. The symbol * shows the distribution obtained by tau-leaping algorithm and ○ shows the distribution obtained by pathwise simulations. Some parameters are λ = 400, A = 500, and X(0) = B/2 = 500. The time step is δt = 0.01 for pathwise simulations and Δt = 0.06 for tau-leaping method. Five thousand samples are simulated.

Image of FIG. 7.
FIG. 7.

Comparison between the pathwise and tau-leaping simulations with adaptive stepsize for double well potential. (a) Plot of the double well potential and (b) the distribution of X at time T = 10. Shown with blue stars is the distribution obtained by tau-leaping algorithm and red circles is the distribution obtained by pathwise algorithm. Five thousand realizations are simulated.

Image of FIG. 8.
FIG. 8.

Comparison between the pathwise and tau-leaping simulations with adaptive switching for periodic forcing. (a) The distribution of X at time T = 10. Shown with blue stars is the distribution obtained by tau-leaping algorithm and red circles is the distribution obtained by pathwise algorithm. Five thousand realizations are simulated. (b) Time history of utilized stepsizes by adaptive tau-leaping algorithm. The red horizontal line corresponds to the threshold Δt 0. The blue curve corresponds to the stepsizes used in tau-leaping steps while the green dots corresponds to the stepsizes used in pathwise simulation steps.

Image of FIG. 9.
FIG. 9.

The distribution of X 3 at time T = 10. The symbol * shows the distribution obtained by tau-leaping algorithm and ○ shows the distribution obtained by pathwise simulations.

Tables

Generic image for table
Table I.

For ODEs (33) and (34) , we use pth and qth order Runge-Kutta methods, respectively. The time stepsize is chosen to be Δt = 0.01, 0.008, 0.005, 0.004, 0.002, and 0.001. Some parameters are λ = 20, σ = 0.2, and T = 1. Two thousand samples are simulated. The numbers shown in the table are the slope by linear fitting compared with the theoretical value in parenthesis.

Generic image for table
Table II.

t r , t f , t g , and t total are the time for generating random variables, solving the drift ODE (33) , solving the jump ODEs (53) and (54) , and the total computation time, respectively. Mean and Std are the mean value and standard deviation of X(T = 10). Five thousand simulations are performed. The parameters are the same as those in Fig. 6 .

Generic image for table
Table III.

Comparison between the pathwise and tau-leaping simulations with adaptive stepsize for double well potential. The computation time is for 5000 simulations. Mean and Std are the mean value and standard deviation of X(10).

Generic image for table
Table IV.

Comparison between the pathwise and adaptive tau-leaping simulations with different thresholds r 0 for periodic forcing. The computation time is for 5000 simulations. Mean and Std are the mean value and standard deviation of X(10).

Generic image for table
Table V.

Comparison between the pathwise and tau-leaping simulations with adaptive stepsize for high dimensional equations. The computation time is for 5000 simulations. In each column of X i , the mean value and standard variation of X i (10) are listed.

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/content/aip/journal/jcp/138/10/10.1063/1.4794780
2013-03-14
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Marcus canonical integral for non-Gaussian processes and its computation: Pathwise simulation and tau-leaping algorithm
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/10/10.1063/1.4794780
10.1063/1.4794780
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