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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]

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]

In this paper we study the following problem: for a given bounded positive function f on a filtered probability space can we find another function (a multiplier) m, 0 ≤ m ≤ 1 , such that the function mf is not “too small” but its square function is bounded? We explicitly show how to construct such multipliers for the usual martingale square function and for so-called conditional square function. Besides that, we show that for the usual square function more general statement can be obtained by application of a non-constructive abstract correction theorem by Kislyakov. [ABSTRACT FROM AUTHOR]

Let 0 < q ≤ ∞ , b be a slowly varying function and Φ : [ 0 , ∞) ⟶ [ 0 , ∞) be an increasing function with Φ (0) = 0 and lim r → ∞ Φ (r) = ∞ . In this paper, we introduce a new class of function spaces L Φ , q , b which unify and generalize the Lorentz-Karamata spaces with Φ (t) = t p and the Orlicz-Lorentz spaces with b ≡ 1 . Based on the new spaces, we introduce five new Hardy spaces containing martingales, the so-called Orlicz-Lorentz-Karamata Hardy martingale spaces and then develop a theory of these martingale Hardy spaces. To be precise, we first investigate several properties of Orlicz-Lorentz-Karamata spaces and then present Doob's maximal inequalities by using Hardy's inequalities. The characterization of these Hardy martingale spaces are constructed via the atomic decompositions. As applications of the atomic decompositions, martingale inequalities and the relation of the different martingale Hardy spaces are presented. The dual theorems and a new John-Nirenberg type inequality for the new framework are also established. Moreover, we study the boundedness of fractional integral operators on Orlicz-Lorentz-Karamata Hardy martingale spaces. The results obtained here generalize the previous results for Lorentz-Karamata Hardy martingale spaces as well as for Orlicz-Lorentz Hardy martingales spaces. Especially, we remove the condition that b is non-decreasing as in [38-39] and the condition q Φ - 1 < 1 / q in [24], respectively. [ABSTRACT FROM AUTHOR]

In this article, some necessary and sufficient conditions are shown for weighted weak type mixed Φ -inequality and weighted extra-weak type mixed Φ -inequality for martingale maximal operator. The obtained results generalize some existing statements. [ABSTRACT FROM AUTHOR]

In this paper, we present the stability result of a spatial semi-discrete scheme to backward stochastic differential equations taking values in a Hilbert space. Under suitable assumptions of the final value and the drift, a convergence rate is established. [ABSTRACT FROM AUTHOR]

In this article, we study a class of reflecting stochastic differential equations whose coefficients depend on image measures of solutions under a given initial measure in Wasserstein space P 2 . By the penalization method, the image process, which is a diffusion process in P 2 , is constrained in a priori given domain O ⊂ P 2 . The large deviation principle for this reflecting image process is also established by weak convergence method. [ABSTRACT FROM AUTHOR]

Let φ : [ 0 , 1) × [ 0 , ∞) → [ 0 , ∞) be a Musielak–Orlicz function and q ∈ (0 , ∞ ] . In this article, the authors characterize the martingale Musielak–Orlicz Hardy space H φ [ 0 , 1) and the martingale Musielak–Orlicz–Lorentz Hardy space H φ , q [ 0 , 1) via dyadic maximal operators. As applications, the authors prove that the maximal Fejér operator is bounded from the space H φ [ 0 , 1) to the Musielak–Orlicz space L φ [ 0 , 1) and from H φ , q [ 0 , 1) to the Musielak–Orlicz–Lorentz space L φ , q [ 0 , 1) , which further implies some convergence results of the Fejér means. [ABSTRACT FROM AUTHOR]

For probability measures μ,ν\mu ,\nu , and ρ\rho , define the cost functionals C(μ,ρ)≔supπ∈Π(μ,ρ)∫⟨x,y⟩π(dx,dy)andC(ν,ρ)≔supπ∈Π(ν,ρ)∫⟨x,y⟩π(dx,dy),C\left(\mu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\mu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y)\hspace{1.0em}{\rm{and}}\hspace{1em}C\left(\nu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\nu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y), where ⟨⋅,⋅⟩\langle \cdot ,\cdot \rangle denotes the scalar product and Π(⋅,⋅)\Pi \left(\cdot ,\cdot ) is the set of couplings. We show that two probability measures μ\mu and ν\nu on Rd{{\mathbb{R}}}^{d} with finite first moments are in convex order (i.e., μ≼cν\mu {\preccurlyeq }_{c}\nu ) iff C(μ,ρ)≤C(ν,ρ)C\left(\mu ,\rho )\le C\left(\nu ,\rho ) holds for all probability measures ρ\rho on Rd{{\mathbb{R}}}^{d} with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of ∫fdν−∫fdμ\int f{\rm{d}}\nu -\int f{\rm{d}}\mu over all 1-Lipschitz functions ff, which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. Building on this result, we derive new proofs of well known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.

