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In this paper a martingale problem for super-Brownian motion with interactive branching is derived. The uniqueness of the solution to the martingale problem is obtained by using the pathwise uniqueness of the solution to a corresponding system of SPDEs with proper boundary conditions. The existence of the solution to the martingale problem and the local Hölder continuity of the density process are also studied. [ABSTRACT FROM AUTHOR]

In this paper we derive generalizations of comparison results for semimartingales. Our results are based on Markov projections and on known comparison results for Markov processes. The first part of the paper is concerned with an alternative method for the construction of Markov projections of semimartingales. In comparison to the construction in Bentata and Cont (2009) which is based on the solution of a well-posed martingale problem, we make essential use of pseudo-differential operators as investigated in Böttcher (2008) and of fundamental solutions of related evolution problems. This approach allows to dismiss with some boundedness assumptions on the differential characteristics in the martingale approach. As consequence of the construction of Markov projections, comparison results for path-independent functions (European options) of semimartingales can be reduced to the well investigated problem of comparison of Markovian semimartingales. The Markov projection approach to comparison results does not require one of the semimartingales to be Markovian, which is a common assumption in literature. An idea of Brunick and Shreve (2013) to mimick updated processes leads to a related reduction result to the Markovian case and thus to the comparison of related generators. As consequence, a general comparison result is also obtained for path-dependent functions of semimartingales. [ABSTRACT FROM AUTHOR]

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part M of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on "the future" of the process M as well as a dependence of its law (McKean–Vlasov-type equations). For such drivers, we prove the existence and uniqueness of a global solution (i.e. for any maturity T > 0) to a BSDE with data satisfying natural integrability conditions. [ABSTRACT FROM AUTHOR]

The paper develops multiplicative compensation for complex-valued semimartingales and studies some of its consequences. It is shown that the stochastic exponential of any complex-valued semimartingale with independent increments becomes a true martingale after multiplicative compensation when such compensation is meaningful. This generalization of the Lévy–Khintchin formula fills an existing gap in the literature. It allows, for example, the computation of the Mellin transform of a signed stochastic exponential, which in turn has practical applications in mean–variance portfolio theory. Girsanov-type results based on multiplicatively compensated semimartingales simplify treatment of absolutely continuous measure changes. As an example, we obtain the characteristic function of log returns for a popular class of minimax measures in a Lévy setting. [ABSTRACT FROM AUTHOR]

In this paper, we study the connections between three concepts — the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure. [ABSTRACT FROM AUTHOR]

We study a stochastic control problem for continuous multidimensional martingales with fixed quadratic variation. In a radially symmetric environment, we are able to find an explicit solution to the control problem and find an optimal strategy. We show that it is optimal to switch between two strategies, depending only on the radius of the controlled process. The optimal strategies correspond to purely radial and purely tangential motion. It is notable that the value function exhibits smooth fit even when switching to tangential motion, where the radius of the optimal process is deterministic. Under sufficient regularity on the cost function, we prove optimality via viscosity solutions of a Hamilton–Jacobi–Bellman equation. We extend the results to cost functions that may become infinite at the origin. Extra care is required to solve the control problem in this case, since it is not clear how to define the optimal strategy with deterministic radius at the origin. Our results generalise some problems recently considered in Stochastic Portfolio Theory and Martingale Optimal Transport. [ABSTRACT FROM AUTHOR]

We study the randomized rumor spreading algorithm pull on complete graphs with n vertices. Starting with one informed vertex and proceeding in rounds, each vertex that is uninformed connects to a neighbor chosen uniformly at random and receives the information, if the vertex it connected to is informed. The goal is to study the number of rounds needed to spread the information to everybody, also known as the runtime. In our main result we provide a description, as n gets large, for the distribution of the runtime that involves a martingale limit. This allows us to establish that in general there is no limiting distribution and that convergence occurs only on suitably chosen subsequences (n i) i ∈ N of N , namely when the fractional part of (log 2 n i + log 2 ln n i) i ∈ N converges. [ABSTRACT FROM AUTHOR]

