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Duality and transform analysis for non-decreasing functionals of stochastic processes and their applications

Part of: Mathematical finance Stochastic systems and control

Published online by Cambridge University Press:  30 September 2025

Zhenyu Cui ,
Chihoon Lee Open the ORCID record for Chihoon Lee [Opens in a new window] ,
Yanchu Liu  and
Lingjiong Zhu
Zhenyu Cui*
Affiliation:
Stevens Institute of Technology
Chihoon Lee*
Affiliation:
Stevens Institute of Technology
Yanchu Liu*
Affiliation:
Sun Yat-sen University
Lingjiong Zhu*
Affiliation:
Florida State University
*
*Postal address: School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA.
*Postal address: School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA.
****Postal address: Lingnan College, Sun Yat-sen University, Guangzhou 510275, China. liuych26@mail.sysu.edu.cn
*****Postal address: Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA. zhu@math.fsu.edu
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Abstract

We establish a novel duality relationship between continuous/discrete non-negative non-decreasing functionals of stochastic (not necessarily Markovian) processes and their right inverses, and further discuss its applications. For general Markov processes, we develop a theoretical and computational framework for the transform analysis via an operator-based approach, i.e. through the infinitesimal generators. More precisely, we characterize the joint double transforms of additive functionals of Markov processes and the terminal values in continuous/discrete time. Under the continuous-time Markov chain (CTMC) setting, we obtain single Laplace transforms for continuous/discrete additive functionals and their inverses. We apply the proposed transform methodology to computing option prices related to the occupation time of the underlying asset price process.

Keywords

Additive functionaloccupation timeMarkov process, continuous-time Markov chain

MSC classification

Primary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Secondary: 93E11: Filtering 93E20: Optimal stochastic control

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 27
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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