For a Hilbert space-valued martingale (f n) and an adapted sequence of positive random variables (w n) , we show the weighted Davis-type inequality E | f 0 | w 0 + 1 4 ∑ n = 1 N | d f n | 2 f n ∗ w n ≤ E (f N ∗ w N ∗). More generally, for a martingale (f n) with values in a (q , δ) -uniformly convex Banach space, we show that E | f 0 | w 0 + δ ∑ n = 1 ∞ | d f n | q (f n ∗) q - 1 w n ≤ C q E (f ∗ w ∗). These inequalities unify several results about the martingale square function. [ABSTRACT FROM AUTHOR]

In this paper, complete moment convergence for maximum of randomly weighted sums and complete convergence for randomly indexed sums of martingale difference sequences (MDS) are investigated under some proper and sufficient conditions. A Marcinkiewicz–Zygmund type strong law of large numbers (MZSLLN) for MDS is obtained. In addition, relationships among weights, weight functions and boundary functions are revealed in a sense. The results obtained in the paper generalize some corresponding ones for independent and some dependent random variables. As an application, strong consistency for estimators in a nonparametric regression model is established. [ABSTRACT FROM AUTHOR]

In this paper, we study the question of representing one class of Brownian functionals as a stochastic Itô integral with an explicit form of the integrand. The considered class of functionals also includes functionals that are not smooth in the sense of Malliavin, to which both the well-known Clark–Ocone formula (1984) and its generalization, the Glonti–Purtukhia formula (2017), are inapplicable. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the optimal investment strategies for an insurer with the liquid constraint. The insurer's surplus is modeled by a Lévy process. The insurer can invest in N risky assets and a risk-free asset. However, in reality, the insurer needs to keep partial amount of his wealth as liquid reserves to meet with the management, which is called as the liquid constraint. We assume that the liquid reserve is a proportion of the insurer's wealth. Using the martingale approach, we derive the explicit optimal investment strategies for both CARA and quadratic utilities. [ABSTRACT FROM AUTHOR]

The objective of this paper is to develop a duality between a novel entropy martingale optimal transport (EMOT) problem and an associated optimisation problem. In EMOT, we follow the approach taken in the entropy optimal transport (EOT) problem developed in Liero et al. (Invent. Math. 211:969–1117, 2018), but we add the constraint, typical of martingale optimal transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures. In the associated problem, the objective functional, related via Fenchel conjugacy to the entropic term in EMOT, is no longer linear as in (martingale) optimal transport. This leads to a novel optimisation problem which also has a clear financial interpretation as a nonlinear subhedging problem. Our theory allows us to establish a nonlinear robust pricing–hedging duality which also covers a wide range of known robust results. We also focus on Wasserstein-induced penalisations and study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT. [ABSTRACT FROM AUTHOR]

The random trigonometric series ∑ n = 1 ∞ ρ n cos (n t + ω n) on the circle T are studied under the conditions ∑∣ρn∣2 = ∞ and ρn → 0, where {ωn} are independent and uniformly distributed random variables on T . They are almost surely not Fourier-Stieltjes series but determine pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed. The behavior of the partial sums of the above series is proved to be multifractal. Our theory holds on the torus T d of dimension d ⩾ 1. [ABSTRACT FROM AUTHOR]

In this paper, we develop the martingale theory in the framework of grand Lebesgue spaces for 0 < p ≤ 1 . Atomic decompositions for the grand martingale Hardy spaces are established. With the help of the atomic decompositions, the dual spaces of the grand martingale Hardy spaces are characterized and some martingale inequalities are deduced. Furthermore, when the stochastic basis is regular, the boundedness of the fractional integrals on grand martingale Hardy spaces is proved. The results obtained here unify and generalize the previous results of grand martingale Hardy spaces. [ABSTRACT FROM AUTHOR]

We introduce atoms for dyadic atomic H 1 for which the equivalence between the atomic and maximal function definitions is dimension independent. We give sharp, up to log (d) factor, estimates for the H 1 → L 1 norm of the special maximal function. [ABSTRACT FROM AUTHOR]

Empirical studies (e.g. Jiang et al. (2015) and Mislove et al. (2007)) show that online social networks have not only in- and out-degree distributions with Pareto-like tails, but also a high proportion of reciprocal edges. A classical directed preferential attachment (PA) model generates in- and out-degree distributions with power-law tails, but the theoretical properties of the reciprocity feature in this model have not yet been studied. We derive asymptotic results on the number of reciprocal edges between two fixed nodes, as well as the proportion of reciprocal edges in the entire PA network. We see that with certain choices of parameters, the proportion of reciprocal edges in a directed PA network is close to 0, which differs from the empirical observation. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates that alternative models need to be considered. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the number of customers in infinite-server systems driven by Hawkes processes. In these systems, the self-exciting arrival process is assumed to be represented by a Hawkes process and the self-exciting service process by a state-dependent Hawkes process (sdHawkes process). Under some suitable conditions, for the Hawkes / sdHawkes / ∞ system, the Markov property of the system is derived. The joint time-dependent distribution of the number of customers in the system, the arrival intensity and the server intensity is characterized by a system of differential equations. Then, the time-dependent results are also deduced for the M / sdHawkes / ∞ system. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an optimal control problem where the system is driven by a G-stochastic differential equation, where an admissible set of controls is convex. We establish necessary as well as sufficient optimality conditions for this model. At the end of this work, we illustrate our main result by giving an example that deals with the linear-quadratic problem. [ABSTRACT FROM AUTHOR]

