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Refractory period (RP), the waiting time between signals, can induce complex signaling dynamics, such as acceleration, adaptation, and oscillation, within many cellular biochemical networks. However, its underlying molecular mechanisms are still unclear. Rigorously estimating the RP distribution may be essential to identify its causal regulatory mechanisms. Traditional methods of estimating the RP distribution depend on solving the underlying Chemical Master Equations (CMEs), the dominant modeling formalism of biochemical systems. However, exact solutions of the CME are only known for simple reaction systems with zero- and first-order reactions or specific systems with second-order reactions. General solutions still need to be derived for systems with bimolecular reactions. It is even more challenging if large state-space and nonconstant reaction rates are involved. Here, we developed a direct method to gain the analytical RP distribution for a class of second-order reaction systems with nonconstant reaction rates and large state space. Instead of using the CME, we used an equivalent path-wise representation, which is the solution to a transformed martingale problem of the CME. This allowed us to bypass solving a CME. We then applied the method to derive the analytical RP distribution of a real complex biochemical network with second-order reactions, the Drosophila phototransduction cascade. Our approach provides an alternative to the CMEs in deriving the analytical RP distributions of a class of second-order reaction systems. Since the bimolecular reactions are common in biological systems, our approach could enhance understanding real-world biochemical processes. [ABSTRACT FROM AUTHOR]

Abstract–When responses of massive data are hard to obtain due to some reasons such as privacy and security, high cost and administrative management, response-free subsampling is considered. In this paper, we propose a response-free decorrelated score subsampling approach to estimate and make statistical inference for a preconceived low-dimensional parameter in high-dimensional generalized linear models. The unconditional consistency and asymptotic normality of the resulting weighted subsample estimator are established using martingale techniques since the subsamples are no longer independent. The optimal response-free subsampling probabilities are derived based on A- and L-optimality criteria. Based on the optimal subsample, we further propose a more efficient and stable unweighted decorrelated score subsample estimator. The satisfactory performance of our proposed subsample estimators are demonstrated by simulation results and two real data applications. [ABSTRACT FROM AUTHOR]

In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant. [ABSTRACT FROM AUTHOR]

In this paper, we study the stochastic maximum principle for a partially observed optimal control problem of forward-backward stochastic differential equations (FBSDEs for short) of McKean–Vlasov type driven by both a family of Teugels martingales and an independent Brownian motion. The coefficients of the system and the cost functional depending on the state of the solution process as well as its probability law and the control variable. We establish partially observed necessary conditions of optimality for this system under assumption that the control domain is supposed to be convex. Our main result is based on Girsavov's theorem and the derivatives with respect to probability law. As an application of the general theory, a partially observed linear-quadratic control problem of McKean–Vlasov type is studied in terms of stochastic filtering. [ABSTRACT FROM AUTHOR]

This paper is about the modeling of cumulative hazard functions using martingale posterior distributions. The focus is on uncertainty quantification from a nonparametric perspective. The foundational Bayesian model in this case is the beta process and the classic estimator is the Nelson–Aalen. We use a sequence of estimators which form a martingale in order to obtain a random cumulative hazard function from the martingale posterior. The connection with the beta process is established and a number of illustrations is presented. [ABSTRACT FROM AUTHOR]

This study builds upon our previous work by introducing a refined Inductive Conformal Martingale (ICM) approach for addressing Concept Drift. Specifically, we enhance our previously proposed CAUTIOUS betting function to incorporate multiple density estimators for improving detection ability. We also combine this betting function with two base estimators that have not been previously utilized within the ICM framework: the Interpolated Histogram and Nearest Neighbor Density Estimators. We assess these extensions using both a single ICM and an ensemble of ICMs. For the latter, we conduct a comprehensive experimental investigation into the influence of the ensemble size on prediction accuracy and the number of available predictions. Our experimental results on four benchmark datasets demonstrate that the proposed approach surpasses our previous methodology in terms of performance while matching or in many cases exceeding that of three contemporary state-of-the-art techniques. [ABSTRACT FROM AUTHOR]

A signal is a description of an information source that specifies both its correlation with the state and its correlation with other signals. Extending Blackwell (1953), we characterize when one signal is more valuable than another regardless of preferences and regardless of access to other signals. This comparison is equivalent to reveal-or-refine: every realization of the more valuable signal reveals the state or refines the realization of the less valuable signal. We also study other comparisons of signals, including sufficiency, martingale dominance, and Lehmann. Reveal-or-refine is also equivalent to making any of these comparisons robust to access to other signals. (JEL C90, D82, D83) [ABSTRACT FROM AUTHOR]

