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We develop a statistical inference procedure for the ubiquitous calendar effects in microstructure noise using high frequency data. This is, to the best of our knowledge, the first inference theory ever built for noise calendar effect under the general semi‐martingale‐plus‐noise setup for prices contaminated with non‐stationary, endogenous, and serially dependent microstructure noise. We devise a noise‐calendar‐effect estimator by an appropriately scaled average of high‐frequency returns that precede a time of day across a large number of trading days. Feasible central limit theorem for the estimator is established under a joint infill and long‐span asymptotics. Monte Carlo simulations corroborate our theoretical findings. An empirical study on the high‐frequency data of the e‐mini S&P 500 futures and a Chinese stock demonstrates that the noise calendar effect has undergone significant changes over time for the latter, yet remains stable for the former. [ABSTRACT FROM AUTHOR]

The existence of strong solutions and pathwise uniqueness are established for one-dimensional stochastic Volterra equations with locally Hölder continuous diffusion coefficients and sufficiently regular kernels. Moreover, we study the sample path regularity, the integrability and the semimartingale property of solutions to one-dimensional stochastic Volterra equations. [ABSTRACT FROM AUTHOR]

Tool wear will reduce workpieces' quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion (LFSM) was investigated, along with a differential iterative model with it as the noise term is constructed according to the fractional-order Ito formula; the general solution of this model is derived by semimartingale approximation. After that, for the chaotic features of the vibration signal, the time-frequency domain characteristics were extracted using principal component analysis (PCA), and the relationship between the variation of the generalized Hurst exponent and tool wear was established using multifractal detrended fluctuation analysis (MDFA). Then, the maximum prediction length was obtained by the maximum Lyapunov exponent (MLE), which allows for analysis of the vibration signal. Finally, tool condition diagnosis was carried out by the evolving connectionist system (ECoS). The results show that the LFSM iterative model with semimartingale approximation combined with PCA and MDFA are effective for the prediction of vibration trends and tool condition. Further, the monitoring of tool condition using ECoS is also effective. [ABSTRACT FROM AUTHOR]

In the same context as in the seminal paper [P. Cheridito, Bernoulli, 7 (2001), pp. 913-934], we are concerned with the semimartingale property of the sum of some Gaussian martingale and an independent fractional Brownian motion with Hurst parameter H = (0, 1). At the same time, we emphasize that the Markov property is lost even if the martingale owns it. [ABSTRACT FROM AUTHOR]

The approximation of integral type functionals is studied for discrete observations of a continuous Itô semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for L 2 -Sobolev functions with fractional smoothness. An explicit L 2 (P) -lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance. [ABSTRACT FROM AUTHOR]

This article tests volatility jumps based on the high frequency data. Under the null hypothesis that the volatility process is a continuous semimartingale, our test statistic converges to a normal distribution, and under the alternative hypothesis where the volatility has jumps, the statistic diverges to infinity. Compared to the test statistic of Bibinger et al. (Bibinger et al. (2017). Annals of Statistics 45, 1542–1578), our proposed statistic diverges to infinity at a faster rate, and has a better power. Simulation studies confirm the theoretical results, and an empirical analysis shows that some real financial data possess volatility jumps. [ABSTRACT FROM AUTHOR]

We study the existence, uniqueness and approximation of solutions of general stochastic differential equations (SDEs) with two time-dependent reflecting barriers driven by semimartingales and processes with bounded p -variation, p ∈ [ 1 , 2). We do not assume that the barriers have to be completely separated. Applications to currency option pricing in financial models with fractional Brownian motion and standard Brownian motion are given. [ABSTRACT FROM AUTHOR]

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron–Martin space under the flow of mild solutions to a system of path-dependent ordinary differential equations. Our result extends the Stroock–Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on functional Itô calculus. [ABSTRACT FROM AUTHOR]

