arXiv subjects

Libo Li

Publications and source records attributed to Libo Li.

At least 19 recordsLinked to original sources

arXiv2026

A Monotone Limit Approach to Entropy-Regularized American Options

Recent advances in continuous-time optimal stopping have been driven by entropy-regularized formulations of randomized stopping problems, with most existing approaches relying on partial differential equation methods. In this paper, we propose a fully probabilistic framework based on the Doob-Meyer-Mertens decomposition of the Snell envelope and its representation through reflected backward stochastic differential equations. We introduce an entropy-regularized penalization scheme yielding a monotone approximation of the value function and establish explicit convergence rates under suitable regularity assumptions. In addition, we develop a policy improvement algorithm based on linear backward stochastic differential equations and illustrate its performance through a simple numerical experiment for an American-style max call option

q-fin.CP↗

arXiv2026

Entropy-regularized penalization schemes and reflected BSDEs with singular generators

This paper extends our previous work to continuous-time optimal stopping, focusing on American options in an exploratory setting. Our first contribution is an entropy-regularized penalization scheme, inspired by classical penalization techniques for reflected BSDEs. It yields a smooth approximation of the stopping rule, promotes exploration, and enables gradient-based learning methods. We prove well-posedness, convergence, and illustrate numerical performance in low-dimensional examples. Our second contribution analyzes the behaviour of the scheme as the penalization parameter grows, showing that the limit solves a reflected BSDE with a logarithmically singular generator, for which we establish existence and uniqueness via a monotone limit argument.

q-fin.MF↗

arXiv2025

An Entropy Regularized BSDE Approach to Bermudan Options and Games

In this paper, we investigate optimal stopping problems in a continuous-time framework where only a discrete set of stopping dates is admissible, corresponding to the Bermudan option, within the so-called exploratory formulation. We introduce an associated control problem for the value function, represented as a non-c\`adl\`ag reflected backward stochastic differential equation (RBSDE) with an entropy regulariser that promotes exploration, and we establish existence and uniqueness results for this entropy-regularised RBSDE. We then compare the entropy-regularised RBSDE with the theoretical value of a Bermudan option and propose a reinforcement learning algorithm based on a policy improvement scheme, for which we prove both monotone improvement and convergence. This methodology is further extended to Bermudan game options, where we obtain analogous results. Finally, drawing on the preceding analysis, we present two numerical approximation schemes - a BSDE solver based on a temporal-difference scheme and neural networks and the policy improvement algorithm - to illustrate the feasibility and effectiveness of our approach.

math.PR↗

arXiv2025

Robust quantum computational advantage with programmable 3050-photon Gaussian boson sampling

The creation of large-scale, high-fidelity quantum computers is not only a fundamental scientific endeavour in itself, but also provides increasingly robust proofs of quantum computational advantage (QCA) in the presence of unavoidable noise and the dynamic competition with classical algorithm improvements. To overcome the biggest challenge of photon-based QCA experiments, photon loss, we report new Gaussian boson sampling (GBS) experiments with 1024 high-efficiency squeezed states injected into a hybrid spatial-temporal encoded, 8176-mode, programmable photonic quantum processor, Jiuzhang 4.0, which produces up to 3050 photon detection events. Our experimental results outperform all classical spoofing algorithms, particularly the matrix product state (MPS) method, which was recently proposed to utilise photon loss to reduce the classical simulation complexity of GBS. Using the state-of-the-art MPS algorithm on the most powerful supercomputer EI Capitan, it would take > $10^{42}$ years to construct the required tensor network for simulation, while our Jiuzhang 4.0 quantum computer takes 25.6 $\mu$s to produce a sample. This work establishes a new frontier of QCA and paves the way to fault-tolerant photonic quantum computing hardware.

quant-ph↗

arXiv2025

Perpetual American Standard and Lookback Options in Insider Models with Progressively Enlarged Filtrations

We derive closed-form solutions to the optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in the extensions of the Black-Merton-Scholes model with progressively enlarged filtrations. More specifically, the information available to the insider is modelled by Brownian filtrations progressively enlarged with the times of either the global maximum or minimum of the underlying risky asset price over the infinite time interval, which is not a stopping time in the filtration generated by the underlying risky asset. We show that the optimal exercise times are the first times at which the asset price process reaches either lower or upper stochastic boundaries depending on the current values of its running maximum or minimum given the occurrence of times of either the global maximum or minimum, respectively. The proof is based on the reduction of the original problems into the necessarily three-dimensional optimal stopping problems and the equivalent free-boundary problems. We apply either the normal-reflection or the normal-entrance conditions as well as the smooth-fit conditions for the value functions to characterise the candidate boundaries as either the maximal or minimal solutions to the associated first-order nonlinear ordinary differential equations and the transcendental arithmetic equations, respectively.