Stochastic approximation (SA) was introduced in the early 1950s and has been an active area of research for several decades. While the initial focus was on statistical questions, it was seen to have applications to signal processing, convex optimization. In later years, SA has found application in reinforced learning (RL) and led to revival of interest.While bulk of the literature is on SA for the case when the observations are from a finite dimensional Euclidian space, there has been interest in extending the same to infinite dimension. Extension to Hilbert spaces is relatively easier to do, but this is not so when we come to a Banach space — since in the case of a Banach space, even law of large numbers is not true in general. We consider some cases where approximation works in a Banach space. Our framework includes case when the Banach space 픹 is C([0, 1], ℝd), as well as ℒ1([0, 1], ℝd), the two cases which do not even have the Radon–Nikodym property. [ABSTRACT FROM AUTHOR]

A derivative is a financial asset whose future payoff is a function of underlying assets. Pricing a financial derivative involves setting up a market model, finding a martingale ("fair game") probability measure for the model from the existing asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. In this work, we first formulate the pricing problem in a linear algebra setting, including the realistic setting of incomplete markets where derivatives do not have a unique price. We show that the problem can be solved with a variety of quantum techniques such as quantum linear programming and the quantum linear systems algorithm. While in previous works the martingale measure is assumed to be given, here it is extracted from market variables akin to bootstrapping, a common practice among financial institutions. We discuss the quantum zero-sum game algorithm and the quantum simplex algorithm as viable subroutines. For quantum linear systems solvers, we formalize a new market assumption milder than market completeness, which allows the potential for large speedups. Towards prototype use cases, we conduct numerical experiments motivated by the Black–Scholes–Merton model. [ABSTRACT FROM AUTHOR]

The functional partially linear regression model comprises a functional linear part and a non-parametric part. Testing the linear relationship between the response and the functional predictor is of fundamental importance. In cases where functional data cannot be approximated with a few principal components, we develop a second-order U-statistic using a pseudo-estimate for the unknown non-parametric component. Under some regularity conditions, the asymptotic normality of the proposed test statistic is established using the martingale central limit theorem. The proposed test is evaluated for finite sample properties through simulation studies and its application to real data. [ABSTRACT FROM AUTHOR]

This research is devoted to studying a geometric Brownian motion with drift switching driven by a 2 χ 2 Markov chain. A discrete-time multiplicative approximation scheme was developed, and its convergence in Skorokhod topology to the continuous-time geometric Brownian motion with switching has been proved. Furthermore, in a financial market where the discounted asset price follows a geometric Brownian motion with drift switching, market incompleteness was established, and multiple equivalent martingale measures were constructed. [ABSTRACT FROM AUTHOR]

The present note is an addition to the author's recent paper [44], concerning general multiplicative systems of random variables. Using some lemmas and the methodology of [13], we obtain a general extremal inequality, with corollaries involving Rademacher chaos sums and those analogues for multiplicative systems. In particular we prove that a system of functions generated by bounded products of a multiplicative system is a convergence system. [ABSTRACT FROM AUTHOR]

In Athreya et al. (2021), models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterized as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set (or more generally, a topological space), an adjacency matrix, and a sampling (Borel) measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Griffiths–Engen–McCloskey (GEM) distribution. In a companion paper (Greven et al. 2023), we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics. [ABSTRACT FROM AUTHOR]

The stochastic food chain model is an important model within the field of ecological research. Since existing models are difficult to describe the influence of cross-diffusion and random factors on the evolution of species populations, this work is concerned with a stochastic cross-diffusion three-species food chain model with prey-taxis, in which the direction of predators' movement is opposite to the gradient of prey, i.e., a higher density of prey. The existence and uniqueness of martingale solutions are established in a Hilbert space by using the stochastic Galerkin approximation method, the tightness criterion, Jakubowski's generalization of the Skorokhod theorem, and the Vitali convergence theorem. Furthermore, asymptotic behaviors around the steady states of the stochastic cross-diffusion three-species food chain model in the time mean sense are investigated. Finally, numerical simulations are carried out to illustrate the results of our analysis. [ABSTRACT FROM AUTHOR]

We establish some \Phi-moment inequalities for noncommutative differentially subordinate martingales. Let \Phi be a p-convex and q-concave Orlicz function with 1

