[Paper Review] A Lie systems approach for the first passage-time of piecewise deterministic processes

Florin Avram, José F. Cariñena+1 more|arXiv (Cornell University)|Aug 16, 2010

Differential Equations and Boundary Problems22 references3 citations

TL;DR

This paper applies Lie systems theory to solve first passage-time problems for piecewise deterministic processes with phase-type jumps, enabling explicit analytical solutions for ruin probabilities in a class of hypergeometric diffusion models with Markovian jump components. The key contribution is a systematic method to reduce the associated integro-differential equations to solvable Riccati and ODE systems via symmetry-based transformations, yielding closed-form expressions for first-passage and ruin probabilities under specific drift and jump structures.

ABSTRACT

Our paper illustrates how the theory of Lie systems allows recovering known results and provide new examples of piecewise deterministic processes with phase-type jumps for which the corresponding first-time passage problems may be solved explicitly.

Motivation & Objective

• To extend analytical solvability of first-passage problems in jump-diffusion processes beyond standard affine models by incorporating phase-type jump distributions.
• To investigate whether the algebraic-geometric framework of Lie systems can provide new tractable models for first-passage time problems in processes with non-Poissonian or structured jump dynamics.
• To demonstrate that the combination of solvable Lie algebras and phase-type distributions allows for explicit solutions to the backward Kolmogorov equation in a class of hypergeometric diffusions with Markovian jumps.
• To provide a systematic method for transforming complex integro-differential equations into solvable ordinary differential equations using symmetry-based reparametrization and change of variables.
• To recover known results in risk theory (e.g., Cramér-Lundberg model) and extend them to new models with analytically tractable ruin probabilities under general drift and jump structures.

Proposed method

• Applies the theory of Lie systems to the backward Kolmogorov equation governing the first-passage time distribution, exploiting the underlying solvable Lie algebra structure of the infinitesimal generator.
• Uses a change of variables based on the Riccati equation derived from the generator, transforming the non-autonomous system into an autonomous one via a reparametrization of the state space.
• Employs a transformation involving the function $ ar{x}(x) = rac{1}{2} extstyleig(rac{ u}{q+ u}ig)^{1/2} extstyleig[ extstyle ext{log}ig(rac{1- extstyleig(1-Kig)^{1/2}}{1+ extstyleig(1-Kig)^{1/2}}ig)ig] $ to linearize the dynamics and reduce the problem to quadrature.
• Introduces a new variable $ ar{ heta}(x) $ such that the Riccati equation becomes $ \frac{d\bar{\eta}}{dx} = \sqrt{\frac{-\lambda\mu}{\varphi_K(x)}}(1 - \bar{\eta}^2) $, which is solvable in terms of hyperbolic tangents.
• Solves the system $ \frac{dM}{dx} = (\mu \eta(x) - \mu)M $ and $ \Psi(x) = \eta(x)M(x) $ using the transformed variable $ \bar{x}(x) $, yielding explicit expressions in terms of exponential and logarithmic functions.
• Imposes boundary conditions $ \Psi(0) = 1 $, $ M(0) = 1 $, and derives the final solution in terms of $ K_1(K) = \frac{\sqrt{\lambda/(q+\lambda)} - \sqrt{1-K}}{\sqrt{\lambda/(q+\lambda)} + \sqrt{1-K}} $, enabling analytical evaluation of ruin probabilities.

Experimental results

Research questions

• RQ1 Can the theory of Lie systems be used to solve first-passage-time problems in piecewise deterministic processes with phase-type jumps, beyond standard diffusion or Poisson jump models?
• RQ2 Under what conditions on the drift and jump intensity does the associated backward Kolmogorov equation admit a Lie system structure that allows for explicit integration?
• RQ3 Can the Riccati equation arising from the first-passage problem be transformed into an autonomous system via a change of variables, enabling analytical solution in terms of elementary functions?
• RQ4 What is the asymptotic behavior of the ruin probability as the initial surplus tends to infinity in the proposed model with negative drift and phase-type jumps?
• RQ5 How do the parameters $ \lambda, \mu, q, K $ affect the shape and decay rate of the ruin probability function in the derived solution?

Key findings

• The first-passage time problem for a class of hypergeometric diffusions with phase-type jumps is analytically solvable using Lie systems, yielding explicit closed-form expressions for the ruin probability $ \Psi(x) $ and the associated function $ M(x) $.
• The solution is derived via a symmetry-based transformation that reparametrizes the state space such that the Riccati equation becomes autonomous, allowing integration by quadrature.
• The asymptotic behavior of the ruin probability satisfies $ \Psi(x) \sim C \cdot e^{\mu\big(\sqrt{\lambda/(\lambda+q)} - 1\big)x} $ as $ x \to \infty $, with the exponent negative when $ \sqrt{\lambda/(\lambda+q)} < 1 $, implying $ \lim_{x\to\infty} \Psi(x) = 0 $.
• For the specific case $ \mu = 1.5, \lambda = q = 0.5, K = 0.75 $, the model exhibits a well-defined, decaying ruin probability that matches the derived analytical form.
• The method recovers known results from risk theory (e.g., Cramér-Lundberg model) and extends them to new models with non-Poissonian jump dynamics and structured phase-type jump distributions.
• The solution depends explicitly on the parameter $ K_1(K) = \frac{\sqrt{\lambda/(q+\lambda)} - \sqrt{1-K}}{\sqrt{\lambda/(q+\lambda)} + \sqrt{1-K}} $, which controls the shape of the ruin probability curve and ensures consistency with boundary conditions at $ x = 0 $.