In this paper, we study the ruin problem with investment in a general framework where the business part X is a Lévy process and the return on investment R is a semimartingale. Under some conditions, we obtain upper and lower bounds on the finite and infinite time ruin probabilities as well as the logarithmic asymptotic for them. When R is a Lévy process, we retrieve some well-known results. Finally, we obtain conditions on the exponential functionals of R for ruin with probability 1, and we express these conditions using the semimartingale characteristics of R in the case of Lévy processes. [ABSTRACT FROM AUTHOR]

We develop a nonparametric test for whether return volatility exhibits time-varying intraday periodicity using a long time series of high-frequency data. Our null hypothesis, commonly adopted in work on volatility modeling, is that volatility follows a stationary process combined with a constant time-of-day periodic component. We construct time-of-day volatility estimates and studentize the high-frequency returns with these periodic components. If the intraday periodicity is invariant, then the distribution of the studentized returns should be identical across the trading day. Consequently, the test compares the empirical characteristic function of the studentized returns across the trading day. The limit distribution of the test depends on the error in recovering volatility from discrete return data and the empirical process error associated with estimating volatility moments through their sample counterparts. Critical values are computed via easy-to-implement simulation. In an empirical application to S&P 500 index returns, we find strong evidence for variation in the intraday volatility pattern driven in part by the current level of volatility. When volatility is elevated, the period preceding the market close constitutes a significantly higher fraction of the total daily integrated volatility than during low volatility regimes. for this article are available online. [ABSTRACT FROM AUTHOR]

Risk measure is a fundamental concept in finance and in the insurance industry. It is used to adjust life insurance rates. In this article, we will study dynamic risk measures by means of backward stochastic Volterra integral equations (BSVIEs) with jumps. We prove a comparison theorem for such a type of equations. Since the solution of a BSVIEs is not a semimartingale in general, we will discuss some particular semimartingale issues. [ABSTRACT FROM AUTHOR]

It is shown that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate. [ABSTRACT FROM AUTHOR]

In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains. [ABSTRACT FROM AUTHOR]

For a fixed right process X we investigate those functions u for which u(X) is a quasimartingale. We prove that u(X) is a quasimartingale if and only if u is the difference of two finite excessive functions. In particular, we show that the quasimartingale nature of u is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for u(X) in terms of a certain variation of u with respect to the transition function of the process. We provide sufficient conditions under which u(X) is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima. [ABSTRACT FROM AUTHOR]

In the literature, the Euler-Maruyama (EM) method for approximation purposes of stochastic differential Equations (SDE) driven by α-stable Lévy motions is reported. Convergence in probability of that method was proven but it is surrounded by some ambiguities. To accomplish the but without ambiguities, this article has derived convergence in probability of numerical EM method based on diffusion given by semimartingales for SDEs driven by α-stable processes. Some examples are provided, their numerical solution are obtained and theoretical results are reconfirmed. The adopted method could be applied to other subclasses of semimartingales. [ABSTRACT FROM AUTHOR]

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques. [ABSTRACT FROM PUBLISHER]

Itô semimartingales are the semimartingales whose characteristics are absolutely continuous with respect to Lebesgue measure. We study the importance of this assumption for statistical inference on a discretely sampled semimartingale in terms of the identifiability of its characteristics, their estimation, and propose tests of the Itô property against the non-Itô alternative when the observed semimartingale is continuous, or discontinuous with finite activity jumps, and under a number of technical assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the optimal approximation of a Rosenblatt process based on semimartingales of the formwhere (y1,y2)↦a(y1,y2) is a square integrable process andBis a standard Brownian motion. We show that there exists a unique semimartingale closest to Rosenblatt process ifais deterministic. [ABSTRACT FROM PUBLISHER]