Let M$\mathcal {M}$ be a semifinite von Neumann algebra equipped with an increasing filtration (Mn)n⩾1$(\mathcal {M}_n)_{n\geqslant 1}$ of (semifinite) von Neumann subalgebras of M$\mathcal {M}$. For 1⩽p⩽∞$1\leqslant p \leqslant \infty$, let Hpc(M)$\mathcal {H}_p^c(\mathcal {M})$ denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration (Mn)n⩾1$(\mathcal {M}_n)_{n\geqslant 1}$ and the index p$p$. We prove the following real interpolation identity: If 0<θ<1$0<\theta <1$ and 1/p=1−θ$1/p=1-\theta$, then H1c(M),H∞c(M)θ,p=Hpc(M).$$\begin{equation*} \hspace*{48pt}{\left(\mathcal {H}_1^c(\mathcal {M}), \mathcal {H}_\infty ^c(\mathcal {M})\right)}_{\theta,p}=\mathcal {H}_p^c(\mathcal {M}).\hspace*{-48pt} \end{equation*}$$This is new even for classical martingale Hardy spaces as it is previously known only under the assumption that the filtration is regular. We also obtain an analogous result for noncommutative column martingale Orlicz–Hardy spaces. [ABSTRACT FROM AUTHOR]

In this paper, we considered the boundedness of Burkholder's martingale transforms for martingale Hardy spaces with variable exponents. In addition, through martingale transforms, some characterizations of predictable variable exponent martingale Hardy spaces were also provided. [ABSTRACT FROM AUTHOR]

Abstract.To check the significance of the regression coefficients in the linear component of high-dimensional partial linear models, we proposed some projection-based test statistics. These test statistics are connected with U-statistics of order two and they are applicable for diverging dimensions and heteroscedastic model errors. By using the martingale central limit theorem, we show the asymptotic normalities of the proposed test statistics under the null hypothesis and local alternative hypotheses. The performance of test statistics are evaluated by simulation studies. The simulation results show that the proposed test statistics are powerful and have the correct type-I error asymptotically under the null hypothesis. [ABSTRACT FROM AUTHOR]

This article investigates the estimation and variable selection procedures of single-index models when the data have non ignorable response. For estimation of the index coefficients, a new procedure based on martingale difference divergence is developed. Consistency and asymptotic normality of the proposed estimators are established. The finite sample performance of the proposed procedure is assessed through extensive simulation studies. Finally, we illustrate the proposed method through a real body fat data set. [ABSTRACT FROM AUTHOR]

In 1733, de Moivre, investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem (CLT). In this review article, we briefly recall the history of classical CLT and martingale CLT, and introduce new directions of CLT, namely Peng's nonlinear CLT and Chen–Epstein's nonlinear CLT, as well as Chen–Epstein's nonlinear normal distribution function. [ABSTRACT FROM AUTHOR]

Recognizing the importance of incorporating different risk measures in the portfolio management model, this paper examines the dynamic mean-risk portfolio optimization problem using both variance and value at risk (VaR) as risk measures. By employing the martingale approach and integrating the quantile optimization technique, we provide a solution framework for this problem. We demonstrate that, under a general market setting, the optimal terminal wealth may exhibit different patterns. When the market parameters are deterministic, we derive the closed-form solution for this problem. Examples are provided to illustrate the solution procedure of our method and demonstrate the benefits of our dynamic portfolio model compared to its static counterpart. [ABSTRACT FROM AUTHOR]

Continuous‐time diffusion models with jumps, especially the jump intensity coefficient, can depict the impact of sudden and large shocks to financial markets. It is possible to disentangle, from the discrete observations, the contributions given by the jumps and those by the diffusion part through threshold functions. Based on this threshold technique, we employ non‐parametric local linear threshold estimator for the unknown jump intensity function of a semimartingale with jumps. The asymptotic normality of our estimator is provided in the presence of finite activity jumps under certain regular conditions. The finite‐sample performance for the underlying estimator has been shown through a Monte Carlo experiment and an empirical analysis on high frequency returns of indexes in the USA and China. [ABSTRACT FROM AUTHOR]

