Monte-Carlo Methods and Stochastic Processes by Emmanuel Gobet

Author:Emmanuel Gobet
Language: eng
Format: epub
Publisher: CRC Press

⊳ Square root process in dimension 1. This process exhibits a mean-reverting effect but with the specificity to stay non-negative: it is defined by

i.e., b(t, x) = −ax and . Although σ is not Lipschitz continuous, it is possible to show the existence and uniqueness of the solution, but it is more delicate. This model is often used in finance to model stochastic volatility (Heston model [77]) and interest rates (CIR model [30]). In population dynamics [39], the equation of X is called the Feller equation and functions like the limit model of the birth-death process. The distribution of Xt is known in the form of special functions (in particular Bessel functions) and its simulation is not simple; see [22].

In genetic model, the Fisher-Wright process describing the asymptotic proportion of individuals carrying a given allele is an example of the square root process in the form

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