斜扩散过程的构造、性质及其应用
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摘要
斜扩散过程是一类特殊的扩散过程,它所满足的随机微分方程中含有局部时项.由于这类过程具有特殊的路径性质,使得它在物理和生物等领域有着广泛的应用.本文先从斜Brown运动出发,概括总结了斜Brown运动的两种构造方式、离散逼近形式和联合概率分布.在斜Brown运动的基础上,本文给出斜Ornstein-Uhlenbeck (OU)过程和斜分支过程的定义,并展示了它们的诸如转移密度、首达时分布等概率性质.最后介绍了斜扩散过程的一些应用.
The skew diffusion process is a special diffusion process. The stochastic differential equation expression of such a special process contains the symmetric local time. Due to its special property, this model is widely used in physics and biology. Starting from skew Brownian motion, this work summaries its two constructions:discrete approximation and jointly probability distribution. Based on the skew Brownian motion, we provide the definitions for the skew Ornstein-Uhlenbeck(OU) process and skew branching process, and derive their properties:transition density, first hitting time, etc. We also introduce some applications under the skew diffusion processes.
The skew diffusion process is a special diffusion process. The stochastic differential equation expression of such a special process contains the symmetric local time. Due to its special property, this model is widely used in physics and biology. Starting from skew Brownian motion, this work summaries its two constructions:discrete approximation and jointly probability distribution. Based on the skew Brownian motion, we provide the definitions for the skew Ornstein-Uhlenbeck(OU) process and skew branching process, and derive their properties:transition density, first hitting time, etc. We also introduce some applications under the skew diffusion processes.
引文
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2 Walsh J B.A diffusion with discontinuous local time.Ast′erisque,1978,52:37-45
3 Harrison J M,Shepp L A.On skew Brownian motion.Ann Probab,1981,9:309-313
4 Le Gall J F.One-dimensional stochastic differential equations involving the local times of the unknown process.In:Stochastic Analysis and Applications.Lecture Notes in Mathematics,vol.1095.New York:Springer-Verlag,1984,51-82
5 Lejay A.On the construction of the skew Brownian motion.Probab Surv,2007,3:413-466
6 Bass R F,Chen Z Q.One-dimensional stochastic differential equations with singular and degenerate coefficients.Sankhyˉa,2005,67:19-45
7 Appuhamillage T,Sheldon D.First passage time of skew Brownian motion.J Appl Probab,2012,49:685-696
8 Abundo M.First-passage problems for asymmetric diffusions and skew-diffusion processes.Open Syst Inf Dyn,2009,16:325-350
9 Bo L,Wang Y,Yang X.First passage times of(reflected)Ornstein-Uhlenbeck processes over random jump boundaries.J Appl Probab,2011,48:723-732
10 Song S,Wang S,Wang Y.First hitting times for doubly skewed Ornstein-Uhlenbeck processes.Statist Probab Lett,2015,96:212-222
11 Song S,Wang S,Wang Y.On some properties of reflected skew Brownian motions and applications to dispersion in heterogeneous media.Phys A,2016,456:90-105
12 Song S,Xu G,Wang Y.On first hitting times for skew CIR processes.Methodol Comput Appl Probab,2016,18:169-180
13 Wang S,Song S,Wang Y.Skew Ornstein-Uhlenbeck processes and their financial applications.J Comput Appl Math,2015,273:363-382
14 Xu G,Song S,Wang Y.Some properties of doubly skewed CIR processes.J Math Anal Appl,2016,434:1194-1210
15 Xu G,Wang Y.On stability of the Markov-modulated skew CIR process.Statist Probab Lett,2016,109:139-144
16 Appuhamillage T,Bokil V,Thomann E,et al.Occupation and local times for skew Brownian motion with applications to dispersion across an interface.Ann Appl Probab,2011,21:183-214
17 Freidlin M I,Wentzell A D.Diffusion processes on graphs and the averaging principle.Ann Probab,1993,21:2215-2245
18 Ramirez J M.Multi-skewed Brownian motion and diffusion in layered media.Proc Amer Math Soc,2011,139:3739-3752
19 Cantrell R,Cosner C.Diffusion models for population dynamics incorporating individual behavior at boundaries:Applications to refuge design.Theor Popul Biol,1999,55:189-207
20 Lejay A.Simulating a diffusion on a graph.Application to reservoir engineering.Monte Carlo Methods Appl,2003,9:241-256
21 Decamps M,de Schepper A,Goovaerts M.Applications ofδ-function perturbation to the pricing of derivative securities.Phys A,2004,342:677-692
22 Zhang M.Calculation of diffusive shock acceleration of charged particles by skew Brownian motion.Astrophys J,2000,541:428-435
