Orthogonal gamma-based expansion for the CIR's first passage time distribution

TL;DR

This work tackles the CIR first-passage time problem by building a Laguerre–Gamma orthogonal expansion that approximates the FPT density $g(t)$ and distribution $G(t)$. It analyzes truncation errors, selects gamma parameters via moment matching, and introduces standardization and positivity-correction strategies to maintain valid densities. It also proposes an acceptance–rejection–type algorithm that uses the approximation to generate FPT samples when the true distribution is unknown, and it compares the approach with Volterra integral equations and Monte Carlo, highlighting strengths and limitations. The methodology extends to other diffusions by relying on moment-derived reference densities and offers practical tools for density estimation and simulation of FPTs.

Abstract

In this paper we analyze a method for approximating the first-passage time density and the corresponding distribution function for a CIR process. This approximation is obtained by truncating a series expansion involving the generalized Laguerre polynomials and the gamma probability density. The suggested approach involves a number of numerical issues which depend strongly on the coefficient of variation of the first passage time random variable. These issues are examined and solutions are proposed also involving the first passage time distribution function. Numerical results and comparisons with alternative approximation methods show the strengths and weaknesses of the proposed method. A general acceptance-rejection-like procedure, that makes use of the approximation, is presented. It allows the generation of first passage time data, even if its distribution is unknown.

Figures (7)

• Figure 1: In a) plots of the approximation $\hat{g}_n$ and of its correction $\hat{g}^{corr}_{n}$\ref{['eq:corr1']} over the interval $(0,t'_2)$ are given for $n = 10$ and parameters $y_0 = 0$, $\mu = 3$, $S=10$, $c = - 10$, $\sigma = 1.2, \tau = 0.2$ (see case C in Section \ref{['numexamples']}). In b), plots of the approximation $\hat{g}_n$ and of its correction $\hat{g}^{corr}_{n}$\ref{['eq:corr2']} over the interval $(t'_1,t'_2)$ are given for $n = 9$ and parameters $y_0 = 0.2$, $\mu = 0.9$, $S =1$, $c = 0$, $\sigma = 1.2, \tau = 2/3$ (see case A in Section \ref{['numexamples']}).
• Figure 2: Plots of the empirical (not standardized) FPT cdfs for cases A (in solid red), B (in dash green), and C (in dashed purple).
• Figure 3: Plots of the approximated $\tilde{G}_n(t)$ (in solid blue) and of the empirical cdf (in dashed red) together with the corresponding absolute error $\varepsilon_{a}.$ The plots refer: to case A in a) with $n = 10$, $\alpha = 0.367$ and $\beta = 1.17$; to case B in b) with $n = 10$, $\alpha = -0.34$ and $\beta = 0.812$; to case C in c) with $n = 9$, $\alpha = 0.7$ and $\beta = 1.306$. Note that $\tilde{G}_n(t)$ is obtained using the stopping criteria \ref{['eq:stop']} and corrected according to Section \ref{['sec:pos1']} while the empirical cdf is obtained from the standardized samples ${\mathcal{T}}_A, {\mathcal{T}}_B$ and ${\mathcal{T}}_C$ respectively.
• Figure 4: Plot of $\tilde{g}_n(t)$ (in solid blue) in case A with $n = 10$, $\alpha = 0.367$ and $\beta = 1.17$, obtained with the stopping criteria \ref{['eq:stop']} and corrected to ensure positivity as in \ref{['eq:corr2']}, together with a KDE (in dashed red) and a histogram both computed with the standardized sample $\mathcal{T}_A$.
• Figure 5: In a), plot of $\tilde{g}_n(t)$ (in solid blue) in case B with $n = 10$, $\alpha = -0.34$ and $\beta = 0.812$, obtained with the stopping criteria \ref{['eq:stop']} together with a KDE (in dashed red) and a histogram both computed with the standardized sample $\mathcal{T}_B$. In b), a plot of $\tilde{g}_n(t)$ (in solid blue) in case B with $n = 55$ and $\alpha = -0.34$, obtained without any stopping criterion, increasing the numerical precision.

Theorems & Definitions (16)

• Remark 2.1
• Proposition 2.2
• Theorem 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Theorem 6.1: Vysochanskij-Petunin inequality
• Remark 6.2
• Lemma 6.3