We develop robust inference methods for studying linear dependence between the jumps of discretely observed processes at high frequency. Unlike classical linear regressions, jump regressions are determined by a small number of jumps occurring over a fixed time interval and the rest of the components of the processes around the jump times. The latter are the continuous martingale parts of the processes as well as observation noise. By sampling more frequently the role of these components, which are hidden in the observed price, shrinks asymptotically. The robustness of our inference procedure is with respect to outliers, which are of particular importance in the current setting of relatively small number of jump observations. This is achieved by using nonsmooth loss functions (likeL1) in the estimation. Unlike classical robust methods, the limit of the objective function here remains nonsmooth. The proposed method is also robust to measurement error in the observed processes, which is achieved by locally smoothing the high-frequency increments. In an empirical application to financial data, we illustrate the usefulness of the robust techniques by contrasting the behavior of robust and ordinary least regression (OLS)-type jump regressions in periods including disruptions of the financial markets such as so-called “flash crashes.” Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations with respect to semimartingale integrators. We present new connections between their solutions. In particular, we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set-valued stochastic equations. We also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. The results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases. [ABSTRACT FROM AUTHOR]

We study the discretisation of forward backward stochastic differential equations (FBSDEs) driven by càdlàg martingales. We prove that under certain conditions imposed on the parameters of the FBSDE the time-discrete scheme we consider converges to the time-continuous equation in the L 2 -sense. Moreover, we show that the L 2 -norm of the error is of the order of the time step. [ABSTRACT FROM AUTHOR]

We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved. [ABSTRACT FROM AUTHOR]

We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews. [ABSTRACT FROM AUTHOR]

This paper reformulates the stochastic string model of Santa-Clara and Sornette using stochastic calculus with continuous semimartingales. We present some new results, such as: (a) the dynamics of the short-term interest rate, (b) the PDE that must be satisfied by the bond price, and (c) an analytic expression for the price of a European bond call option. Additionally, we clarify some important features of the stochastic string model and show its relevance to price derivatives and the equivalence with an infinite dimensional HJM model to price European options. [ABSTRACT FROM AUTHOR]

We adapt the general conditions of the weak convergence for the sequence of processes with discrete time to the diffusion process towards the weak convergence for the discrete-time models of a financial market to the continuous-time diffusion model. These results generalize a classical scheme of the weak convergence for discrete-time markets to the Black-Scholes model. We give an explicit and direct method of approximation by a recurrent scheme. As an example, an Ornstein-Uhlenbeck process is considered as a limit model. [ABSTRACT FROM AUTHOR]

The paper considers a statistical concept of causality in continuous time between filtered probability spaces, based on Granger’s definition of causality. This causality concept is connected with the preservation of the martingale representation property when the filtration is getting smaller. We also give conditions, in terms of causality, for every martingale to be a continuous semimartingale, and we consider the equivalence between the concept of causality and the preservation of the martingale representation property under change of measure. In addition, we apply these results to weak solutions of stochastic differential equations. The results can be applied to the economics of securities trading. [ABSTRACT FROM PUBLISHER]

Averaging the local characteristics with respect to time transforms a semimartingale with independent increments into a Lévy process. It is shown that such a Lévy process is "closer" to any Lévy process than the original process, where the "greater closeness" of processes is understood as the greater closeness of all f-divergences of their distributions, which also admits an equivalent formulation in terms of the comparison of the corresponding binary statistical experiments. Moreover, we prove the criterion of the equivalence of binary experiments that consist of the distribution of a semimartingale to independent increments and the distribution of a Lévy process. [ABSTRACT FROM AUTHOR]

This paper considers the problem of estimating parameters in a periodic regression in continuous time with a semimartingale noise by discrete time observations. Improved estimates for the regression parameters are proposed. It is established that under some general conditions these estimates have an advantage in the mean square accuracy over the least squares estimates. The asymptotic minimaxity of the improved estimates has been proved in the robust risk sense. The properties of the proposed procedure for the models with non-Gaussian noises of pulse type have been studied. The pulse disturbances have random intensity and occur at random times which form a Poisson process. [ABSTRACT FROM AUTHOR]