We give an explicit description of the law of terminal value W of additive martingales in a remarkable branching stable process. We show that the right tail probability of the terminal value decays exponentially fast and the left tail probability follows that − log P (W < x) ∼ 1 2 (log x) 2 as x → 0 +. These are in sharp contrast with results in the literature such as Liu (2000, 2001) and Buraczewski (2009). We further show that the law of W is self-decomposable, and therefore, possesses a unimodal density. We specify the asymptotic behavior at 0 and at + ∞ of the latter. [ABSTRACT FROM AUTHOR]

In order to characterize quantitatively the fluctuations between the ergodic limit and the time-averaging estimator, we establish a central limit theorem for a full discretization of the parabolic SPDE, which shows that the normalized time-averaging estimator converges weakly to a normal distribution as the time stepsize tends to 0. A key ingredient in the proof is to extract an appropriate martingale difference series sum from the normalized time-averaging estimator via the Poisson equation, so that convergences of such a sum and the remainder are well balanced. [ABSTRACT FROM AUTHOR]

Suppose X = { X t , t ≥ 0 } is a supercritical superprocess. Let ϕ be the non-negative eigenfunction of the mean semigroup of X corresponding to the principal eigenvalue λ > 0. Then M t (ϕ) = e − λ t 〈 ϕ , X t 〉 , t ≥ 0 , is a non-negative martingale with almost sure limit M ∞ (ϕ). In this paper we study the rate at which M t (ϕ) − M ∞ (ϕ) converges to 0 as t → ∞ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the convergence rate in L p with p ∈ (1 , 2) are also obtained. [ABSTRACT FROM AUTHOR]

The model of directed polymer in a random environment is a fundamental model of interaction between a simple random walk and ambient disorder. This interaction gives rise to complex phenomena and transitions from a central limit theory to novel statistical behaviours. Despite its intense study, there are still many aspects and phases which have not yet been identified. In this review we focus on the current status of our understanding of the transition between weak and strong disorder phases, give an account of some of the methods that the study of the model has motivated and highlight some open questions. [ABSTRACT FROM AUTHOR]

We study the stochastic Camassa–Holm equation with pure jump noise. We establish the existence of the global martingale solution by the regularization method, the tightness criterion, the generalization of the Skorokhod theorem for nonmetric spaces and the stochastic renormalized formulations. [ABSTRACT FROM AUTHOR]

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent 3 / 4. The self-interacting process originated in discussions with Francis Comets. [ABSTRACT FROM AUTHOR]

We show a deviation inequality inequalities for multi-indexed martingale We then provide applications to kernel regression for random fields and rates in the law of large numbers for orthomartingale difference random fields. [ABSTRACT FROM AUTHOR]

Given d ≥ 1 , we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as e 2 d X d μ where X is a log -correlated Gaussian field and μ is a locally finite measure on R d. Our construction generalizes the one performed in the case where μ is the Lebesgue measure. It requires that the measure μ is sufficiently spread out, namely that for μ almost every x we have ∫ B (x , 1) μ (d y) | x − y | d e ρ log 1 | x − y | < ∞ , where ρ : R + → R + can be chosen to be any lower envelope function for the 3-Bessel process (this includes ρ (x) = x α with α ∈ (0 , 1 / 2)). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure μ is in a sense optimal. [ABSTRACT FROM AUTHOR]

We introduce a class of measure-valued processes, which – in analogy to their finite dimensional counterparts – will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators obtaining a representation analogous to polynomial diffusions on R + m , in cases where their domain is large enough. In general the infinite dimensional setting allows for richer specifications strictly beyond this representation. As a special case, we recover measure-valued affine diffusions, sometimes also called Dawson–Watanabe superprocesses. From a mathematical finance point of view, the polynomial framework is especially attractive since it allows to transfer many famous finite dimensional models and their tractability properties to an infinite dimensional measure-valued setting. [ABSTRACT FROM AUTHOR]

