An Analytical Formula for the Transition Density of a Conic Combination of Independent Squared Bessel Processes with Time-Dependent Dimensions and Financial Applications

Nopporn Thamrongrat, Chhaunny Chhum, Sanae Rujivan () and Boualem Djehiche
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Nopporn Thamrongrat: Research Center in Data Science for Health Study, Division of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand
Chhaunny Chhum: Research Center in Data Science for Health Study, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand
Sanae Rujivan: Research Center in Data Science for Health Study, Division of Mathematics and Statistics, School of Science, Walailak University, Nakhon Si Thammarat 80161, Thailand
Boualem Djehiche: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

Mathematics, 2025, vol. 13, issue 13, 1-34

Abstract: The squared Bessel process plays a central role in stochastic analysis, with broad applications in mathematical finance, physics, and probability theory. While explicit expressions for its transition probability density function (TPDF) under constant parameters are well known, analytical results in the case of time-dependent dimensions remain scarce. In this paper, we address a significantly challenging problem by deriving an analytical formula for the TPDF of a conic combination of independent squared Bessel processes with time-dependent dimensions. The result is expressed in terms of a Laguerre series expansion. Furthermore, we obtain closed-form expressions for the conditional moments of such conic combinations, represented via generalized hypergeometric functions. These results also yield new analytical formulas for the TPDF and conditional moments of both squared Bessel processes and Bessel processes with time-dependent dimensions. The proposed formulas provide a unified analytical framework for modeling and computation involving a broad class of time-inhomogeneous diffusion processes. The accuracy and computational efficiency of our formulas are verified through Monte Carlo simulations. As a practical application, we provide an analytical valuation of an interest rate swap, where the underlying short rate follows a conic combination of independent squared Bessel processes with time-dependent dimensions, thereby illustrating the theoretical and practical significance of our results in mathematical finance.

Keywords: squared Bessel process; transition probability density function; conditional moments; time-dependent dimension; Laguerre series expansion; financial applications (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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