This article studies the problem of optimal control with state constraints for mean‐field type stochastic systems, which is governed by a fully coupled forward‐backward stochastic differential equation with Teugels martingales. In this system, the coefficients contain not only the state processes but also its expectation value, and the cost function is of mean‐field type as well. We use an equivalent backward formulation to deal with the terminal state constraint, and then we obtain a stochastic maximum principle by Ekeland's variational principle. In addition, we discuss a stochastic linear‐quadratic control problem with state constraints. [ABSTRACT FROM AUTHOR]

In this study, we establish a Burkholder-Davis-Gundy (BDG) inequality type for certain nonlinear martingales arising from backward stochastic differential equations (BSDE) with generalized Lipschitz generator. As a consequence, we attempt to prove the equivalence between the convergence in probability of a g-martingale sequence and the associated quadratic variation sequence. Using a counterexample, we prove that BDG fails when g is quadratic. [ABSTRACT FROM AUTHOR]

Let X be a linear diffusion taking values in (ℓ , r) and consider the standard Euler scheme to compute an approximation to E [ g (X T) 1 { T < ζ } ] for a given function g and a deterministic T , where ζ = inf { t ≥ 0 : X t ∉ (ℓ , r) } . It is well known since Gobet (Stoch. Process. Appl. 87:167–197, 2000) that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1 / N with N being the number of discretisations. We introduce a drift-implicit Euler method to bring the convergence rate back to 1 / N , i.e., the optimal rate in the absence of killing, using the theory of recurrent transformations developed in Çetin (Ann. Appl. Probab. 28:3102–3151, 2018). Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions. [ABSTRACT FROM AUTHOR]

The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorokhod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework. [ABSTRACT FROM AUTHOR]

In this paper, we study the real interpolation theory for variable Lorentz–Karamata spaces as well as for the corresponding martingale Hardy spaces. As a consequence, the generalization of Doob's maximal inequality will be proved. Moreover, the atomic decompositions of the variable martingale Hardy–Lorentz–Karamata spaces are also presented. The results obtained here are new even for martingale Hardy–Lorentz–Karamata spaces. [ABSTRACT FROM AUTHOR]

This paper studies the zero-viscosity limit for the Navier-Stokes equations with rotation, an additive white noise and nonslip boundary condition in dimension three. By establishing the well-posedness of the limit problems and constructing approximate solutions, we obtain the convergence in the mean square for the Oseen type equation with nonlinear source term, and the convergence in probability of the martingale solution of the stochastic Navier-Stokes equations with rotation. [ABSTRACT FROM AUTHOR]

The general kinetic theory of coarse-grained systems is presented in the abstract formalism of communication theory developed by Shannon and Weaver, Khinchin and Kolmogorov. The martingale theory shows that, under reasonable, general hypotheses, coarse-grained systems can be approximated by generalized Markov systems. For mixing systems, the Kolmogorov entropy production can be defined for nonstationary processes as Kolmogorov defined it for stationary processes. [ABSTRACT FROM AUTHOR]

The detection of seismic activity precursors as part of an alarm system will provide opportunities for minimization of the social and economic impact caused by earthquakes. It has long been envisaged, and a growing body of empirical evidence suggests that the Earth's electromagnetic field could contain precursors to seismic events. The ability to capture and monitor electromagnetic field activity has increased in the past years as more sensors and methodologies emerge. Missions such as Swarm have enabled researchers to access near-continuous observations of electromagnetic activity at second intervals, allowing for more detailed studies on weather and earthquakes. In this paper, we present an approach designed to detect anomalies in electromagnetic field data from Swarm satellites. This works towards developing a continuous and effective monitoring system of seismic activities based on SWARM measurements. We develop an enhanced form of a probabilistic model based on the Martingale theories that allow for testing the null hypothesis to indicate abnormal changes in electromagnetic field activity. We evaluate this enhanced approach in two experiments. Firstly, we perform a quantitative comparison on well-understood and popular benchmark datasets alongside the conventional approach. We find that the enhanced version produces more accurate anomaly detection overall. Secondly, we use three case studies of seismic activity (namely, earthquakes in Mexico, Greece, and Croatia) to assess our approach and the results show that our method can detect anomalous phenomena in the electromagnetic data. [ABSTRACT FROM AUTHOR]

