In this paper we present the asymptotic analysis of the realised quadratic variation for multivariate symmetric $β$-stable Lévy processes, $β\in (0,2)$, and certain pure jump semimartingales. The main focus is on derivation of functional limit theorems for the realised quadratic variation and its spectrum. We will show that the limiting process is a matrix-valued $β$-stable Lévy process when the original process is symmetric $β$-stable, while the limit is conditionally $β$-stable in case of integrals with respect to symmetric $β$-stable motions. These asymptotic results are mostly related to the work [5], which investigates the univariate version of the problem. Furthermore, we will show the implications for estimation of eigenvalues and eigenvectors of the quadratic variation matrix, which is a useful result for the principle component analysis. Finally, we propose a consistent subsampling procedure in the Lévy setting to obtain confidence regions.
In this paper, we study a class of Quadratic Backward Stochastic Differential Equations (QBSDE in short) with jumps and unbounded terminal condition. We extend the class of quadratic semimartingales introduced by Barrieu and El Karoui (2013) in the jump diffusion model. The properties of these class of semimartingales lead us to prove existence result for the solution of a quadratic BSDEs.
We prove that typical (in the model-free finance setting) price paths with jumps may be uniformly approximated with accuracy $c>0$ by paths whose total variation is of order $1/c.$ A more precise result is obtained for semimartingales with jumps.
A new integral with respect to an integer-valued random measure is introduced. In contrast to the finite variation integral ubiquitous in semimartingale theory (Jacod and Shiryaev, 2003, II.1.5), the new integral is closed under stochastic integration, composition, and smooth transformations. The new integral gives rise to a previously unstudied class of pure-jump processes: the sigma-locally finite variation pure-jump processes. As an application, it is shown that every semimartingale $X$ has a unique decomposition $$X = X_0 + X^{\mathrm{qc}}+X^{\mathrm{dp}},$$ where $X^{\mathrm{qc}}$ is quas…
The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard task, in particular when the underlying process contains jumps. For this reason, several authors have recently discussed methods to automatically estimate the conditional variance, i.e. they build a consistent estimator from the original statistics, but computed at various different time scales. Their methods work in several situations, but are essentially restr…
In this paper we show that if large jumps of an Itô-semimartingale $X$ have a finite $p$-moment, $p>0$, the radial part of its drift is dominated by $-|X|^κ$ for some $κ\geq -1$, and the balance condition $p+κ>1$ holds true, then under some further natural technical assumptions $\sup_{t\geq 0} \mathbf{E} |X_t|^{p_X}<\infty$ for each $p_X\in(0,p+κ-1)$. The upper bound $p+κ-1$ is generically optimal. The proof is based on the extension of the method of Lyapunov functions to the semimartingale framework. The uniform moment estimates obtained in this paper are indispensable for the analysis of erg…
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general càdlàg semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries of SDEs, the new concepts of gauge and time symmetries for semimartingales on Lie groups are introduced. Markovian and non-Markovian examples of gauge and time symmetric processes are provided. The considered set of SDEs includes affine and Marcus type SDEs as well as smooth SDEs driven by Lévy processes. Non trivial invariance results concerning a class of itera…
We study multivalued stochastic differential equations (MSDEs) with maximal monotone operators driven by semimartingales with jumps. We discuss in detail some methods of approximation of solutions of MSDEs based on discretization of processes and Yosida approximation of the monotone operator. We also study the general problem of stability of solutions of MSDEs with respect to the convergence of driving semimartingales.
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmet…
In this paper, we consider a diffusion process with jumps whose drift and jump coefficient depend on an unknown parameter. We then give a self-contained proof of the local asymptotic mixed normality (LAMN) property when the process is observed continuously in a time interval $[0; T]$ as $T\to\infty$, and derive, as a consequence, the local asymptotic normality (LAN) property in the ergodic case. For this, we give a proof of a Girsanov's theorem and a Central Limit theorem for a pure jump martingale. Our results could be viewed as a consequence of the LAMN property for semimartingales proved by…
This article investigates discrete-time approximations of stochastic integrals driven by semimartingales with jumps via weighted bounded mean oscillation (BMO) approach. This approach enables $L_p$-estimates, $p \in (2, \infty)$, for the approximation error depending on the weight, and it allows a change of the underlying measure which leaves the error estimates unchanged. To take advantage of this approach, we propose a new approximation scheme obtained from a correction for the Riemann approximation based on tracking jumps of the underlying semimartingale. We also discuss a way to optimize t…
A well-known Itô formula for finite dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite dimensional Itô formula for continuous semimartingales proved by Krylov to a class of $L_p$-valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.