In this paper, based on the theory of regularly varying functions we study central limit theorems for the weighted sum S n = ∑ j = 1 m n c nj X nj , where (X nj ; 1 ≤ j ≤ m n , n ≥ 1) is a Hilbert-space-valued identically distributed martingale difference array and (c nj ; 1 ≤ j ≤ m n , n ≥ 1) is an array of real numbers. As an application, we present a central limit theorem for moving average processes of martingale differences. [ABSTRACT FROM AUTHOR]

This paper contributes to the study of stochastic processes of the class (Σ) . First, we extend the notion of the above-mentioned class to càdlàg semi-martingales, whose finite variation part is considered càdlàg instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class (Σ) . More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class (Σ) . [ABSTRACT FROM AUTHOR]

We prove a version of the Ambrosio–Figalli–Trevisan superposition principle for a restricted subclass of solutions to the Fokker–Planck–Kolmogorov equation that is valid on separable infinite-dimensional Hilbert spaces. Furthermore, we transfer this restricted superposition principle into a nonlinear setting. [ABSTRACT FROM AUTHOR]

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot)}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot)}^{s}$ with $0 can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a martingale representation theorem in the progressive enlargement G by a random time τ of the filtration F L generated by a Lévy process. The assumptions on the random time are that F L is immersed in G and that τ avoids F L stopping times. We also study the multiplicity of a progressively enlarged filtration. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce weak martingale Hardy-type spaces associated with a quasi-Banach function lattice. The authors then establish the atomic characterizations of these weak martingale Hardy-type spaces. As applications, the authors give the sufficient conditions for the boundedness of σ-sublinear operators from weak martingale Hardy-type spaces to a quasi-Banach function lattice. Furthermore, the authors clarify the relation among different weak martingale Hardy-type spaces in the framework of a rearrangement-invariant quasi-Banach function space. Finally, the authors apply these results to the weighted Lorentz space and the generalized grand Lebesgue space. [ABSTRACT FROM AUTHOR]

We prove explicit L p bounds for second order Riesz transforms of the sub-Laplacian and of the Laplacian in the Lie groups H , SU (2) , and SL ~ (2) . Our proof makes use of martingale transform techniques and specific commutation properties between the complex gradient and the sub-Laplacian in those Lie groups. [ABSTRACT FROM AUTHOR]

Contributions of the present paper consist of two parts. In the first one, we contribute to the theory of stochastic calculus for signed measures. For instance, we provide some results permitting to characterize martingales and Brownian motion both defined under a signed measure. We also prove that the uniformly integrable martingales (defined with respect to a signed measure) can be expressed as relative martingales and we provide some new results to the study of the class Σ (H) . The second part is devoted to the construction of solutions for the homogeneous skew Brownian motion equation and for the inhomogeneous skew Brownian motion equation. To do this, our ingredients are the techniques and results developed in the first part that we apply on some stochastic processes borrowed from the theory of stochastic calculus for signed measures. Our methods are inspired by those used by Bouhadou and Ouknine in [2013]. Moreover, their solution of the inhomogeneous skew Brownian motion equation is a particular case of those we propose in this paper. [ABSTRACT FROM AUTHOR]

Let (X n j ; 1 ≤ j ≤ m n , n ≥ 1) be an array of rowwise R d -valued martingale difference ( d ≥ 1) with respect to σ-fields (F n j ; 0 ≤ j ≤ m n , n ≥ 1) and let (C n j ; 1 ≤ j ≤ m n , n ≥ 1) be an array of m × d matrices of real numbers, where (m n ; n ≥ 1) is a sequence of positive integers such that m n → ∞ as n → ∞. The aim of this paper is to establish convergence in mean and central limit theorems for weighted sums type S n = ∑ j = 1 m n C n j X n j under some conditions of slow variation at infinity. We also apply the obtained results to study the asymptotic properties of estimates in some statistical models. In addition, two illustrative examples and their simulation are given. This study is motivated by models arising in economics, telecommunications, hydrology, and physics applications where the innovations are often dependent on each other and have infinite variances. [ABSTRACT FROM AUTHOR]

In this article, we are interested in studying the transition density function of the couple given by the first hitting time of a fixed threshold by a one-dimensional uniformly elliptic diffusion process and the associated stopped process. Our strategy relies on the parametrix technique applied to the related semigroup. It notably allows us to extend standard PDEs results on Green and Poisson functions, see e.g. Garroni and Menaldi [Green functions for second order parabolic integro-differential problems, volume 275 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992], by establishing the infinite expansion of the corresponding transition density for irregular coefficients, here bounded measurable drift coefficient and bounded η-Hölder-continuous diffusion coefficient. As a by-product, we establish some Gaussian upper estimates. [ABSTRACT FROM AUTHOR]