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The probabilistic symbol is defined as the right-hand side derivative at time zero of the characteristic functions corresponding to the one-dimensional marginals of a time-homogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving time-homogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the non-homogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of p-variationa. In contrast to many situations where non-homogeneous Markov processes are involved, the space-time process cannot be utilized when considering maximal inequalities. [ABSTRACT FROM AUTHOR]

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]

We study the response of the optimal investment problem to small changes of the stock price dynamics. Starting with a multidimensional semimartingale setting of an incomplete market, we suppose that the perturbation process is also a general semimartingale. We obtain second-order expansions of the value functions, first-order corrections to the optimisers, and provide the adjustments to the optimal control that match the objective function up to the second order. We also give a characterisation in terms of the risk-tolerance wealth process, if it exists, by reducing the problem to the Kunita–Watanabe decomposition under a change of measure and numéraire. Finally, we illustrate the results by examples of base models that allow closed-form solutions, but where this structure is lost under perturbations of the model where our results allow an approximate solution. [ABSTRACT FROM AUTHOR]

Using rough path theory, we provide a pathwise foundation for stochastic Itô integration which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called property (RIE) for càdlàg paths, which is shown to imply the existence of a càdlàg rough path and of quadratic variation in the sense of Föllmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Cover's universal portfolio are admissible integrands, and that property (RIE) is satisfied by both (Young) semimartingales and typical price paths. [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]

This article proposes more efficient estimators for the leverage effect than the existing ones. The idea is to allow for nonuniform kernel functions in the spot volatility estimates or the aggregated returns. This finding highlights a critical difference between the leverage effect and integrated volatility functionals, where the uniform kernel is optimal. Another distinction between these two cases is that the overlapping estimators of the leverage effect are more efficient than the nonoverlapping ones. We offer two perspectives to explain these differences: one is based on the "effective kernel" and the other on the correlation structure of the nonoverlapping estimators. The simulation study shows that the proposed estimator with a nonuniform kernel substantially increases the estimation efficiency and testing power relative to the existing ones. [ABSTRACT FROM AUTHOR]

In a standard continuous time asset pricing model, this paper provides an explosion time characterization of asset price bubbles that extends the existing characterization theorems in the literature from diffusion processes to general semimartingales (which can include jumps). This characterization has a nice economic interpretation, not emphasized in the existing literature. [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]

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]

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]

We propose using a permutation test to detect discontinuities in an underlying economic model at a known cutoff point. Relative to the existing literature, we show that this test is well suited for event studies based on time‐series data. The test statistic measures the distance between the empirical distribution functions of observed data in two local subsamples on the two sides of the cutoff. Critical values are computed via a standard permutation algorithm. Under a high‐level condition that the observed data can be coupled by a collection of conditionally independent variables, we establish the asymptotic validity of the permutation test, allowing the sizes of the local subsamples to be either be fixed or grow to infinity. In the latter case, we also establish that the permutation test is consistent. We demonstrate that our high‐level condition can be verified in a broad range of problems in the infill asymptotic time‐series setting, which justifies using the permutation test to detect jumps in economic variables such as volatility, trading activity, and liquidity. These potential applications are illustrated in an empirical case study for selected FOMC announcements during the ongoing COVID‐19 pandemic. [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]

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.

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]

We introduce the Realized moMents of Disjoint Increments (ReMeDI) paradigm to measure microstructure noise (the deviation of the observed asset prices from the fundamental values caused by market imperfections). We propose consistent estimators of arbitrary moments of the microstructure noise process based on high-frequency data, where the noise process could be serially dependent, endogenous, and nonstationary. We characterize the limit distributions of the proposed estimators and construct confidence intervals under infill asymptotics. Our simulation and empirical studies show that the ReMeDI approach is very effective to measure the scale and the serial dependence of microstructure noise. Moreover, the estimators are quite robust to model specifications, sample sizes, and data frequencies. [ABSTRACT FROM AUTHOR]

We present a new theory for the conduct of nonparametric inference about the latent spot volatility of a semimartingale asset price process. In contrast to existing theories based on the asymptotic notion of an increasing number of observations in local estimation blocks, our theory treats the estimation block size k as fixed. While the resulting spot volatility estimator is no longer consistent, the new theory permits the construction of asymptotically valid and easy‐to‐calculate pointwise confidence intervals for the volatility at any given point in time. Extending the theory to a high‐dimensional inference setting with a growing number of estimation blocks further permits the construction of uniform confidence bands for the volatility path. An empirically realistically calibrated simulation study underscores the practical reliability of the new inference procedures. An empirical application based on intraday data for the S&P 500 equity index reveals highly significant abrupt changes, or jumps, in the market volatility at FOMC news announcement times, validating recent uses of various high‐frequency‐based identification schemes in asset pricing finance and monetary economics. [ABSTRACT FROM AUTHOR]

We mainly tackle the optimal portfolio problem when investors have partial information in the financial market, and the inflation risk is taken into account, the risky asset price is suddenly changed by impacts of major events which result in the asset return disorder at a random time. Applying semimartingale and backward stochastic differential equation theory, we deduce the optimal investment strategy and value process which satisfy the exponential utility maximization of terminal real wealth. Moreover, numerical simulations are presented to illustrate the effects of the expected inflation rate and inflation volatility on the optimal investment strategy, and to compare the asset return disorder with the normal situation.