We derive limit theorems for the empirical distribution function of "devolatilized" increments of an Itô semimartingale observed at high frequencies. These "devolatilized" increments are formed by suitably rescaling and truncating the raw increments to remove the effects of stochastic volatility and "large" jumps. We derive the limit of the empirical c.d.f. of the adjusted increments for any Itô semimartingale whose dominant component at high frequencies has activity index of $1<β\le2$, where $β=2$ corresponds to diffusion. We further derive an associated CLT in the jump-diffusion case. We use…
In this paper we examine the asymptotic theory for U-statistics and V-statistics of discontinuous Ito semimartingales that are observed at high frequency. For different types of kernel functions we show laws of large numbers and associated stable central limit theorems. In most of the cases the limiting process will be conditionally centered Gaussian. The structure of the kernel function determines whether the jump and/or the continuous part of the semimartingale contribute to the limit.
We compare two definitions of multistable Lévy motions. Such processes are extensions of classical Lévy motion where the stability index is allowed to vary in time. We show that the two multistable Lévy motions have distinct properties: in particular, one is a pure-jump Markov process, while the other one satisfies neither of these properties. We prove that both are semimartingales and provide semimartingale decompositions.
We provide verification theorems (at different levels of generality) for infinite horizon stochastic control problems in continuous time for semimartingales. The control framework is given as an abstract "martingale formulation", which encompasses a broad range of standard control problems. Under appropriate conditions we show that the set of admissible controls gives rise to a certain class of controlled special semimartingales. Our results generalise both the standard controlled Itô- and Lévy-diffusion settings as we allow ourselves to locally control not only the drift and diffusion coeffic…
This paper derives the asymptotic behavior of realized power variation of pure-jump Itô semimartingales as the sampling frequency within a fixed interval increases to infinity. We prove convergence in probability and an associated central limit theorem for the realized power variation as a function of its power. We apply the limit theorems to propose an efficient adaptive estimator for the activity of discretely-sampled Itô semimartingale over a fixed interval.
We consider the moderate deviations behaviors for two (co-) volatility estima-tors: generalised bipower variation, Hayashi-Yoshida estimator. The results are obtained by using a new result about the moderate deviations principle for m-dependent random variables based on the Chen-Ledoux type condition. In the last decade there has been a considerable development of the asymptotic theory for processes observed at a high frequency. This was mainly motivated by financial applications , where the data, such as stock prices or currencies, are observed very frequently. As under the no-arbitrage assum…
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 which is new also for semimartingales: 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…
In this paper, we study a Backward Stochastic Differential Equation with Jumps (BSDEJs in short) where the jumps have infinite activity. Following a forward approach based on Exponential Quadratic semimartingale, we prove the existence of solution of Quadratic BSDEJs with unbounded terminal condition and quadratic growth in z.
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.
In this paper we introduce the \textit{multivariate} Brownian semistationary (BSS) processes and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general stationary multivariate Gaussian processes, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.
A transformation relation between multivariate ARMA and CARMA processes is derived through a discretization procedure. This gives a direct relationship between the discrete time and continuous time analogues, serving as the basis for an estimation method for multivariate CARMA models. We will see that the autoregressive coefficients, making up the deterministic part of a multivariate CARMA model, are entirely given by the transformation relation. An Euler discretization convergence rate of jump diffusions is found for the case of small jumps of infinite variation. This substantiates applying t…
We develop and analyze a class of unbiased Monte Carlo estimators for multivariate jump-diffusion processes with state-dependent drift, volatility, jump intensity and jump size. A change of measure argument is used to extend existing unbiased estimators for the inter-arrival diffusion to include state-dependent jumps. Under standard regularity conditions on the coefficient and target functions, we prove the unbiasedness and finite variance properties of the resulting jump-diffusion estimators. Numerical experiments illustrate the efficiency of our estimators.
Motivated by entropic optimal transport, time reversal of Markov jump processes in $\mathbb{R}^n$ is investigated. Relying on an abstract integration by parts formula for the carré du champ of a Markov process recently obtained by Cattiaux, Gentil and the authors, and using an entropic improvement strategy discovered by Föllmer in the eighties, we compute the semimartingale characteristics of the time reversed process for a wide class of jump processes with possibly unbounded variation sample paths and singular intensities of jump.
We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance dominates its quadratic variation in increasing convex order. The results have immediate applications to the pricing of options on realized variance. For a class of models including time-changed Levy models and Sato processes with symmetric jumps our results show that options on…
This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.
