[Paper Review] Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions

Claudio Albanese, Alexey Kuznetsov|ArXiv.org|Oct 8, 2007

Stochastic processes and financial applications18 references17 citations

TL;DR

This paper introduces a novel classification scheme for analytically solvable one-dimensional diffusion processes by leveraging stochastic transformations that eliminate drift through a combination of Doob’s h-transform and diffeomorphisms. The key contribution is a systematic method to generate new families of solvable driftless diffusions—specifically hypergeometric and confluent hypergeometric R-families—by transforming known processes, with all such processes characterized by transition densities expressible in terms of hypergeometric functions via invariant Bose invariants and Liouville transformations.

ABSTRACT

We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction makes use of transformations of diffusion processes to eliminate the drift which combine a measure change given by Doob's h-transform and a diffeomorphism. Such transformations have the important property of preserving analytic solvability of the process: the transition probability density for the driftless process can be expressed through the transition probability density of original process. We also make use of tools from the theory of ordinary differential equations such as Liouville transformations, canonical forms and Bose invariants. Beside recognizing all analytically solvable diffusion process known in the previous literature fall into this scheme and we also discover rich new families of analytically solvable processes.

Motivation & Objective

• To develop a comprehensive classification scheme for one-dimensional Markov processes whose backward Kolmogorov equation is solvable in closed form via reduction to hypergeometric equations.
• To establish a framework for transforming general diffusion processes into driftless forms while preserving analytic solvability, using measure changes (Doob’s h-transform) and diffeomorphisms.
• To identify and characterize new families of analytically solvable diffusions beyond known classes such as Ornstein-Uhlenbeck, Bessel, CIR, and Jacobi processes.
• To introduce invariants—Bose invariants and Liouville transformations—that preserve solvability under transformation and enable systematic generation of new solvable processes.
• To unify and generalize existing results in the literature by showing that all previously known analytically solvable diffusions fall into this classification, while discovering richer new families.

Proposed method

• Utilizes stochastic transformations composed of Doob’s h-transform (measure change) and diffeomorphisms (state space transformation) to eliminate drift from a diffusion process.
• Applies Liouville transformations and canonical forms to reduce second-order differential operators (generators) to standard hypergeometric or confluent hypergeometric equations.
• Employs Bose invariants as invariants under stochastic transformations, ensuring that solvability is preserved across the transformation class.
• Derives the volatility function of the transformed process using the Jacobian of the transformation and the Wronskian of fundamental solutions to the associated ODE.
• Constructs the transition density of the driftless process from the original process via the transformation relationship, leveraging the fact that the density transforms via the change of measure and state space.
• Uses the theory of second-order linear ODEs with regular singular points to classify all possible solvable generators based on their canonical forms and associated special functions (hypergeometric or confluent hypergeometric).

Experimental results

Research questions

• RQ1 Which classes of one-dimensional diffusion processes admit analytically closed-form solutions to their backward Kolmogorov equation?
• RQ2 How can stochastic transformations be systematically constructed to remove drift from a diffusion process while preserving analytic solvability?
• RQ3 What invariants remain unchanged under such transformations, and how can they be used to classify solvable processes?
• RQ4 Can the framework generate new families of solvable diffusions beyond the classical ones (e.g., CIR, Jacobi, Ornstein-Uhlenbeck)?
• RQ5 What is the precise relationship between the generator of the original process and the volatility function of the transformed driftless process?

Key findings

• All known analytically solvable diffusions—such as Ornstein-Uhlenbeck, Bessel, CIR, and Jacobi processes—are subsumed within the proposed classification scheme.
• The paper constructs two new families of solvable diffusions: the hypergeometric R-family (for $A(x) = x(1-x)$) and the confluent hypergeometric R-family (for $A(x) = x$), which generalize the Jacobi and CIR processes.
• The volatility function of the transformed driftless process is explicitly given by $\sigma_Y(Y(x)) = y'(x) / \gamma(y(x))$, where $y(x)$ is a Möbius transformation of solutions to a hypergeometric-type ODE.
• The transition density of the driftless process is expressible in terms of hypergeometric functions through the transformation of the original process’s density, preserving analytic solvability.
• The Bose invariants are invariant under stochastic transformations and serve as a complete classification tool for solvable generators, enabling the identification of new solvable classes.
• The second classification theorem establishes that all processes in the (confluent) hypergeometric R-family arise as stochastic transformations of a single base process with dynamics $dX_t = (a + bX_t) \frac{A(X_t)}{R(X_t)} dt + \frac{A(X_t)}{\sqrt{R(X_t)}} dW_t$, where $R(x)$ is a quadratic polynomial with no zeros in the domain.