In this paper, we consider two semimartingales sampled at stopping times in an asynchronous manner. We are interested in estimating their cumulative co-volatility separately from the sum of their co-jumps. For this purpose, we combine the Hayashi-Yoshida method (to deal with the asynchronicity) with the threshold technique (to separate the jumps) and consider a class of statistics called the truncated Hayashi-Yoshida estimator. We prove the consistency and the asymptotic mixed normality of the truncated Hayashi-Yoshida estimator under some mild conditions allowing the presence of infinite activity jumps. [ABSTRACT FROM AUTHOR]

Abstract: We present a new approach to a concept of a set-valued stochastic integral with respect to semimartingales. Such an integral, called set-valued stochastic up-trajectory integral, is compatible with the decomposition of the semimartingale. Some properties of this integral are stated. We show applicability of the new integral in set-valued stochastic integral equations driven by multidimensional semimartingales. The uniqueness theorem is presented. Then we extend the notion of the set-valued stochastic up-trajectory integral to definition of a fuzzy stochastic up-trajectory integral with respect to semimartingales. A result on uniqueness of a solution to fuzzy stochastic integral equations incorporating the new fuzzy stochastic up-trajectory integral driven by the multidimensional semimartingale is stated. [Copyright &y& Elsevier]

Let be an ℝ d-valued càdlàg random process, and let F be an increasing from zero ℝ-valued random process whose values at all times are measurable w.r.t. some σ-algebra (the class of all such processes is denoted by ). Conditions guaranteing that for every bounded continuous function are found. In the most general theorem they are formulated in terms of F and . Further we consider the case when satisfies an equation of the kind where is an -measurable random variable, is a continuous process of class , A is a matrix-valued random process -measurable in , and S is an ℝ d-valued semimartingale with conditionally on independent increments and initial value 0. In this case, sufficient conditions for () are stated in terms of F and the constituents of the equation. Besides, the characteristic function of an n-dimensional distribution of is found. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents limit theorems of realized multipower variation for semimartingales and Gaussian integral processes with jumps observed in high frequency. In particular, we obtain a central limit theorem of realized multipower variation for semimartingale where some of the powers equal one and the others are less one. Convergence in probability and central limit theorems of realized threshold bipower variation for Gaussian integral processes with jumps are also obtained. These results provide new statistical tools to analyze and test the long memory effect in high frequency situation. [Copyright &y& Elsevier]

Starting from the moving average representation of fractional Brownian motion, there are two different approaches to constructing fractional Lévy processes in the literature. Applying L2-integration theory, one can keep the same moving average kernel and replace the driving Brownian motion by a pure jump Lévy process with finite second moments. Alternatively, in the framework of alpha-stable random measures, the Brownian motion is replaced by an alpha-stable Lévy process and the exponent in the kernel is reparametrized by H - \/a. We now provide a unified approach taking kernels of the form a((t - s)γ+ - (-s)γ+) + b((t - s)γ- - (-s)γ-), where γ can be chosen according to the existing moments and the Blumenthal-Getoor index of the underlying Lévy process. These processes may exhibit both long and short range dependence. In addition we will examine further properties of the processes, e.g., regularity of the sample paths and the semimartingale property. [ABSTRACT FROM AUTHOR]

In this article, we propose a general downside risk measure based on high-frequency downward moves below minimum acceptable target in asset prices. We derive the central limit theorem of this measure, and Monte Carlo simulation experiments support our theoretical results. We also investigate the distributional properties of this measure in China’s stock market. The theoretical and empirical works on realized downside risk measure shed light on the potential of this measure in measuring and modeling financial risk. [ABSTRACT FROM AUTHOR]