This work is primarily the product of the first listed author, who completed it as a student under the direction of the second and third listed authors. In this paper we analyze three progressive betting strategies, each applied to three casino games. The betting strategies considered are Martingale, Paroli, and Fibonacci, each applied to the casino games blackjack, craps, and roulette. The aim of this analysis is to try to identify optimal betting strategies for each game with a given or maximum number of bets placed that pay 1:1. The purpose of this work is not to try to discover methods for “beating the house,” which are already known to not exist, but rather to search for methods for advancing gameplay through a given or maximum number of bets while retaining the possibility of earning a profit. Programming in the computer algebra system Maple, more specifically through the use of a Maplet, which is like an applet but uses (and requires) Maple’s engine, will be used for calculations. [ABSTRACT FROM AUTHOR]

In this paper, we establish the complete moment convergence for a field of martingale difference random variables, weakly upper bounded by a random variable. The results generalize and extend the complete convergences for non-identically distributed martingale-difference random fields to the case of complete moment convergences. [ABSTRACT FROM AUTHOR]

This article establishes a comprehensive theory of the optimality, robustness, and cross-validation selection consistency for the ridge regression under factor-augmented models with possibly dense idiosyncratic information. Using spectral analysis for random matrices, we show that the ridge regression is asymptotically efficient in capturing both factor and idiosyncratic information by minimizing the limiting predictive loss among the entire class of spectral regularized estimators under large-dimensional factor models and mixed-effects hypothesis. We derive an asymptotically optimal ridge penalty in closed form and prove that a bias-corrected k-fold cross-validation procedure can adaptively select the best ridge penalty in large samples. We extend the theory to the autoregressive models with many exogenous variables and establish a consistent cross-validation procedure using the what-we-called double ridge regression method. Our results allow for nonparametric distributions for, possibly heavy-tailed, martingale difference errors and idiosyncratic random coefficients and adapt to the cross-sectional and temporal dependence structures of the large-dimensional predictors. We demonstrate the performance of our ridge estimators in simulated examples as well as an economic dataset. All the proofs are available in the , which also includes more technical discussions and remarks, extra simulation results, and useful lemmas that may be of independent interest. [ABSTRACT FROM AUTHOR]

We propose a sequential, anytime-valid method to test the conditional independence of a response Y and a predictor X given a random vector Z. The proposed test is based on e-statistics and test martingales, which generalize likelihood ratios and allow valid inference at arbitrary stopping times. In accordance with the recently introduced model-X setting, our test depends on the availability of the conditional distribution of X given Z, or at least a sufficiently sharp approximation thereof. Within this setting, we derive a general method for constructing e-statistics for testing conditional independence, show that it leads to growth-rate optimal e-statistics for simple alternatives, and prove that our method yields tests with asymptotic power one in the special case of a logistic regression model. A simulation study is done to demonstrate that the approach is competitive in terms of power when compared to established sequential and nonsequential testing methods, and robust with respect to violations of the model-X assumption. for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, we give an overview of (nonlinear) pricing-hedging duality and of its connection with the theory of entropy martingale optimal transport (EMOT), recently developed, and that of convex risk measures. Similarly to Doldi and Frittelli (Finance Stoch 27(2):255–304, 2023), we here establish a duality result between a convex optimal transport and a utility maximization problem. Differently from Doldi and Frittelli (Finance Stoch 27(2):255–304, 2023), we provide here an alternative proof that is based on a compactness assumption. Subhedging and superhedging can be obtained as applications of the duality discussed above. Furthermore, we provide a dual representation of the generalized optimized certainty equivalent associated with indirect utility. [ABSTRACT FROM AUTHOR]

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan, Prob. Eng. Inf. Sci. 36 , 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes. [ABSTRACT FROM AUTHOR]

We establish the asset pricing and hedging principle in a financial market model, which is a specific case of the von Neumann-Gale dynamical system, with both fixed and proportional transaction costs and trading constraints. The main results are hedging criteria stated in terms of consistent valuation systems, generalizing the notion of an equivalent martingale measure. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic strongly damped wave equation with critical nonlinearity driven by a spatially homogeneous Winer process on the whole space. We first establish the existence of martingale solutions for the equation. Moreover, we prove that the martingale solution of the equation with deterministic initial data is pathwise unique and obtain a pathwise unique strong solution in the sense of probability. Finally, we construct an invariant measure for the equation with critical nonlinearity by using a weak Feller approach and construct an invariant measure for the equation with subcritical nonlinearity by using a decomposition method. We mention that the invariant measure constructed by the decomposition method is more regular than that constructed by the weak Feller approach. [ABSTRACT FROM AUTHOR]

