Showing papers on "Semimartingale published in 2016"

Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems

Abstract: Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.

A general HJM framework for multiple yield curve modelling

Abstract: We propose a general framework for modelling multiple yield curves which have emerged after the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and, in addition, we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor’s length. When the driving semimartingale is an affine process, we obtain a flexible and tractable Markovian structure. Finally, we show that the proposed framework allows unifying and extending several recent approaches to multiple yield curve modelling.

Generalized Method of Integrated Moments for High‐Frequency Data

Abstract: We propose a semiparametric two-step inference procedure for a finite-dimensional parameter based on moment conditions constructed from high-frequency data. The population moment conditions take the form of temporally integrated functionals of state-variable processes that include the latent stochastic volatility process of an asset. In the first step, we nonparametrically recover the volatility path from high-frequency asset returns. The nonparametric volatility estimator is then used to form sample moment functions in the second-step GMM estimation, which requires the correction of a high-order nonlinearity bias from the first step. We show that the proposed estimator is consistent and asymptotically mixed Gaussian and propose a consistent estimator for the conditional asymptotic variance. We also construct a Bierens-type consistent specification test. These infill asymptotic results are based on a novel empirical-process-type theory for general integrated functionals of noisy semimartingale processes.

Modelling electricity prices: a time change approach

Abstract: To capture mean reversion and sharp seasonal spikes observed in electricity prices, this paper develops a new stochastic model for electricity spot prices by time changing the Jump Cox-Ingersoll-Ross (JCIR) process with a random clock that is a composite of a Gamma subordinator and a deterministic clock with seasonal activity rate. The time-changed JCIR process is a time-inhomogeneous Markov semimartingale which can be either a jump-diffusion or a pure-jump process, and it has a mean-reverting jump component that leads to mean reversion in the prices in addition to the smooth mean-reversion force. Furthermore, the characteristics of the time-changed JCIR process are seasonal, allowing spikes to occur in a seasonal pattern. The Laplace transform of the time-changed JCIR process can be efficiently computed by Gauss–Laguerre quadrature. This allows us to recover its transition density through efficient Laplace inversion and to calibrate our model using maximum likelihood estimation. To price electricity deri...

Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing

Abstract: This paper develops a spectral theory of Markovian asset pricing models where the underlying economic uncertainty follows a continuous-time Markov process X with a general state space (Borel right process, or BRP) and the stochastic discount factor (SDF) is a positive semimartingale multiplicative functional of X. A key result is the uniqueness theorem for a positive eigenfunction of the pricing operator such that X is recurrent under a new probability measure associated with this eigenfunction (recurrent eigenfunction). As economic applications, we prove uniqueness of the Hansen and Scheinkman factorization of the Markovian SDF corresponding to the recurrent eigenfunction; extend the Recovery Theorem from discrete time, finite state irreducible Markov chains to recurrent BRPs; and obtain the long-maturity asymptotics of the pricing operator. When an asset pricing model is specified by given risk-neutral probabilities together with a short rate function of the Markovian state, we give sufficient condition...

Long Term Risk: A Martingale Approach

Abstract: This paper extends the long-term factorization of the stochastic discount factor introduced and studied by Alvarez and Jermann (2005) in discrete-time ergodic environments and by Hansen and Scheinkman (2009) and Hansen (2012) in Markovian environments to general semimartingale environments. The transitory component discounts at the stochastic rate of return on the long bond and is factorized into discounting at the long-term yield and a positive semimartingale that extends the principal eigenfunction of Hansen and Scheinkman (2009) to the semimartingale setting. The permanent component is a martingale that accomplishes a change of probabilities to the long forward measure, the limit of T-forward measures. The change of probabilities from the data generating to the long forward measure absorbs the long-term risk-return trade-off and interprets the latter as the long-term risk-neutral measure.

Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

Abstract: We continue the analysis of our previous paper (Czichowsky/Schachermayer/Yang 2014) pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process $S=(S_t)_{0\leq t\leq T}$ satisfying the condition $(NUPBR)$ of "no unbounded profit with bounded risk". This condition requires that $S$ is a semimartingale and therefore is too restrictive for applications to models driven by fractional Brownian motion. In the present paper, we derive the same conclusion under the weaker condition $(TWC)$ of "two way crossing", which does not require $S$ to be a semimartingale. Using a recent result of R.~Peyre, this allows us to show the existence of a shadow price for exponential fractional Brownian motion and $all$ utility functions defined on the positive half-line having reasonable asymptotic elasticity. Prime examples of such utilities are logarithmic or power utility.

