Generalized Integral Transforms in Mathematical Finance

Multilayer heat equations and their solutions via oscillating integral transforms

“…In many cases, even the time-dependent models can be reduced to the MHE (or to the Multilayer Bessel equations, (Decamps et al)). However, the boundaries inherent to those problems and constant in the original variables become time-dependent in the new variables; see various examples in (Itkin et al, 2021a). Therefore, our framework covers both problems with time-dependent interfaces, as well as models with time-dependent parameters.…”

“…Therefore, to proceed, we need to sacrifice the beauty of elegant and straightforward classical transforms and instead build a new class of integral transforms. Of course, these transforms have to be adapted to the specific structure at hand; see, e.g., (Itkin et al, 2021a) where various time-dependent problems of mathematical finance and physics with moving boundaries are solved by doing precisely that. Usually, those transforms are constructed by using some basis in eigenfunctions found by solving the corresponding Sturm-Liouville (SL) problem, (Antimirov et al).…”

“…In this paper, we extend the approach of (Itkin et al, 2021a) by applying its main idea to solving various multilayer problems for the one-dimensional heat equation. Let us introduce the time t ∈ [0, ∞), the space coordinate x ∈ R, and a function u(t, x) : R + × R → R + , u ∈ C 2 .…”

“…Once this is done, the problem Eq. ( 5) can be solved by using a direct/inverse integral transform in line of (Itkin et al, 2021a). Surprisingly, in many cases, the corresponding integral transforms can be explicitly obtained.…”

“…for processes associated with Eq. ( 3) with discontinuous coefficients, can be solved by either the HP or GIT methods, (Itkin et al, 2021a).…”