arXiv subjects

Kei Noba

Publications and source records attributed to Kei Noba.

At least 19 recordsLinked to original sources

arXiv2026

Continuous-time multi-armed bandits under random intervention times

This paper examines multi-armed bandits with $J$ independent arms in which actions are taken at random discrete times. When an arm is operated, it must remain active for a renewal inter-arrival time. For arms evolving as a Lévy process, we provide an explicit characterization of the Gittins index, known to yield an optimal strategy. Furthermore, when the inter-arrival times are exponential and the arms evolve as a spectrally negative Lévy process, a reflected spectrally negative Lévy process, or a diffusion process, the Gittins index is explicitly characterized in terms of the scale function or diffusion characteristics, respectively. Convergence analysis and numerical experiments are performed to support the theoretical results.

math.OC↗

arXiv2026

Large deviations related to Dynkin--Lamperti arcsine laws for last visit times of Markov processes

We establish large deviation estimates related to the Dynkin--Lamperti arcsine laws for subordinators whose Laplace exponents are either regularly varying or comparable to regularly varying functions. By applying these results to inverse local times, we derive large deviation estimates for the last visit times to the starting point of various Hunt processes satisfying suitable on-diagonal resolvent density estimates, such as one-dimensional generalized diffusion processes, one-dimensional Lévy processes, and Brownian motion and its subordinate process on the Sierpiński gasket.

math.PR↗

arXiv2026

Stochastic control with dividend payments and capital injections for Markov additive processes

Motivated by de Finetti's optimal dividend problem with capital injections, we study a stochastic control problem for the additive component of a Markov additive process (MAP). In contrast to previous studies, the modulating component is allowed to be a general right process on a Radon space, so the model is not restricted to finite-state regime switching and cannot in general be reduced to a finite collection of Lévy process control problems. Capital injections are allowed at arbitrary times. We first consider the case in which dividend payments are allowed only at prescribed discrete times and establish necessary and sufficient conditions for the optimality of a strategy. These conditions then yield the optimality of a class of Markov-modulated periodic--classical barrier strategies. Combining this optimality result with an approximation argument, we obtain insight into the possible form of optimal strategies in the case where dividend payments, like capital injections, may be made at arbitrary times. Because of the generality of the MAPs considered here, the proof techniques used in previous studies of similar problems are not directly applicable. We therefore develop an alternative argument based on the additive structure of MAPs and dynamic programming between dividend opportunities. The argument also suggests a possible approach to other stochastic control problems involving general MAPs.

math.PR↗

arXiv2026

Fluctuation theory for spectrally negative Lévy processes killed by additive functionals

In this paper, we study fluctuation identities for spectrally negative Lévy processes killed by a general class of additive functionals. We consider positive co-natural additive functionals (PcNAFs), which include as special cases both absolutely continuous functionals and finite mixtures of local times. Our main result shows that the associated fluctuation identities, such as two-sided exit problems and resolvent measures, retain the same structure as in the classical case and can be expressed in terms of generalized scale functions. These scale functions are characterized as the unique solutions to Volterra-type integral equations driven by Radon measures, thereby extending the results of Li and Palmowski and Li and Zhou. Our approach is based on representing the additive functional as a mixture of local times with respect to its Revuz measure, combined with classical fluctuation identities and an approximation scheme for general Radon measures using Poisson random measures.

math.PR↗

arXiv2025

On optimal periodic dividend and capital injection strategies for general Lévy models

We consider a version of de Finetti's dividend problem, with the bail-out contraint to keep the surplus non-negative, and where dividend payments can only be made at the arrival times of an independent Poisson process. For a general Lévy process with positive and negative jumps, we show the optimality of a periodic-classical reflection strategy that pays the excess above a given level at each Poisson arrival time, and also reflects below at 0 in the classical sense.

math.PR↗

arXiv2024

On stochastic control under Poisson observations: optimality of a barrier strategy in a general Lévy model

We study a version of the stochastic control problem of minimizing the sum of running and controlling costs, where control opportunities are restricted to independent Poisson arrival times. Under a general setting driven by a general Lévy process, we show the optimality of a periodic barrier strategy, which moves the process upward to the barrier whenever it is observed to be below it. The convergence of the optimal solutions to those in the continuous-observation case is also shown.

math.OC↗

arXiv2024

Optimal dividends and capital injection: A general Lévy model with extensions to regime-switching models

This paper studies a general Lévy process model of the bail-out optimal dividend problem with an exponential time horizon, and further extends it to the regime-switching model. We first show the optimality of a double barrier strategy in the single-regime setting with a concave terminal payoff function. This is then applied to show the optimality of a Markov-modulated double barrier strategy in the regime-switching model via contraction mapping arguments. We solve these for a general Lévy model with both positive and negative jumps, greatly generalizing the existing results on spectrally one-sided models.

math.PR↗

arXiv2024

Analytic property of generalized scale functions for standard processes with no negative jumps and its application to quasi-stationary distributions

For a generalized scale function of standard processes, we characterize it as a unique solution to a Volterra type integral equation. This allows us to extend it to an entire function and to derive a useful identity that we call the resolvent identity. We apply this result to study the existence of a quasi-stationary distribution for the processes killed at hitting boundaries. A new classification of the boundary, which is a natural extension of Feller's for one-dimensional diffusions, is introduced and plays a central role to characterize the existence.