On infinitely divisible semimartingales

Abstract: 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 Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). 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 generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosinski (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of cadlag infinitely divisible processes given in Basse-O’Connor and Rosinski (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.

Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization

Abstract: The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping. Our results have natural extensions to the case of general multidimensional continuous semimartingales.

Quadratic covariation estimation of an irregularly observed semimartingale with jumps and noise

Abstract: This paper presents a central limit theorem for a pre-averaged version of the realized covariance estimator for the quadratic covariation of a discretely observed semimartingale with noise. The semimartingale possibly has jumps, while the observation times show irregularity, non-synchronicity, and some dependence on the observed process. It is shown that the observation times’ effect on the asymptotic distribution of the estimator is only through two characteristics: the observation frequency and the covariance structure of the noise. This is completely different from the case of the realized covariance in a pure semimartingale setting.

The functional Itō formula under the family of continuous semimartingale measures

Abstract: Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

Weak convergence to stochastic integrals for econometric applications

Abstract: Limit theory involving stochastic integrals is now widespread in time series econometrics and relies on a few key results on functional weak convergence. In establishing such convergence, the literature commonly uses martingale and semimartingale structures. While these structures have wide relevance, many applications involve a cointegration framework where endogeneity and nonlinearity play major roles and complicate the limit theory. This paper explores weak convergence limit theory to stochastic integral functionals in such settings. We use a novel decomposition of sample covariances of functions of I (1) and I (0) time series that simplifies the asymptotics and our limit results for such covariances hold for linear process, long memory, and mixing variates in the innovations. These results extend earlier findings in the literature, are relevant in many applications, and involve simple conditions that facilitate practical implementation. A nonlinear extension of FM regression is used to illustrate practical application of the methods.

Optimal insider control and semimartingale decompositions under enlargement of filtration

Abstract: We combine stochastic control methods, white noise analysis, and Hida–Malliavin calculus applied to the Donsker delta functional to obtain explicit representations of semimartingale decompositions under enlargement of filtrations. Some of the expressions are more explicit than previously known. The results are illustrated by examples.

Estimating the volatility occupation time via regularized laplace inversion

Abstract: We propose a consistent functional estimator for the occupation time of the spot variance of an asset price observed at discrete times on a finite interval with the mesh of the observation grid shrinking to zero. The asset price is modeled nonparametrically as a continuous-time Ito semimartingale with nonvanishing diffusion coefficient. The estimation procedure contains two steps. In the first step we estimate the Laplace transform of the volatility occupation time and, in the second step, we conduct a regularized Laplace inversion. Monte Carlo evidence suggests that the proposed estimator has good small-sample performance and in particular it is far better at estimating lower volatility quantiles and the volatility median than a direct estimator formed from the empirical cumulative distribution function of local spot volatility estimates. An empirical application shows the use of the developed techniques for nonparametric analysis of variation of volatility.

Sentiment Lost: The Effect of Projecting the Empirical Pricing Kernel Onto a Smaller Filtration Set

Abstract: Supported by empirical examples, this paper provides a theoretical analysis to show what is the impact of an improper calibration of the physical measure on the estimation of the empirical pricing kernel. While extracting the risk-neutral measure from option data provides a naturally forward looking measure, extracting the real world probability from a stream of historical returns is only partially informative, thus suboptimal with respect to investors’ future beliefs. In virtue of this disalignment, most of papers present in literature are then affected by a non-homogeneity bias.From a probabilistic viewpoints, the missing beliefs are totally unaccessible stopping times on the coarser filtration set. As a consequence, an absolutely continuous local or strict local martingale, once projected on it, becomes continuous with jumps.As a result of a non fully informative physical measure, the proposed empirical pricing kernel is no longer a true martingale, as required by the classical theory, but a strict local martingale with consequences on the probabilistic nature of the relative risk neutral measure.Finally we show how the implied options’ moments help in reducing the degree of inaccessibility and shorten the distance between what is theoretically required and empirically accessible.