In this article, we seek to solve the problem of stochastic filtering of the unobserved drift of the stock price in the presence of privileged information. Working within a finite time investment horizon, the privileged information which is a function of the future value of the stock price, is modeled such that its quality improves as we move towards the information reveal date. The hidden/unobserved drift is modeled as a Gaussian process. Combining the techniques of progressive enlargement of filtration and stochastic filtering of linear state-space models, we obtain explicit analytic results for the insider’s estimates of the unobserved drift process. In addition, we obtain the optimal portfolio strategy for an insider having the log utility function. Our numerical results reveal that when the quality of privileged information is high, the insider would require less initial capital as compared to the regular trader who has no access to the privileged information. Further, we show how the stock price volatility influences the value of the insider’s privileged information, with period of high volatility pointing to increased value of the privileged information.

We examine a Wong-Zakai type approximation of a family of stochastic differential equations driven by a general càdlàg semimartingale. For such an approximation, compared with the pointwise convergence result by Kurtz, Pardoux and Protter [10,Theorem 6.5], we establish stronger convergence results under the Skorokhod -topology, which, among other possible applications, implies the convergence of the first passage time of the solution to the approximating stochastic differential equation. [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 general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of multiple curve markets under minimal assumptions in an extended HJM framework and provide a fundamental theorem of asset pricing based on NAFLVR. The approach with stochastic discontinuities permits to embed market models directly, unifying seemingly different modelling philosophies. We also develop a tractable class of models, based on affine semimartingales, going beyond the requirement of stochastic continuity. [ABSTRACT FROM AUTHOR]

We mainly tackle the optimal portfolio problem when investors have partial information in the financial market, and the inflation risk is taken into account, the risky asset price is suddenly changed by impacts of major events which result in the asset return disorder at a random time. Applying semimartingale and backward stochastic differential equation theory, we deduce the optimal investment strategy and value process which satisfy the exponential utility maximization of terminal real wealth. Moreover, numerical simulations are presented to illustrate the effects of the expected inflation rate and inflation volatility on the optimal investment strategy, and to compare the asset return disorder with the normal situation. [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]

Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell's renewal theorem holds. [ABSTRACT FROM AUTHOR]

We introduce and study the notion of sure profits via flash strategies, consisting of a high-frequency limit of buy-and-hold trading strategies. In a fully general setting, without imposing any semimartingale restriction, we prove that there are no sure profits via flash strategies if and only if asset prices do not exhibit predictable jumps. This result relies on the general theory of processes and provides the most general formulation of the well-known fact that, in an arbitrage-free financial market, asset prices (including dividends) should not exhibit jumps of a predictable direction or magnitude at predictable times. We furthermore show that any price process is always right-continuous in the absence of sure profits. Our results are robust under small transaction costs and imply that, under minimal assumptions, price changes occurring at scheduled dates should only be due to unanticipated information releases. [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]

We develop tests for deciding whether a large cross‐section of asset prices obey an exact factor structure at the times of factor jumps. Such jump dependence is implied by standard linear factor models. Our inference is based on a panel of asset returns with asymptotically increasing cross‐sectional dimension and sampling frequency, and essentially no restriction on the relative magnitude of these two dimensions of the panel. The test is formed from the high‐frequency returns at the times when the risk factors are detected to have a jump. The test statistic is a cross‐sectional average of a measure of discrepancy in the estimated jump factor loadings of the assets at consecutive jump times. Under the null hypothesis, the discrepancy in the factor loadings is due to a measurement error, which shrinks with the increase of the sampling frequency, while under an alternative of a noisy jump factor model this discrepancy contains also nonvanishing firm‐specific shocks. The limit behavior of the test under the null hypothesis is nonstandard and reflects the strong‐dependence in the cross‐section of returns as well as their heteroskedasticity which is left unspecified. We further develop estimators for assessing the magnitude of firm‐specific risk in asset prices at the factor jump events. Empirical application to S&P 100 stocks provides evidence for exact one‐factor structure at times of big market‐wide jump events. [ABSTRACT FROM AUTHOR]

We propose a test for the rank of a cross-section of processes at a set of jump events. The jump events are either specific known times or are random and associated with jumps of some process. The test is formed from discretely sampled data on a fixed time interval with asymptotically shrinking mesh. In the first step, we form nonparametric estimates of the jump events via thresholding techniques. We then compute the eigenvalues of the outer product of the cross-section of increments at the identified jump events. The test for rank r is based on the asymptotic behavior of the sum of the squared eigenvalues excluding the largest r. A simple resampling method is proposed for feasible testing. The test is applied to financial data spanning the period 2007–2015 at the times of stock market jumps. We find support for a one-factor model of both industry portfolio and Dow 30 stock returns at market jump times. This stands in contrast with earlier evidence for higher-dimensional factor structure of stock returns during "normal" (nonjump) times. We identify the latent factor driving the stocks and portfolios as the size of the market jump. [ABSTRACT FROM AUTHOR]

Stochastic control of exchange rates when a central bank employs anti-inflationary stochastic differential equation (SDE) monetary policy is the key topic of our paper. Despite low money growth SDE policy means exchange rates invariably violate the central bank's targets. Monetary policy also incorporates interventions reflected by sudden money supply jumps that moderate deviations from targets. Controlling exchange rates involves minimizing target deviation and intervention costs. Restrictions on these costs ensure intervention vanishes under the optimal control, implying the central bank engineers freely floating exchange rates instead of managed floating or fixed exchange rates. Econometric evidence suggests discretionary interventions may be ineffective or generate excess volatility and speculation in currency markets. Our result demonstrates mathematically that such collateral damage discourages intervention. [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]