Stricker's theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see [1983, Z. Wahrsch. Verw. Gebiete 64(3)]. We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker's theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generat…
We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the limit process. This holds for both diffusion and deterministic limit processes. The almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved. The particular case of a multivariate Stein pro…
The paper discusses multivariate self- and cross-exciting processes. We define a class of multivariate point processes via their corresponding stochastic intensity processes that are driven by stochastic jumps. Essentially, there is a jump in an intensity process whenever the corresponding point process records an event. An attribute of our modelling class is that not only a jump is recorded at each instance, but also its magnitude. This allows large jumps to influence the intensity to a larger degree than smaller jumps. We give conditions which guarantee that the process is stable, in the sen…
The dynamics of the eigenvalues (semimartingales) of a Lévy process $X$ with values in Hermitian matrices is described in terms of Itô stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian motion. The simultaneity of the jumps of the eigenvalues of $X$ is also studied. If $X$ has a jump at time $t$ two different situations are considered, depending on the commutativity of $X(t)$ and $X(t-)$. In the commutative case all the eigenvalues jump at time $t$ only when the jump of $X$ is of full rank. In the noncommutative case, $X$ jumps at time $t$ if and only …
For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the Stein--Chen method, approximation can often be achieved with error bounds of the same order as those for the CLT. In this paper, an analogous theory, again based on Stein's method, is developed in the multivariate context. The approximating family consists of the equilibrium distributions of a collection of Markov jump processes, whose analogues in one dimens…
We consider a process $X_t$, which is observed on a finite time interval $[0,T]$, at discrete times $0,Δ_n,2Δ_n,\ldots.$ This process is an Itô semimartingale with stochastic volatility $σ_t^2$. Assuming that $X$ has jumps on $[0,T]$, we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of $X_t$. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh $Δ_n$ goes to $0$. We show on some simulations that these tests perform reasonably …
We consider a mean-reverting stochastic volatility model which satisfies some relevant stylized facts of financial markets. We introduce an algorithm for the detection of peaks in the volatility profile, that we apply to the time series of Dow Jones Industrial Average and Financial Times Stock Exchange 100 in the period 1984-2013. Based on empirical results, we propose a bivariate version of the model, for which we find an explicit expression for the decay over time of cross-asset correlations between absolute returns. We compare our theoretical predictions with empirical estimates on the same…
For any real-valued stochastic process $X$ with cŕdlŕg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on compacts. The application of the defined class is the definition of stochastic integral with semimartingale integrand and integrator as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term (different from the Stratonovich integral). We compare the obtained result with classical results of Wong-…
This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and non-semimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multi-dimensional Riccati-Volterra equations. This framew…
The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process $(X,D)$ of a diffusion state variable $X$ driving default intensity and a default indicator process $D$ and time change it with a Lévy subordinator ${\mathcal{T}}$. We characterize the time-changed process $(X^ϕ_t,D^ϕ_t)=(X({\mathcal{T}}_t),D({\mathcal{T}}_t))$ as a Markovian--Itô semimartingale and show from the Doob--Meyer decomposition of $D^ϕ$ that the default time in the time-changed model has a j…
This paper considers population processes in which general, not necessarily Markovian, multivariate Hawkes processes dictate the stochastic arrivals. We establish results to determine the corresponding time-dependent joint probability distribution, allowing for general intensity decay functions, general intensity jumps, and general sojourn times. We obtain an exact, full characterization of the time-dependent joint transform of the multivariate population process and its underlying intensity process in terms of a fixed-point representation and corresponding convergence results. We also derive …
Quadratic Hawkes (QHawkes) processes have proved effective at reproducing the statistics of price changes, capturing many of the stylised facts of financial markets. Motivated by the recently reported strong occurrence of endogenous co-jumps (simultaneous price jumps of several assets) we extend QHawkes to a multivariate framework (MQHawkes), that is considering several financial assets and their interactions. Assuming that quadratic kernels write as the sum of a time-diagonal component and a rank one (trend) contribution, we investigate endogeneity ratios and the resulting stationarity condit…
Our motivation comes from the large population approximation of individual based models in population dynamics and population genetics. We propose a general method to investigate scaling limits of finite dimensional population size Markov chains to diffusion with jumps. The statements of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the transition of the chain and strongly exploit the structure of the population processes defined recursively as sums of independent random variables. These results allow to reduce the convergence of c…
The paper deals with disorders detection in the multivariate stochastic process. We consider the multidimensional Poisson process or the multivariate renewal process. This class of processes can be used as a description of the distributed detection system. The multivariate renewal process can be seen as the sequence of random vectors, where parts of its coordinates are holding times, others are the size of jumps and the index of stream, at which the new event appears. It is assumed that at each stream two kinds of changes are possible: in the holding time or in the size of jumps distribution. …
Let $X=\{X_t: t\ge 0\}$ be a stationary piecewise continuous $\R^d$-valued process that moves between jumps along the integral curves of a given continuous vector field, and let $S\subset\R^d$ be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of $S$ by $X$ to the distribution of $X_0$. Our result is illustrated by examples relating to queueing networks and stress release network models.