math.PR↗

arXiv2023

Scale functions of space-time changed processes with no positive jumps

The scale functions were defined for spectrally negative Lévy processes and other strong Markov processes with no positive jumps, and have been used to characterize their behavior. In particular, I defined the scale functions for standard processes with no positive jumps using the excursion measures in Noba(2020). In this paper, we consider a standard process $X$ with no positive jumps and a standard process $Y$ defined by the space-time change of $X$. We express the scale functions of $Y$ using the scale functions of $X$ defined in Noba(2020) and the Volterra integral equation. From this result, we can express the scale functions of some important processes, such as positive or negative self-similar Markov processes with no positive jumps and continuous-state branching processes, using the scale function of spectrally negative Lévy processes and the Volterra integral equations.

math.PR↗

arXiv2023

Refraction strategies in stochastic control: optimality for a general Lévy process model

We revisit an absolutely-continuous version of the stochastic control problem driven by a Lévy process. A strategy must be absolutely continuous with respect to the Lebesgue measure and the running cost function is assumed to be convex. We show the optimality of a refraction strategy, which adjusts the drift of the state process at a constant rate whenever it surpasses a certain threshold. The optimality holds for a general Lévy process, generalizing the spectrally negative case presented in Hernández-Hernández et al.(2016).

math.PR↗

arXiv2022

On the optimality of the refraction--reflection strategy for Lévy processes

In this paper, we study de Finetti's optimal dividend problem with capital injection under the assumption that the dividend strategies are absolutely continuous. In many previous studies, the process before being controlled was assumed to be a spectrally one-sided Lévy process, however in this paper we use a Lévy process that may have both positive and negative jumps. In the main theorem, we show that a refraction--reflection strategy is an optimal strategy. We also mention the existence and uniqueness of solutions of the stochastic differential equations that define refracted Lévy processes.

math.PR↗

arXiv2022

On singular control for Lévy processes

We revisit the classical singular control problem of minimizing running and controlling costs. The problem arises in inventory control, as well as in healthcare management and mathematical finance. Existing studies have shown the optimality of a barrier strategy when driven by the Brownian motion or Lévy processes with one-side jumps. Under the assumption that the running cost function is convex, we show the optimality of a barrier strategy for a general class of Lévy processes.

math.PR↗

arXiv2022

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative \lev process with a final payoff at an exponential terminal time and characterise the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.

math.PR↗

arXiv2022

On Boolean selfdecomposable distributions

This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and the singular continuous part is zero. We then analyze how shifting probability measures changes Boolean selfdecomposability. Several examples are presented to supplement the above results. Finally, we prove that the standard normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one $N(m,1)$ is not for sufficiently large $|m|$.

math.PR↗

arXiv2020

On the bail-out dividend problem for spectrally negative Markov additive models

This paper studies the bail-out optimal dividend problem with regime switching under the constraint that the cumulative dividend strategy is absolutely continuous. We confirm the optimality of the regime-modulated refraction-reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. To verify the conjecture of a barrier type optimal control, we first introduce and study an auxiliary problem with the final payoff at an exponential terminal time and characterize the optimal threshold explicitly using fluctuation identities of the refracted-reflected Levy process. Second, we transform the problem with regime-switching into an equivalent local optimization problem with a final payoff up to the first regime switching time. The refraction-reflection strategy with regime-modulated thresholds can be shown as optimal by using results in the first step and some fixed point arguments for auxiliary recursive iterations.

q-fin.MF↗

arXiv2019

On the optimality of double barrier strategies for Lévy processes

This paper studies de Finetti's optimal dividend problem with capital injection. We confirm the optimality of a double barrier strategy when the underlying risk model follows a Lévy process that may have positive and negative jumps. The main result in this paper is a generalization of Theorem 3 in Avram et al.(2007), which is the spectrally negative case, and Theorem 3.1 in Bayraktar et al.(2013), which is the spectrally positive case. In contrast with the spectrally one-sided cases, double barrier strategies cannot be handled by using scale functions to obtain some properties of the expected net present values (NPVs) of dividends and capital injections. Instead, to obtain these properties, we observe changes in the sample path (and the associated NPV) when there is a slight change to the initial value or the barrier value.

math.PR↗

arXiv2019

Generalized refracted Lévy process and its application to exit problem

Generalizing Kyprianou--Loeffen's refracted Lévy processes, we define a new refracted Lévy process which is a Markov process whose positive and negative motions are Lévy processes different from each other. To construct it we utilize the excursion theory. We study its exit problem and the potential measures of the killed processes. We also discuss approximation problem.

math.PR↗

arXiv2018

Approximation and duality problems of refracted processes

For given two standard processes with no positive jumps, we construct, using the excursion theory, a Markov process whose positive and negative motions have the same law as the two processes. The resulting process is a generalization of Kyprianou--Loeffen's refracted Lévy processes. We discuss approximation problem for our refracted processes coming from Lévy processes by removing small jumps and taking the limit as the removal level tends to zero. We also discuss conditions for refracted processes to have dual processes.

math.PR↗