We study Markov-modulated affine processes (abbreviated MMAPs), a class of Markov processes that are created from affine processes by allowing some of their coefficients to be a function of an exogenous Markov process X. MMAPs largely preserve the tractability of standard affine processes, as their characteristic function has a computationally convenient functional form. Our setup is a substantial generalization of earlier work, since we consider the case where the generator of X is an unbounded operator. We prove existence of MMAPs via a martingale problem approach, we derive the formula for their characteristic function and we study various mathematical properties. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in L p , 1 ≤ p ≤ 2 , under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the L p -distances (1 ≤ p ≤ 2) between the common distribution function and the corresponding empirical distribution function of stationary random fields. [ABSTRACT FROM AUTHOR]

We introduce a class of Markov chains that includes models of stochastic approximation by averaging and non-averaging. Using a martingale approximation method, we establish various deviation inequalities for separately Lipschitz functions of such a chain, with different moment conditions on some dominating random variables of martingale differences. Finally, we apply these inequalities to stochastic approximation by averaging and empirical risk minimization. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension d ≤ 2 , as the domain of the integral gets large (for fixed time t). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that Itô's martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein's method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process. [ABSTRACT FROM AUTHOR]

This paper considers an initial market model, specified by its underlying assets S and its flow of information F , and an arbitrary random time τ which might not be an F -stopping time. As the death time and the default time (that τ might represent) can be seen when they occur only, the progressive enlargement of F with τ sounds tailor-fit for modeling the new flow of information G that incorporates both F and τ. In this setting of informational market, the first principal goal resides in describing as explicitly as possible the set of all deflators for (S τ , G) , while the second principal goal lies in addressing the No-Free-Lunch-with-Vanishing-Risk concept (NFLVR hereafter) for (S τ , G). Besides this direct application to NFLVR, the set of all deflators constitutes the dual set of all "admissible" wealth processes for the stopped model (S τ , G) , and hence it is vital in many hedging and pricing related optimization problems. Thanks to the results of Choulli et al. (2020), on martingales classification and representation for progressive enlarged filtration, our two main goals are fully achieved in different versions, when the survival probability never vanishes. The results are illustrated on the two particular cases when (S , F) follows the jump-diffusion model and the discrete-time model. [ABSTRACT FROM AUTHOR]

There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar–Parisi–Zhang (KPZ) equation in dimension d ≥ 3 as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in Mukherjee et al. (2016) to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards–Wilkinson regime of the SHE and KPZ equation studied in Gu et al. (2018), Magnen and Unterberger (2018), Dunlap et al. (2020) to the full L 2 -region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers. [ABSTRACT FROM AUTHOR]

The Ewens–Pitman model refers to a distribution for random partitions of [ n ] = { 1 , ... , n } , which is indexed by a pair of parameters α ∈ [ 0 , 1) and θ > − α , with α = 0 corresponding to the Ewens model in population genetics. The large n asymptotic properties of the Ewens–Pitman model have been the subject of numerous studies, with the focus being on the number K n of partition sets and the number K r , n of partition subsets of size r , for r = 1 , ... , n. While for α = 0 asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for α ∈ (0 , 1) only almost-sure convergences are available, with the proof for K r , n being given only as a sketch. In this paper, we make use of martingales to develop a unified and comprehensive treatment of the large n asymptotic behaviours of K n and K r , n for α ∈ (0 , 1) , providing alternative, and rigorous, proofs of the almost-sure convergences of K n and K r , n , and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for K n and K r , n. [ABSTRACT FROM AUTHOR]

A step-reinforced random walk is a discrete-time process with long range memory. At each step, with a fixed probability p , the positively step-reinforced random walk repeats one of its preceding steps chosen uniformly at random, and with complementary probability 1 − p , it has an independent increment. The negatively step-reinforced random walk follows the same reinforcement algorithm but when a step is repeated its sign is also changed. Strong laws of large numbers and strong invariance principles are established for positively and negatively step-reinforced random walks in this work. Our approach relies on two general theorems on the invariance principles for martingale difference sequences and a truncation argument. As by-products of our main results, the law of iterated logarithm and the functional central limit theorem are also obtained for step-reinforced random walks. [ABSTRACT FROM AUTHOR]