Exponentially concave functions and high dimensional stochastic portfolio theory

Abstract: We consider the following problem in stochastic portfolio theory. Are there portfolios that are relative arbitrages with respect to the market portfolio over very short periods of time under realistic assumptions? We answer a slightly relaxed question affirmative in the following high dimensional sense, where dimension refers to the number of stocks being traded. Very roughly, suppose that for every dimension we have a continuous semimartingale market such that (i) the vector of market weights in decreasing order has a stationary regularly varying tail with an index between $-1$ and $-1/2$ and (ii) zero is not a limit point of the relative volatilities of the stocks. Then, given a probability $\eta 0$, and an arbitrarily high positive amount $M$, for all high enough dimensions, it is possible to construct a functionally generated portfolio such that, with probability at least $\eta$, its relative value with respect to the market at time $\delta$ is at least $M$, and never goes below $(1-\epsilon)$ during $[0, \delta]$. There are two phase transitions; if the index of the tail is less than $-1$ or larger than $-1/2$. The construction uses properties of regular variation, high-dimensional convex geometry and concentration of measure under Dirichlet distributions. We crucially use the notion of $(K,N)$ convex functions introduced by Erbar, Kuwada, Sturm in the context of curvature-dimension conditions and Bochner's inequalities.

Compactness Criterion for Semimartingale Laws and Semimartingale Optimal Transport

Abstract: We provide a compactness criterion for the set of laws $\mathfrak{P}^{ac}_{sem}(\Theta)$ on the Skorokhod space for which the canonical process $X$ is a semimartingale having absolutely continuous characteristics with differential characteristics taking values in some given set $\Theta$ of Levy triplets. Whereas boundedness of $\Theta$ implies tightness of $\mathfrak{P}^{ac}_{sem}(\Theta)$, closedness fails in general, even when choosing $\Theta$ to be additionally closed and convex, as a sequence of purely discontinuous martingales may converge to a diffusion. To that end, we provide a necessary and sufficient condition that prevents the purely discontinuous martingale part in the canonical representation of $X$ to create a diffusion part in the limit. As a result, we obtain a sufficient criterion for $\mathfrak{P}^{ac}_{sem}(\Theta)$ to be compact, which turns out to be also a necessary one if the geometry of $\Theta$ is similar to a box on the product space. As an application, we consider a semimartingale optimal transport problem, where the transport plans are elements of $\mathfrak{P}^{ac}_{sem}(\Theta)$. We prove the existence of an optimal transport law $\widehat{\mathbb{P}}$ and obtain a duality result extending the classical Kantorovich duality to this setup.

Mixed-Sub Fractional Brownian Motion

Abstract: A new extension of the sub-fractional Brownian motion, and thus of the Brownian motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian motions, that we have chosen to call the mixed sub-fractional Brownian motion. In this paper, we study some basic properties of this process, its non-Markovian and non-stationarity characteristics, the conditions under which it is a semimartingale, and the main features of its sample paths. We also show that this process could serve to get a good model of certain phenomena, taking not only the sign (like in the case of the sub-fractional Brownian motion), but also the strength of dependence between the increments of this phenomena into account.

On Natural and Predictable Processes

Abstract: We give short, nontechnical proofs of the equivalence of predictability and naturality of stochastic processes of integrable variation, as well as of the properties that any natural martingale of integrable variation is indistinguishable from zero, any local predictable martingale of locally integrable variation is constant a.s., any local martingale is predictable if and only if it is continuous a.s. These statements are well known, but their proofs are very complicated since they depend on deep results from the advanced theory of stochastic integration and local martingales. Our proofs use only basic concepts of stochastic calculus.

On an inversion theorem for Stratonovich's signatures of multidimensional diffusion paths

Abstract: Dans ce papier, nous prouvons qu’avec probabilite egale a 1, les trajectoires d’un processus de diffusion multi-dimensionnel (eventuellement degenere) sur $[0,1]$ sont determinees par ses signatures de Stratonovich, i.e. par la famille de toutes les integrales iterees de Stratonovich du processus.

Arbitrage without borrowing or short selling

Abstract: We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann-Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.

Martingale Property in Terms of Semimartingale Problems

Abstract: Starting from the seventies mathematicians face the question whether a non-negative local martingale is a true or a strict local martingale. In this article we answer this question from a semimartingale perspective. We connect the martingale property to existence, uniqueness and topological properties of semimartingale problems. This not only leads to valuable characterizations of the martingale property, but also reveals new existence and uniqueness results for semimartingale problems. As a case study we derive explicit conditions for the martingale property of stochastic exponentials driven by infinite-dimensional Brownian motion.