We present and solve an optimal asset allocation problem under a weighted limited expected loss (WLEL) constraint. This formulation encompasses the risk management problem with a limited expected loss (LEL) constraint as a specialized instance and offers a pertinent internal risk management instrument for firms. We observe that a WLEL constraint makes the optimizing investor pursue less volatile payoffs than the unconstrained Merton solution. Compared to the LEL-constrained problem with the same weighted default threshold, the WLEL optimal terminal wealth displays a less dispersed distribution with a smaller variance, suggesting a more secure risk management framework. Conducting a comprehensive equilibrium analysis in the presence of a WLEL risk manager, we validate the relatively conservative investment approach undertaken by the WLEL manager. Subsequently, we expand our findings to encompass broader incomplete market settings, wherein the uniqueness of the equivalent local martingale measure is not assured. [ABSTRACT FROM AUTHOR]

This paper posits that when an asset exhibits a bubble, its price process can be unbounded from above in finite time with positive probability if a quadratic variation (QV) risk premium is large enough. Based on the local martingale theory of bubbles, we provide sufficient conditions under which price explosion occurs via the QV channel provided that bubbles are present. This QV channel of price explosions is new to the literature and distinct from the explosive autoregressive (AR) dynamics, which is often identified with the presence of bubbles as defined in the time series literature. Using the S&P 500 index and a sample of individual stocks over 1996–2021, we document the existence of price explosions during periods when bubbles occur. Almost all price explosion episodes discovered are associated with the QV and not AR drift channel. [ABSTRACT FROM AUTHOR]

Assume that we are given a filtration (Fn)$(\mathcal F_n)$ on a probability space (Ω,F,P)$(\Omega,\mathcal F,\mathbb {P})$ of the form that each Fn$\mathcal F_n$ is generated by the partition of one atom of Fn−1$\mathcal F_{n-1}$ into two atoms of Fn$\mathcal F_n$ having positive measure. Additionally, assume that we are given a finite‐dimensional linear space S$S$ of F$\mathcal F$‐measurable, bounded functions on Ω$\Omega$ so that on each atom A$A$ of any σ$\sigma$‐algebra Fn$\mathcal F_n$, all Lp$L^p$‐norms of functions in S$S$ are comparable independently of n$n$ or A$A$. Denote by Sn$S_n$ the space of functions that are given locally, on atoms of Fn$\mathcal F_n$, by functions in S$S$ and by Pn$P_n$ the orthoprojector (with respect to the inner product in L2(Ω)$L^2(\Omega)$) onto Sn$S_n$. Since S=span{1Ω}$S = \operatorname{span}\lbrace \mathbbm 1_\Omega \rbrace$ satisfies the above assumption and Pn$P_n$ is then the conditional expectation En$\mathbb {E}_n$ with respect to Fn$\mathcal F_n$, for such filtrations, martingales (Enf)$(\mathbb {E}_n f)$ are special cases of our setting. We show in this article that certain convergence results that are known for martingales (or rather martingale differences) are also true in the general framework described above. More precisely, we show that the differences (Pn−Pn−1)f$(P_n - P_{n-1})f$ form an unconditionally convergent series and are democratic in Lp$L^p$ for 1

In visible light communication (VLC), the precise latency evaluation of wireless access networks and the efficient forwarding strategy of core networks are the crux for end-to-end reliability provisioning. Leveraging martingale theory, an elegant latency-bounded reliability analysis framework is studied for the VLC network. Considering the characteristic that VLC links are easy to block, the martingale of the service process is constructed. Based on the time shift features, the martingale process related to latency is proposed for the VLC system, which models the influence of entanglement between aggregate arrivals and random service on latency. A stopping time event about latency is defined. Renting the stopping time theory, a tight upper bound of the unreliability with regard to latency is derived. In the core network, a dynamic forward backhaul framework is proposed, which uses the relay selection algorithm based on back-pressure theory to improve data transmission quality. The theoretical latency-bounded reliability matches the simulation results well, which verifies the effectiveness of the proposed analysis framework, and the proposed relay selection algorithm can also improve network performance under data-intensive transmission. [ABSTRACT FROM AUTHOR]

This work mainly investigates the asymptotic properties for the wavelet estimator in a non parametric regression model with martingale difference random errors. The moment consistency, strong consistency, complete consistency, the rate of strong consistency and the asymptotic normality for the wavelet estimator are established under some mild conditions, which were not obtained before. Some numerical analysis is carried out to support the theoretical results. [ABSTRACT FROM AUTHOR]