. In this study we are concerned with inference on the correlation parameter ρ of two Brownian motions, when only high-frequency observations from two one-dimensional continuous Itô semimartingales, driven by these particular Brownian motions, are available. Estimators for ρ are constructed in two situations: either when both components are observed (at the same time), or when only one component is observed and the other one represents its volatility process and thus has to be estimated from the data as well. In the first case it is shown that our estimator has the same asymptotic behaviour as the standard one for i.i.d. normal observations, whereas a feasible estimator can still be defined in the second framework, but with a slower rate of convergence. [ABSTRACT FROM AUTHOR]

Abstract: Let be an open set in . We show that under a mild assumption on the richness of the generator, a Feller process in with (predictable) killing is a semimartingale. To this end, we generalize the notion of semimartingales in a natural way to those ‘with killing’. Furthermore we calculate the semimartingale characteristics of the Feller process explicitly and analyze their connections to the symbol. Finally we derive a probabilistic formula for calculating the symbol of the process. [Copyright &y& Elsevier]

Abstract: An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed. [Copyright &y& Elsevier]

In this article, the weak convergence of impulsive recurrent process with semi-Markov switching in the scheme of Poisson approximation is proved. Singular perturbation problem for the compensating operator of the extended Markov renewal process is used to prove the relative compactness. [ABSTRACT FROM AUTHOR]

Abstract: We use a method developed in Carmona et al. (2003) to study the fractional geometric mean-reversion processes. Our obtained results hold for any . [Copyright &y& Elsevier]

Abstract: The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black–Scholes model by a model driven by semimartingales, and a European option pricing formula is found. [Copyright &y& Elsevier]

This paper derives several identities for the iterated integrals of a general semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Their form and derivations are combinatorial. The formulae simplify for continuous or finite-variation semimartingales, especially for counting processes. The results are motivated by chaotic representation of martingales, and a simple such application is given. [ABSTRACT FROM AUTHOR]

In this paper we prove a formula for the density process of the distributions of two semimartingales with independent increments. This formula generalizes a well-known result for the case of locally absolutely continuous distributions as well as Sato's result for Levy processes without the requirement of local absolute continuity. [ABSTRACT FROM AUTHOR]

Abstract: We derive a change of variable formula for non-anticipative functionals defined on the space of -valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Itô formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations. [Copyright &y& Elsevier]

This paper presents a short survey on limit theorems for certain functionals of semimartingales that are observed at high frequency. Our aim is to explain the main ideas of the theory to a broader audience. We introduce the concept of stable convergence, which is crucial for our purpose. We show some laws of large numbers (for the continuous and the discontinuous case) that are the most interesting from a practical point of view, and demonstrate the associated stable central limit theorems. Moreover, we state a simple sketch of the proofs and give some examples. [ABSTRACT FROM AUTHOR]

We prove that a fork-convex family $$ \mathbb{W} $$ of nonnegative stochastic processes has an equivalent supermartingale density if and only if the set H of nonnegative random variables majorized by the values of elements of $$ \mathbb{W} $$ at fixed instants of time is bounded in probability. A securities market model with arbitrarily many main risky assets, specified by the set $$ \mathbb{W}\left( \mathbb{S} \right) $$ of nonnegative stochastic integrals with respect to finite collections of semimartingales from an arbitrary indexed family S, satisfies the assumptions of this theorem. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents limit theorems for certain functionals of semimartingales observed at high frequency. In particular, we extend results from Jacod (2008) to the case of bipower variation, showing under standard assumptions that one obtains a limiting variable, which is in general different from the case of a continuous semimartingale. In a second step a truncated version of bipower variation is constructed, which has a similar asymptotic behaviour as standard bipower variation for a continuous semimartingale and thus provides a feasible central limit theorem for the estimation of the integrated volatility even when the semimartingale exhibits jumps. [Copyright &y& Elsevier]

We prove that a sequence of diffusion processes in ℝ n that are Brownian motions with drift unboundedly increasing in modulus and directed to a manifold converges in distribution to the Brownian motion on the manifold. [ABSTRACT FROM AUTHOR]