arXiv subjects

Jorge Ignacio González Cázares

Publications and source records attributed to Jorge Ignacio González Cázares.

10 recordsLinked to original sources

arXiv2024

Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function

We construct a fast exact algorithm for the simulation of the first-passage time, jointly with the undershoot and overshoot, of a tempered stable subordinator over an arbitrary non-increasing absolutely continuous function. We prove that the running time of our algorithm has finite exponential moments and provide bounds on its expected running time with explicit dependence on the characteristics of the process and the initial value of the function. The expected running time grows at most cubically in the stability parameter (as it approaches either $0$ or $1$) and is linear in the tempering parameter and the initial value of the function. Numerical performance, based on the implementation in the dedicated GitHub repository, exhibits a good agreement with our theoretical bounds. We provide numerical examples to illustrate the performance of our algorithm in Monte Carlo estimation.

math.PR↗

arXiv2023

Fast exact simulation of the first-passage event of a subordinator

This paper provides an exact simulation algorithm for the sampling from the joint law of the first-passage time, the undershoot and the overshoot of a subordinator crossing a non-increasing boundary. We prove that the running time of this algorithm has finite moments of all positive orders and give an explicit bound on the expected running time in terms of the Lévy measure of the subordinator. This bound provides performance guarantees that make our algorithm suitable for Monte Carlo estimation. We provide a GitHub repository with an implementation of the algorithm in Python and Julia.

math.PR↗

arXiv2022

Monte Carlo algorithm for the extrema of tempered stable processes

We develop a novel Monte Carlo algorithm for the vector consisting of the supremum, the time at which the supremum is attained and the position at a given (constant) time of an exponentially tempered Lévy process. The algorithm, based on the increments of the process without tempering, converges geometrically fast (as a function of the computational cost) for discontinuous and locally Lipschitz functions of the vector. We prove that the corresponding multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $\varepsilon^{-2}$ if the mean squared error is at most $\varepsilon^2$) and provide its central limit theorem (CLT). Using the CLT we construct confidence intervals for barrier option prices and various risk measures based on drawdown under the tempered stable (CGMY) model calibrated/estimated on real-world data. We provide non-asymptotic and asymptotic comparisons of our algorithm with existing approximations, leading to rule-of-thumb guidelines for users to the best method for a given set of parameters. We illustrate the performance of the algorithm with numerical examples.

q-fin.MF↗

arXiv2022

When is the convex hull of a Lévy path smooth?

We characterise, in terms of their transition laws, the class of one-dimensional Lévy processes whose graph has a continuously differentiable (planar) convex hull. We show that this phenomenon is exhibited by a broad class of infinite variation Lévy processes and depends subtly on the behaviour of the Lévy measure at zero. We introduce a class of strongly eroded Lévy processes, whose Dini derivatives vanish at every local minimum of the trajectory for all perturbations with a linear drift, and prove that these are precisely the processes with smooth convex hulls. We study how the smoothness of the convex hull can break and construct examples exhibiting a variety of smooth/non-smooth behaviours. Finally, we conjecture that an infinite variation Lévy process is either strongly eroded or abrupt, a claim implied by Vigon's point-hitting conjecture. In the finite variation case, we characterise the points of smoothness of the hull in terms of the Lévy measure.

math.PR↗

arXiv2022

Convex minorants and the fluctuation theory of Lévy processes

We establish a novel characterisation of the law of the convex minorant of any Lévy process. Our self-contained elementary proof is based on the analysis of piecewise linear convex functions and requires only very basic properties of Lévy processes. Our main result provides a new simple and self-contained approach to the fluctuation theory of Lévy processes, circumventing local time and excursion theory. Easy corollaries include classical theorems, such as Rogozin's regularity criterion, Spitzer's identities and the Wiener-Hopf factorisation, as well as a novel factorisation identity.

math.PR↗

arXiv2021

A Gaussian approximation theorem for Lévy processes

Without higher moment assumptions, this note establishes the decay of the Kolmogorov distance in a central limit theorem for Lévy processes. This theorem can be viewed as a continuous-time extension of the classical random walk result by Friedman, Katz and Koopmans.

math.PR↗

arXiv2021

Asymptotic shape of the concave majorant of a Lévy process

We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at $T$) of any Lévy process on $[0,T]$ as $T\to\infty$. The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the Lévy measure. The key tool in the proofs is the recent representation of the concave majorant for all Lévy processes using a stick-breaking representation.

math.PR↗

arXiv2021

Geometrically Convergent Simulation of the Extrema of Lévy Processes

We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general Lévy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\geq 1$) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct non-asymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $ε^{-2}$ if the mean squared error is at most $ε^2$) for locally Lipschitz and barrier-type functionals of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.

math.PR↗

arXiv2019

$\varepsilon$-strong simulation of the convex minorants of stable processes and meanders

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon$SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.

math.PR↗

arXiv2019

Exact Simulation of the Extrema of Stable Processes

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).

math.PR↗