In a network of reinforced stochastic processes, for certain values of the parameters, all the agents' inclinations synchronize and converge almost surely toward a certain random variable. The present work aims at clarifying when the agents can asymptotically polarize , i.e. when the common limit inclination can take the extreme values, 0 or 1, with probability zero, strictly positive, or equal to one. Moreover, we present a suitable technique to estimate this probability that, along with the theoretical results, has been framed in the more general setting of a class of martingales taking values in [ 0 , 1 ] and following a specific dynamics. • Results for a general class of bounded martingales: M n + 1 = (1 − r n) M n + r n Y n + 1. • Networks of reinforced stochastic processes E [ X n + 1 | F n ] = W ⊤ Z n Z n + 1 = (1 − r n) Z n + r n X n + 1 with asymptotic polarization event { Z n → 0 } ∪ { Z n → 1 }. • Sufficient and necessary conditions for the probability of asymptotic polarization. • Confidence intervals for the probability of asymptotic polarization. [ABSTRACT FROM AUTHOR]

In this paper, we study the functional convergence in law of the fluctuations of the derivative martingale of branching random walk on the real line. Our main result strengthens the results of Buraczewski et al. (2021) and is the branching random walk counterpart of the main result of Maillard and Pain (2019) for branching Brownian motion. [ABSTRACT FROM AUTHOR]

Let G be a finite Abelian group of order d. We consider an urn in which, initially, there are labeled balls that generate the group G. Choosing two balls from the urn with replacement, observe their labels, and perform a group multiplication on the respective group elements to obtain a group element. Then, we put a ball labeled with that resulting element into the urn. This model was formulated by P. Diaconis while studying a group theoretic algorithm called MeatAxe (Holt and Rees, 1994). Siegmund and Yakir (2005) partially investigated this model. In this paper, we further investigate and generalize this model. More specifically, we allow a random number of balls to be drawn from the urn at each stage in the Diaconis urn model. For such a case, we verify that the normalized urn composition converges almost surely to the uniform distribution on the group G. Moreover, we obtain the asymptotic joint distribution of the urn composition by using the martingale central limit theorem. [ABSTRACT FROM AUTHOR]

We extend the Itô-Wentzell formula for the evolution along a continuous semimartingale of a time-dependent stochastic field driven by a continuous semimartingale to tensor field-valued stochastic processes on manifolds. More concretely, we investigate how the pull-back (respectively, the push-forward) by a stochastic flow of diffeomorphisms of a time-dependent stochastic tensor field driven by a continuous semimartingale evolves with time, deriving it under suitable regularity conditions. We call this result the Kunita-Itô-Wentzell (KIW) formula for the advection of tensor-valued stochastic processes. Equations of this nature bear significance in stochastic fluid dynamics and well-posedness by noise problems, facilitating the development of certain geometric extensions within existing theories. [ABSTRACT FROM AUTHOR]

We provide a novel characterization of the solutions of a quadratic BSDE, which is analogous to the characterization of local martingales by convex functions. We then use our main result to show that BSDE solutions are closed under ucp convergence. Finally, we use our closure result obtain a sufficient condition for existence, and discuss specific cases in which this sufficient condition can be verified. [ABSTRACT FROM AUTHOR]

We consider a general set X of adapted nonnegative stochastic processes in infinite continuous time. X is assumed to satisfy mild convexity conditions, but in contrast to earlier papers need not contain a strictly positive process. We introduce two boundedness conditions on X — DSV corresponds to an asymptotic L 0 -boundedness at the first time all processes in X vanish, whereas NUPBR loc states that X t = { X t : X ∈ X } is bounded in L 0 for each t ∈ [ 0 , ∞). We show that both conditions are equivalent to the existence of a strictly positive adapted process Y such that X Y is a supermartingale for all X ∈ X , with an additional asymptotic strict positivity property for Y in the case of DSV. [ABSTRACT FROM AUTHOR]

In this work we extend the characterization of semimartingale functions in Çinlar et al. (1980) to the non-Markovian setting. We prove that if a function of a semimartingale remains a semimartingale, then under certain conditions the function must have intervals where it is a difference of two convex functions. Under suitable conditions this property also holds for random functions. As an application, we prove that the median process defined in Prokaj et al. (2011) is not a semimartingale. The same process appears also in Hu and Warren (2000) where the question of the semimartingale property is raised but not settled. [ABSTRACT FROM AUTHOR]