For any two real-valued continuous-path martingales X = {Xt}t≥0 and Y = {Yt}t≥0, with X and Y being orthogonal and Y being differentially subordinate to X, we obtain sharp Lp inequalities for martingales of the form aX + bY with a, b real numbers. The best Lp constant is equal to the norm of the operator aI + bH from Lp to Lp, where H is the Hilbert transform on the circle or real line. The values of these norms were found by Hollenbeck, Kalton and Verbitsky [Studia Math., 2003, 157(3): 237–278]. We also give applications of our martingale inequalities to Riesz transforms and some discrete operators. [ABSTRACT FROM AUTHOR]

The integral representation theorem for martingales has been widely used in probability theory. In this work, we propose and prove a general representation theorem for a class of set-valued submartingales. We also extend the stochastic integral representation for non-trivial initial set-valued martingales. Moreover, we show that this result covers the existing ones in the literature for both degenerated and non-degenerated set-valued martingales. [ABSTRACT FROM AUTHOR]

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or Lp regularization, general transport costs, and multimarginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for noncompact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for marginals with a finite (2+δ) -moment. Funding: This work was supported by the Alfred P. Sloan Foundation [Grant FG-2016-6282] and the Division of Mathematical Sciences [Grants DMS-1812661 and DMS-2106056]. [ABSTRACT FROM AUTHOR]

Current status data are a type of failure time data that arise when the failure time of study subject cannot be determined precisely but is known only to occur before or after a random monitoring time. Variable selection methods for the failure time data have been discussed extensively in the literature. However, the statistical inference of the model selected based on the variable selection method ignores the uncertainty caused by model selection. To enhance the prediction accuracy for risk quantities such as survival probability, we propose two optimal model averaging methods under semiparametric additive hazards models. Specifically, based on martingale residuals processes, a delete‐one cross‐validation (CV) process is defined, and two new CV functional criteria are derived for choosing model weights. Furthermore, we present a greedy algorithm for the implementation of the techniques, and the asymptotic optimality of the proposed model averaging approaches is established, along with the convergence of the greedy averaging algorithms. A series of simulation experiments demonstrate the effectiveness and superiority of the proposed methods. Finally, a real‐data example is provided as an illustration. [ABSTRACT FROM AUTHOR]

The martingale representation theorem (along with Girsanov's measure change theorem) plays an important role in modern stochastic financial mathematics. The first proof of the martingale representation theorem was implicitly provided by Ito himself (1951). Subsequently, many other works have been written about the existence of the martingale representation and its applications, but one of the pioneering works in this direction is certainly the work of Clark (1970). On the other hand, given the needs of modern financial mathematics, it is not enough to know only the existence of an integral representation; it is necessary to be able to find an explicit form of the integrand of the integral representation. It is known that for stochastically smooth functionals, the integrand is calculated using Ocone's formula (1984) as an optional projection of the stochastic derivative of the considered functional F ( E [ D t F | ℑ t ] ). Ocone's formula was later generalized by Glonti and Purtukhia (2017), when only the filter g t : = E [ F | ℑ t ] (conditional mathematical expectation) of the functional is stochastically smooth, according to which the integrand is calculated as the limit of the integrand corresponding to the filter ( lim s ↑ T E [ D t g s | ℑ t ] in L 2 ([ 0 , T ] × Ω) ). Here, we study functionals whose filter is no longer smooth and propose a method for finding the integrand. [ABSTRACT FROM AUTHOR]

We investigate a stochastic version of the Allen–Cahn–Navier–Stokes system in a smooth two- or three-dimensional domain with random initial data. The system consists of a Navier–Stokes equation coupled with a convective Allen–Cahn equation, with two independent sources of randomness given by general multiplicative-type Wiener noises. In particular, the Allen–Cahn equation is characterized by a singular potential of logarithmic type as prescribed by the classical thermodynamical derivation of the model. The problem is endowed with a no-slip boundary condition for the (volume averaged) velocity field, as well as a homogeneous Neumann condition for the order parameter. We first prove the existence of analytically weak martingale solutions in two and three spatial dimensions. Then, in two dimensions, we also establish pathwise uniqueness and the existence of a unique probabilistically-strong solution. Eventually, by exploiting a suitable generalisation of the classical De Rham theorem to stochastic processes, existence and uniqueness of a pressure is also shown. [ABSTRACT FROM AUTHOR]

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term. [ABSTRACT FROM AUTHOR]