Analysis of complex Brownian motion

Yuh-Jia Lee, Kuang-Ghieh Yen
2008 Communications on Stochastic Analysis  
A theory of generalized functions based on the complex Brownian motion {Z(t) : t ∈ R}, for which each Z(t) is N (0, |t|), is established on the probability space (S c , B(S c ), ν(dz)), where S is the dual of the Schwartz space S, S c , the complexification of S , identified as the product space S ×S , B(S c ) the Borel field of S × S and ν(dz) denotes the product measure µ 1 (dx)µ 1 (dy). Using the representation of the complex Brownian motion and employing the technique of white noise
more » ... white noise calculus initiated by Hida ( see, e.g. [2] and [4]), we analyze functionals of complex Brownian motion. To define generalized complex Brownian functionals, we adopt the space of CKS entire functionals as test functions. As applications, the stochastic integral with respect to a complex Brownian motion are defined and studied. The Itô formula for complex Brownian functionals is obtained and it is shown that the evaluation of stochastic integral with respect to a complex Brownian motion follows the rule of Stratonovich integral.
doi:10.31390/cosa.2.1.07 fatcat:zwbc2wfqmvaofno3bu2232hdh4