q-fin.MF↗

arXiv2024

Variable martingale Hardy-Lorentz-Karamata spaces and their applications in Fourier Analysis

In this paper, we introduce a new class of function spaces, which unify and generalize Lorentz-Karamata spaces, variable Lorentz spaces and other several classical function spaces. Based on the new spaces, we develop the theory of variable martingale Hardy-Lorentz-Karamata spaces and apply it to Fourier Analysis. To be precise, we discuss the basic properties of Lorentz-Karamata spaces with variable exponents. We introduce five variable martingale Hardy-Lorentz-Karamata spaces and characterize them via simple atoms as well as via atoms. As applications of the atomic decompositions, dual theorems and the generalized John-Nirenberg theorem for the new framework are presented. Moreover, we obtain the boundedness of $\sigma$-sublinear operator defined on variable martingale Hardy-Lorentz-Karamata spaces, which leads to martingale inequalities and the relation of the five variable martingale Hardy-Lorentz-Karamata spaces. Also, we investigate the boundedness of fractional integral operators in this new framework. Finally, we deal with the applications of variable martingale Hardy-Lorentz-Karamata spaces in Fourier analysis by using the previous results. More precisely, we show that the partial sums of the Walsh-Fourier series converge to the function in norm if $f\in L_{p(\cdot),q,b}$ with $1<p_-\le p_+<\infty$. The Fej\'{e}r summability method is also studied and it is proved that the maximal Fej\'{e}r operator is bounded from variable martingale Hardy-Lorentz-Karamata spaces to variable Lorentz-Karamata spaces. As a consequence, we get conclusions about almost everywhere and norm convergence of Fej\'{e}r means. The results obtained in this paper generalize the results for martingale Hardy-Lorentz-Karamata spaces and variable martingale Hardy-Lorentz spaces. Especially, we remove the condition that $b$ is nondecreasing in previous literature.

math.FA↗

arXiv2023

Integration by parts formula for exit times of one dimensional diffusions

In line with the methodology introduced in our recent article for formulating probabilistic representations of integration by parts involving killed diffusion, we establish an integration by parts formula for the first exit time of one-dimensional diffusion processes. However, our approach diverges from the conventional differential calculus applied to the associated space Markov chain; instead, we employ calculus techniques that focus on the underlying time variables.

math.PR↗

arXiv2022

Well-posedness and penalization schemes for generalized BSDEs and reflected generalized BSDEs

The paper is directly motivated by the pricing of vulnerable European and American options in a general hazard process setup and a related study of the corresponding pre-default backward stochastic differential equations (BSDE) and pre-default reflected backward stochastic differential equations (RBSDE). We work with a generic filtration $\FF$ for which the martingale representation property is assumed to hold with respect to a square-integrable martingale $M$ and the goal of this work is of twofold. First, we aim to establish the well-posedness results and comparison theorems for a generalized BSDE and a reflected generalized BSDE with a continuous and nondecreasing driver $A$. Second, we study extended penalization schemes for a generalized BSDE and a reflected generalized BSDE in which we penalize against the driver in order to obtain in the limit either a particular optimal stopping problem or a Dynkin game in which the set of admissible exercise time is constrained to the right support of the measure generated by $A$.

math.PR↗

arXiv2022

Vulnerable European and American Options in a Market Model with Optional Hazard Process

We study the upper and lower bounds for prices of European and American style options with the possibility of an external termination, meaning that the contract may be terminated at some random time. Under the assumption that the underlying market model is incomplete and frictionless, we obtain duality results linking the upper price of a vulnerable European option with the price of an American option whose exercise times are constrained to times at which the external termination can happen with a non-zero probability. Similarly, the upper and lower prices for an vulnerable American option are linked to the price of an American option and a game option, respectively. In particular, the minimizer of the game option is only allowed to stop at times which the external termination may occur with a non-zero probability.

q-fin.MF↗

arXiv2022

Defaultable perpetual Russian option Under a last passage time model

In this article we provide a valuation formula for a defaultable perpetual Russian option in the Black-Scholes market where the default time is modelled as the last passage time of the running maximum of the stock price. In this setting, default occurs when the stock price fails to exceed its historical maximum, leading to a non-stopping time that depends on the path of the underlying asset.

math.PR↗

arXiv2022

American Options with Last Exit Times: A Free-Boundary Approach

We study the valuation of an American put option with a random time horizon given by the last exit time of the underlying asset from a fixed level. Since this random time is not a stopping time, the problem falls outside the classical optimal stopping framework. Using enlargement of filtrations and the associated Az\'{e}ma supermartingale, we transform the problem into an equivalent optimal stopping problem with a semi-continuous, time-dependent gain function whose partial derivatives exhibit singular behaviour. The resulting formulation introduces significant analytical challenges, including the loss of smoothness of the optimal stopping boundary. We develop new arguments to characterise the continuation and stopping regions, establishing monotonicity of the free boundary under suitable conditions, and analyse the regularity of the value function. In particular, we derive nonlinear integral equations that uniquely characterise both the free-boundary and the value function. Our results extend the classical theory of American options to a class of problems with random horizons and provide a framework for incorporating default-type features modelled by last exit times.

