A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes
Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/11/1907 |
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| author | Jorge E. Macías-Díaz |
| author_facet | Jorge E. Macías-Díaz |
| author_sort | Jorge E. Macías-Díaz |
| collection | DOAJ |
| description | Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work. |
| first_indexed | 2024-03-10T14:42:03Z |
| format | Article |
| id | doaj.art-80dd6f727cf84f0cb1c5fe60d1644db6 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T14:42:03Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-80dd6f727cf84f0cb1c5fe60d1644db62023-11-20T21:41:43ZengMDPI AGSymmetry2073-89942020-11-011211190710.3390/sym12111907A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random ProcessesJorge E. Macías-Díaz0Department of Mathematics, School of Digital Technologies, Tallinn University, 10120 Tallinn, EstoniaDeparting from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.https://www.mdpi.com/2073-8994/12/11/1907moving boundary problemprobability distribution function of the first hitting timeadvection–diffusion problemimplicit finite-difference modelspatially symmetric discrete modeltheoretical analysis |
| spellingShingle | Jorge E. Macías-Díaz A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes Symmetry moving boundary problem probability distribution function of the first hitting time advection–diffusion problem implicit finite-difference model spatially symmetric discrete model theoretical analysis |
| title | A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes |
| title_full | A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes |
| title_fullStr | A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes |
| title_full_unstemmed | A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes |
| title_short | A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes |
| title_sort | numerical schemefor the probability density of the first hitting time for some random processes |
| topic | moving boundary problem probability distribution function of the first hitting time advection–diffusion problem implicit finite-difference model spatially symmetric discrete model theoretical analysis |
| url | https://www.mdpi.com/2073-8994/12/11/1907 |
| work_keys_str_mv | AT jorgeemaciasdiaz anumericalschemefortheprobabilitydensityofthefirsthittingtimeforsomerandomprocesses AT jorgeemaciasdiaz numericalschemefortheprobabilitydensityofthefirsthittingtimeforsomerandomprocesses |