23 Nikiforov A F,Uvarov V B.Special Functions of Mathematical Physics:A Unified Introduction with Applications.New York:Springer,1988
24 Li Y,Wang S,Zhou X,et al.Diffusion occupation time before exiting.Front Math China,2014,9:843-861
25 Cox J C,Ingersoll J E Jr,Ross S A.A theory of the term structure of interest rates.Econometrica,1999,53:385-408
26 Durrett R.Stochastic Calculus:A Practical Introduction.Boca Raton:CRC Press,1996
27 Trutnau G.Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic timedependent curve.Stochastic Process Appl,2010,120:381-402
28 Trutnau G.Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve.Stochastic Process Appl,2011,121:1845-1863
29 Xu G,Song S,Wang Y.The valuation of options on foreign exchange rate in a target zone.Int J Theor Appl Finance,2016,19:1-19
30 Karlin S,Taylor H M.A Second Course in Stochastic Processes.New York:Academic Press,1981
2 Walsh J B.A diffusion with discontinuous local time.Ast′erisque,1978,52:37-45
3 Harrison J M,Shepp L A.On skew Brownian motion.Ann Probab,1981,9:309-313
4 Le Gall J F.One-dimensional stochastic differential equations involving the local times of the unknown process.In:Stochastic Analysis and Applications.Lecture Notes in Mathematics,vol.1095.New York:Springer-Verlag,1984,51-82
5 Lejay A.On the construction of the skew Brownian motion.Probab Surv,2007,3:413-466
6 Bass R F,Chen Z Q.One-dimensional stochastic differential equations with singular and degenerate coefficients.Sankhyˉa,2005,67:19-45
7 Appuhamillage T,Sheldon D.First passage time of skew Brownian motion.J Appl Probab,2012,49:685-696
8 Abundo M.First-passage problems for asymmetric diffusions and skew-diffusion processes.Open Syst Inf Dyn,2009,16:325-350
9 Bo L,Wang Y,Yang X.First passage times of(reflected)Ornstein-Uhlenbeck processes over random jump boundaries.J Appl Probab,2011,48:723-732
10 Song S,Wang S,Wang Y.First hitting times for doubly skewed Ornstein-Uhlenbeck processes.Statist Probab Lett,2015,96:212-222
11 Song S,Wang S,Wang Y.On some properties of reflected skew Brownian motions and applications to dispersion in heterogeneous media.Phys A,2016,456:90-105
12 Song S,Xu G,Wang Y.On first hitting times for skew CIR processes.Methodol Comput Appl Probab,2016,18:169-180
13 Wang S,Song S,Wang Y.Skew Ornstein-Uhlenbeck processes and their financial applications.J Comput Appl Math,2015,273:363-382
14 Xu G,Song S,Wang Y.Some properties of doubly skewed CIR processes.J Math Anal Appl,2016,434:1194-1210
15 Xu G,Wang Y.On stability of the Markov-modulated skew CIR process.Statist Probab Lett,2016,109:139-144
16 Appuhamillage T,Bokil V,Thomann E,et al.Occupation and local times for skew Brownian motion with applications to dispersion across an interface.Ann Appl Probab,2011,21:183-214
17 Freidlin M I,Wentzell A D.Diffusion processes on graphs and the averaging principle.Ann Probab,1993,21:2215-2245
18 Ramirez J M.Multi-skewed Brownian motion and diffusion in layered media.Proc Amer Math Soc,2011,139:3739-3752
19 Cantrell R,Cosner C.Diffusion models for population dynamics incorporating individual behavior at boundaries:Applications to refuge design.Theor Popul Biol,1999,55:189-207
20 Lejay A.Simulating a diffusion on a graph.Application to reservoir engineering.Monte Carlo Methods Appl,2003,9:241-256
21 Decamps M,de Schepper A,Goovaerts M.Applications ofδ-function perturbation to the pricing of derivative securities.Phys A,2004,342:677-692
22 Zhang M.Calculation of diffusive shock acceleration of charged particles by skew Brownian motion.Astrophys J,2000,541:428-435
23 Nikiforov A F,Uvarov V B.Special Functions of Mathematical Physics:A Unified Introduction with Applications.New York:Springer,1988
24 Li Y,Wang S,Zhou X,et al.Diffusion occupation time before exiting.Front Math China,2014,9:843-861
25 Cox J C,Ingersoll J E Jr,Ross S A.A theory of the term structure of interest rates.Econometrica,1999,53:385-408
26 Durrett R.Stochastic Calculus:A Practical Introduction.Boca Raton:CRC Press,1996
27 Trutnau G.Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic timedependent curve.Stochastic Process Appl,2010,120:381-402
28 Trutnau G.Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve.Stochastic Process Appl,2011,121:1845-1863
29 Xu G,Song S,Wang Y.The valuation of options on foreign exchange rate in a target zone.Int J Theor Appl Finance,2016,19:1-19
30 Karlin S,Taylor H M.A Second Course in Stochastic Processes.New York:Academic Press,1981