math.PR↗

arXiv2021

Generalized BSDEs with random time horizon in a progressively enlarged filtration

We study generalized backward stochastic differential equations (BSDEs) up to a random time horizon $\vartheta$, which is not a stopping time, under minimal assumptions regarding the properties of $\vartheta$. In contrast to existing works in this area, we do not impose specific assumptions on the random time $\vartheta$ and we study the existence of solutions to BSDEs and reflected BSDEs with a random time horizon through the method of reduction. In addition, we also examine BSDEs and reflected BSDEs with a l\`adl\`ag driver where the driver is allowed to have a finite number of common jumps with the martingale part.

math.PR↗

arXiv2021

A positivity preserving numerical scheme for the alpha-CEV process

In this article, we present a method to construct a positivity-preserving numerical scheme for a jump-extended CEV (Constant Elasticity of Variance) process, whose jumps are governed by a spectrally positive $\alpha$-stable process with $\alpha \in (1,2)$. The numerical scheme is obtained by making the diffusion coefficient $x^\gamma$, where $\gamma \in (\frac{1}{2},1)$, partially implicit and then finding the appropriate adjustment factor. We show that, for sufficiently small step size, the proposed scheme converges and theoretically achieves a strong convergence rate of at least $\frac{1}{2}\left(\frac{\alpha_-}{2} \wedge \frac{1}{\alpha}\wedge \rho\right)$, where $\rho \in (\frac{1}{2},1)$ is the H\"older exponent of the jump coefficient $x^\rho$ and the constant $\alpha_- < \alpha$ can be chosen arbitrarily close to $\alpha \in (1,2)$.

math.NA↗

arXiv2020

Well-posedness and approximation of some one-dimensional L\'evy-driven non-linear SDEs

In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean-Vlasov, driven by a spectrally-positive L{\'e}vy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada-Watanabe type without non-degeneracy assumption. The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler-Maruyama scheme are also investigated. In particular, the strong convergence rate of the Euler-Maruyama scheme exhibits an interplay between the regularity of the coefficients and the order of singularity of the L{\'e}vy measure around zero.

math.PR↗

arXiv2019

Integration by parts formula for killed processes: A point of view from approximation theory

In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li's type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.

math.PR↗

arXiv2019

Evaluating software defect prediction performance: an updated benchmarking study

Accurately predicting faulty software units helps practitioners target faulty units and prioritize their efforts to maintain software quality. Prior studies use machine-learning models to detect faulty software code. We revisit past studies and point out potential improvements. Our new study proposes a revised benchmarking configuration. The configuration considers many new dimensions, such as class distribution sampling, evaluation metrics, and testing procedures. The new study also includes new datasets and models. Our findings suggest that predictive accuracy is generally good. However, predictive power is heavily influenced by the evaluation metrics and testing procedure (frequentist or Bayesian approach). The classifier results depend on the software project. While it is difficult to choose the best classifier, researchers should consider different dimensions to overcome potential bias.

cs.SE↗

arXiv2018

On a positivity preserving numerical scheme for jump-extended CIR process: the alpha-stable case

We propose a positivity preserving implicit Euler-Maruyama scheme for a jump-extended Cox-Ingersoll-Ross (CIR) process where the jumps are governed by a compensated spectrally positive $\alpha$-stable process for $\alpha \in (1,2)$. Different to the existing positivity preserving numerical schemes for jump-extended CIR or CEV (Constant Elasticity Variance) process, the model considered here has infinite activity jumps. We calculate, in this specific model, the strong rate of convergence and give some numerical illustrations. Jump extended models of this type were initially studied in the context of branching processes and was recently introduced to the financial mathematics literature to model sovereign interest rates, power and energy markets.

math.PR↗

arXiv2018

Characterisation of honest times and optional semimartingales of class-($\Sigma$)

Given a finite honest time, we first show that the associated Az\'ema optional supermartingale can be expressed as the drawdown and the relative drawdown of some local optional supermartingales with continuous running supremum. The relative drawdown representation then allows us to provide a characterisation of finite honest times using a family of non-negative local optional supermartingales with continuous running supremum which converges to zero at infinity. Then we extend the notion of semimartingales of class-$(\Sigma)$ by allowing for jumps in its finite variation part of the semimartingale decomposition. This enables one to establish the Madan-Roynette-Yor option pricing formula for a larger class of processes, and finally, we apply the extended formula to the construction of finite honest times.

math.PR↗