Strong and Weak Solutions of a Stochastic General Linear-Quadratic Diffusion Model

Abstract

This paper is concerned with a one-dimensional general linear-quadratic diffusion model (LQD model) consisting of a linear drift term and a quadratic diffusion term. A strong solution is found, conditioned on two dependent Wiener processes, using the fundamental solution proposed by Shaw and Schofield (2015, p. 982) where the LQD is decomposed into the sum of two underlying dependent processes: a gBm and an aBm. The strong solution embodies the sum of (i) a gBm, (ii) a time-integral of a gBm, and (iii) a standard Wiener process. An approximative weak solution is inspired by Ioffe and Nishnianidze (2020). Three distinctive cases of the LQD model are expressed as canonical equations with a unity diffusion function. Subsequently, they are turned into corresponding Schrödinger diffusion equations. A set of intertwining relationships produces accurate approximations related to the heat equation. Known solutions for each case are then transformed back into probability density functions in the original state-variable. The result constitutes the product of three case-specific components: (i) a stationary solution associated with one of the three fundamental Pearson distributions, (ii) a time-dependent density numerically approximated by an orthogonal polynomial series expansion (Legendre, Laguerre, or Hermite), and (iii) a time-dependent log-normal distribution function. Keywords: linear-quadratic stochastic model, Fokker–Planck equation, Lamperti transform, Schrödinger diffusion equation, heat equation, supersymmetry intertwining operators, Pearson family of probability distributions, classical orthogonal polynomial series expansion. Presentation [here](https://web.tresorit.com/l/noonE#BdHMXaDmOQRltEPGhhSA-g) Paper [here](https://web.tresorit.com/l/3gKiV#mhSpRlwf_UJSmHRwoqLnyw)