Fundamental equations with higher order Malliavin operators

Abstract

We consider three fundamental equations of the Malliavin calculus: the equation involving the Malliavin derivative, the Skorokhod integral and the Ornstein–Uhlenbeck operator of order , . These equations provide a complete characterization of the domain and range of the aforementioned operators. Applying the chaos expansion method in white noise spaces we solve these equations and obtain an explicit form of the solutions in the space of Kondratiev generalized stochastic processes.

Keywords:

• generalized stochastic processes
• white noise
• chaos expansion
• Malliavin derivative
• Skorokhod integral
• Ornstein–Uhlenbeck operator

AMS Subject Classification:

• 60H40
• 60H07
• 60H10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The paper was supported by the projects Modeling and harmonic analysis methods and PDEs with singularities [grant number 174024], and Modeling and research methods of operational control of traffic based on electric traction vehicles optimized by power consumption criterion [grant number TR36047], both financed by the Ministry of Science, Republic of Serbia.