We derive an Itō-formula for the Dawson–Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Itō-formula with respect to two aspects. Firstly, we extend the state–space of the underlying process (X (t)) t ∈ [ 0 , T ] to an infinite-dimensional one — the space of finite measure. Secondly, we extend the formula to functions F (t , X t) depending on the entire paths X t = (X (s ∧ t)) s ∈ [ 0 , T ] up to times t. This later extension is usually called functional Itō-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses. [ABSTRACT FROM AUTHOR]

We extend the continuous dual Hahn process (T t) of Corwin and Knizel from a finite time interval to the entire real line by taking a limit of a closely related Markov process (T t). We also characterize processes (T t) by conditional means and variances under bidirectional conditioning, and we prove that continuous dual Hahn polynomials are orthogonal martingale polynomials for both processes. [ABSTRACT FROM AUTHOR]

Using a new notion of path-derivative, we study the well-posedness of backward stochastic differential equation driven by a continuous martingale M when f (s , γ , y , z) is locally Lipschitz in (y , z) : Y t = ξ (M [ 0 , T ] ) + ∫ t T f (s , M [ 0 , s ] , Y s − , Z s m s) d tr [ M , M ] s − ∫ t T Z s d M s − N T + N t. Here, M [ 0 , t ] is the path of M from 0 to t and m is defined by [ M , M ] t = ∫ 0 t m s m s ∗ d tr [ M , M ] s . When the BSDE is one-dimensional, we show the existence and uniqueness of the solution. On the contrary, when the BSDE is multidimensional, we show the existence and uniqueness only when [ M , M ] T is small enough: otherwise, we provide a counterexample. Then, we investigate the applications to optimal control of diffusion and optimal portfolio selection under various restrictions. [ABSTRACT FROM AUTHOR]

A density dependent Markov chain (DDMC) introduced in Kurtz (1978) is a special continuous time Markov process. Examples are considered in fields like epidemics and processes which describe chemical reactions. Moreover the Yule process is a further example. In this paper we prove a moderate deviation principle for the paths of a certain class of DDMC. The proofs of the bounds utilize an exponential martingale as well as a generalized version of Girsanov's theorem. The exponential martingale is defined according to the generator of the DDMC. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behavior of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctuations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales. We also discuss extensions of our results to multi-class systems. [ABSTRACT FROM AUTHOR]

Large deviations for additive path functionals of stochastic processes have attracted significant research interest, in particular in the context of stochastic particle systems and statistical physics. Efficient numerical 'cloning' algorithms have been developed to estimate the scaled cumulant generating function, based on importance sampling via cloning of rare event trajectories. So far, attempts to study the convergence properties of these algorithms in continuous time have led to only partial results for particular cases. Adapting previous results from the literature of particle filters and sequential Monte Carlo methods, we establish a first comprehensive and fully rigorous approach to bound systematic and random errors of cloning algorithms in continuous time. To this end we develop a method to compare different algorithms for particular classes of observables, based on the martingale characterization of stochastic processes. Our results apply to a large class of jump processes on compact state space, and do not involve any time discretization in contrast to previous approaches. This provides a robust and rigorous framework that can also be used to evaluate and improve the efficiency of algorithms. [ABSTRACT FROM AUTHOR]

We prove existence and uniqueness for semimartingale reflecting diffusions in 2-dimensional piecewise smooth domains with varying, oblique directions of reflection on each "side", under geometric, easily verifiable conditions. Our conditions are optimal in the sense that, in the case of a convex polygon, they reduce to the conditions of Dai and Williams (1996), which are necessary for existence of reflecting Brownian motion. Moreover our conditions allow for cusps. Our argument is based on a new localization result for constrained martingale problems which holds quite generally: as an additional example, we show that it holds for diffusions with jump boundary conditions. • Obliquely reflecting semimartingale Brownian motion in piecewise curved domains. • General existence and uniqueness result lacking: we provide it in dimension 2. • We allow for cusps and points of smooth boundary with discontinuous reflection. • Locally we use a reverse ergodic theorem for inhomogeneous killed Markov chains. • We prove a localization theorem for general constrained martingale problems. [ABSTRACT FROM AUTHOR]

