Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve
| Autor: | Gerald Trutnau |
|---|---|
| Rok vydání: | 2010 |
| Předmět: |
Statistics and Probability
Continuous function (set theory) Secondary: Dirichlet spaces Applied Mathematics Analytical chemistry Local time and additive functionals Positive function Dirichlet space Potentials and capacities Primary: Stochastic ordinary differential equations symbols.namesake Reflection (mathematics) Boundary value problems for second-order parabolic equations Modelling and Simulation Modeling and Simulation symbols Diffusion processes Bessel function Mathematics |
| Zdroj: | Stochastic Processes and their Applications. 120:381-402 |
| ISSN: | 0304-4149 |
| DOI: | 10.1016/j.spa.2010.01.005 |
| Popis: | Let σ > 0 , δ ≥ 1 , b ≥ 0 , 0 p 1 . Let λ be a continuous and positive function in H l o c 1 , 2 ( R + ) . Using the technique of moving domains (see Russo and Trutnau (2005) [9] ), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales ( R , R ) with d R t = σ R t d W t + σ 2 4 ( δ − b R t ) d t + ( 2 p − 1 ) d l t 0 ( R − λ 2 ) , and d R t = σ 2 d W t + σ 2 8 ( δ − 1 R t − b R t ) d t + ( 2 p − 1 ) d l t 0 ( R − λ ) + I { δ = 1 } 2 d l t 0 + ( R ) , where the symmetric local times l 0 ( R − λ 2 ) , l 0 ( R − λ ) , of the respective semimartingales R − λ 2 , R − λ are related through the formula 2 R d l 0 ( R − λ ) = d l 0 ( R − λ 2 ) . Well-known special cases are the (squared) Bessel processes (choose σ = 2 , b = 0 , and λ 2 ≡ 0 , or equivalently p = 1 2 ), and the Cox–Ingersoll–Ross process (i.e. R , with λ 2 ≡ 0 , or equivalently p = 1 2 ). The case 0 δ 1 can also be handled, but is different. If | p | > 1 , then there is no solution. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|
Full Text from ScienceDirect