In this paper we explain how the notion of weak Dirichlet process is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition: in particular we introduce characteristics for weak Dirichlet processes. We also introduce a weak concept (in law) of finite quadratic variation. We investigate a set of new useful chain rules and we discuss a general framework of (possibly path-dependent with jumps) martingale problems with a set of examples of SDEs with jumps driven by a distributional drift. [ABSTRACT FROM AUTHOR]

This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic and naturally includes time-inhomogeneous and non-stationary processes as well as initial laws concentrated on a single point. The class of processes studied includes martingales, Markov processes and general square integrable càdlàg processes. The general approach is complemented by a simple and direct method for martingales, diffusions and discrete-time Markov processes. The method is illustrated by deriving concentration inequalities for the Polyak–Ruppert algorithm, SDEs with time-dependent drift coefficients "contractive at infinity" with both Lipschitz and squared Lipschitz observables, some classical martingales and non-elliptic SDEs. [ABSTRACT FROM AUTHOR]

We focus on a class of BSDEs driven by a càdlàg martingale and the corresponding Markovian BSDEs which arise when the randomness of the driver appears through a Markov process. To those BSDEs we associate a deterministic equation which, when the Markov process is a Brownian diffusion, is nothing else but a parabolic semi-linear PDE. We prove existence and uniqueness of a decoupled mild solution of the deterministic problem, and give a probabilistic representation of this solution through the aforementioned BSDEs. [ABSTRACT FROM AUTHOR]

Our goal is to investigate the asymptotic behavior of the center of mass of the elephant random walk, which is a discrete-time random walk on integers with a complete memory of its whole history. In the diffusive and critical regimes, we establish the almost sure convergence, the law of iterated logarithm and the quadratic strong law for the center of mass of the elephant random walk. The asymptotic normality, properly normalized, is also provided. Finally, we prove a strong limit theorem for the center of mass in the superdiffusive regime. All our analysis relies on asymptotic results for multi-dimensional martingales. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a certain type of martingale problems associated to general operators corresponding to processes which have finite lifetime. We analyse several properties and in particular the weak convergence of sequences of solutions for an appropriate Skorokhod topology setting. We point out the Feller-type features of the associated solutions to this type of martingale problem. Then localisation theorems for well-posed martingale problems or for corresponding generators are proved. [ABSTRACT FROM AUTHOR]

Let X be a point process and let X denote the filtration generated by X. In this paper we study martingale representation theorems in the filtration G obtained as an initial and progressive enlargement of the filtration X. The progressive enlargement is done here by means of a whole point process H. We do not require further assumptions on the point process H nor on the dependence between X and H. In particular, we recover the special case of the progressive enlargement by a random time τ. [ABSTRACT FROM AUTHOR]

Let E ˆ be the upper expectation of a weakly compact but possibly non-dominated family P of probability measures. Assume that Y is a d -dimensional P -semimartingale under E ˆ. Given an open set Q ⊂ R d , the exit time of Y from Q is defined by τ Q ≔ inf { t ≥ 0 : Y t ∈ Q c }. The main objective of this paper is to study the quasi-continuity properties of τ Q under the nonlinear expectation E ˆ. Under some additional assumptions on the growth and regularity of Y , we prove that τ Q ∧ t is quasi-continuous if Q satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we obtain the quasi-continuity of exit times for multi-dimensional G -martingales, which nontrivially generalizes the previous one-dimensional result of Song (2011). [ABSTRACT FROM AUTHOR]

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise in one space dimension and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter–Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous work, no interface potential has to be included, the initial data and the solution can have de-wetted regions of positive measure, and the Trotter–Kato scheme allows for a simpler proof of existence than in case of Itô noise. [ABSTRACT FROM AUTHOR]

We study a family of mean field games with a state variable evolving as a multivariate jump–diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and give conditions under which the optimal strategies are in fact Markovian, hence extending to a jump–diffusion setting previous results established in Lacker (2015). The proofs rely upon the notions of relaxed controls and martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results. [ABSTRACT FROM AUTHOR]

We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established. [ABSTRACT FROM AUTHOR]