Showing papers on "Semimartingale published in 2017"

Remaining Useful Life Prediction for Degradation Processes With Long-Range Dependence

Abstract: A prerequisite for the existing remaining useful life prediction methods based on stochastic processes is the assumption of independent increments. However, this is in sharp contrast to some practical systems including batteries and blast furnace walls, in which the degradation processes have the property of long-range dependence. Based on the fractional Brownian motion, we adopt a degradation process with long-range dependence to predict the remaining useful life of the above systems. Because the degradation process with long-range dependence is neither a Markovian process nor a semimartingale, the exact analytical first passage time is difficult to derive directly. To address this problem, a weak convergence theorem is first adopted to approximately transform a fractional Brownian motion-based degradation process into a Brownian motion-based one with a time-varying coefficient. Then, with a space-time transformation, the first passage time of the degradation process with long-range dependence can be obtained in a closed form. Unknown parameters in the degradation model can be identified using discrete dyadic wavelet transform and maximum likelihood estimation. Numerical simulations and a practical example of a blast furnace wall are given to verify the effectiveness of the proposed method.

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.

Canonical RDEs and general semimartingales as rough paths

Abstract: In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lepingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.

When Moving-Average Models Meet High-Frequency Data: Uniform Inference on Volatility

Abstract: We propose uniformly valid inference on volatility with noisy high-frequency data. We assume the observed transaction price follows a continuous-time Ito-semimartingale, contaminated by a discrete-time moving-average noise process associated with the arrival of trades. We estimate the quadratic variation of the semimartingale by maximizing the likelihood of a misspecified moving-average model, with its order selected based on the information criteria. Our inference is uniformly valid over a large class of noise processes whose magnitude and dependence structure vary with sample size. Our implementation is tuning free barring order selection, and it yields positive estimates in finite samples. Finally, we provide consistent estimators of noise autocovariances as byproducts, which also play a critical role in achieving uniformity.

Stochastic Minimum Principle for Partially Observed Systems Subject to Continuous and Jump Diffusion Processes and Driven by Relaxed Controls

Abstract: In this paper, we consider nonconvex control problems of stochastic differential equations driven by relaxed controls adapted, in the weak star sense, to a current of sigma algebras generated by observable processes. We cover in a unified way both continuous diffusion and jump processes. We present existence of optimal controls before we construct the necessary conditions of optimality (unlike some papers in this area) using only functional analysis. We develop a stochastic Hamiltonian system of equations on a rigorous basis using the semimartingale representation theory and the Riesz representation theorem, leading naturally to the existence of the adjoint process which satisfies a backward stochastic differential equation. In other words, our approach predicts the existence of the adjoint process as a natural consequence of Riesz representation theory ensuring at the same time the (weak star) measurability. This is unlike other papers, where the adjoint process is introduced before its existence is prov...

Asymptotic Theory of Transaction Costs

Abstract: The present lecture notes are based on several advanced courses which I gave at the University of Vienna between 2011 and 2013. In 2015 I gave a similar course (“Nachdiplom-Vorlesung”) at ETH Zürich. The purpose of these lectures was to present and organize the recent progress on portfolio optimization under proportional transaction costs λ ą 0. Special emphasis is given to the asymptotic behaviour when λ tends to zero. The theme of portfolio optimization is a classical topic of Mathematical Finance, going back to the seminal work of Robert Merton in the early seventies (considering the frictionless case without transaction costs). Mathematically speaking, this question leads to a concave optimization problem under linear constraints. A technical challenge arises from the fact that – except for the case of finite probability spaces Ω – the optimization takes place over infinite–dimensional sets. There are essentially two ways of attacking such an optimization problem. The primal method consists in directly addressing the problem at hand. Following the path initiated by Robert Merton, this leads to a partial differential equation of Hamilton–Jacobi–Bellman type. This PDE method can also be successfully extended to the case of proportional transaction costs. Important work on this line was done by G. Constantinides [42], B. Dumas and E. Luciano [77], M. Taksar, M. J. Klass, D. Assaf [240], M. Davis and A. Norman [59], St. Shreve and M. Soner [230], just to name some of the early work on this topic.

Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion

Abstract: While absence of arbitrage in frictionless nancial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an It^ o process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an It^ o process in a one-sided manner without reection.

Stochastic integration with respect to cylindrical Lévy processes

Abstract: A cylindrical Levy process does not enjoy a cylindrical version of the semimartingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Levy processes in Hilbert spaces. The space of admissible integrands consists of caglad, adapted stochastic processes with values in the space of Hilbert–Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semimartingale with cadlag trajectories.

BSE’s, BSDE’s and fixed-point problems

Abstract: In this paper, we introduce a class of backward stochastic equations (BSEs) that extend classical BSDEs and include many interesting examples of generalized BSDEs as well as semimartingale backward equations. We show that a BSE can be translated into a fixed-point problem in a space of random vectors. This makes it possible to employ general fixed-point arguments to establish the existence of a solution. For instance, Banach’s contraction mapping theorem can be used to derive general existence and uniqueness results for equations with Lipschitz coefficients, whereas Schauder-type fixed-point arguments can be applied to non-Lipschitz equations. The approach works equally well for multidimensional as for one-dimensional equations and leads to results in several interesting cases such as equations with path-dependent coefficients, anticipating equations, McKean–Vlasov-type equations and equations with coefficients of superlinear growth.

Polynomial Jump-Diffusion Models

Abstract: We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is preserved under polynomial transformations and Levy time change. We present a generic method for option pricing based on moment expansions. As an application, we introduce a large class of novel financial asset pricing models with excess log returns that are conditional Levy based on polynomial jump-diffusions.

Almost sure stability of the Euler---Maruyama method with random variable stepsize for stochastic differential equations

Abstract: In this paper, the Euler---Maruyama (EM) method with random variable stepsize is studied to reproduce the almost sure stability of the true solutions of stochastic differential equations. Since the choice of the time step is based on the current state of the solution, the time variable is proved to be a stopping time. Then the semimartingale convergence theory is employed to obtain the almost sure stability of the random variable stepsize EM solution. To our best knowledge, this is the first paper to apply the random variable stepsize (with clear proof of the stopping time) to the analysis of the almost sure stability of the EM method.

Portfolio optimisation beyond semimartingales: shadowprices and fractional Brownian motion

Abstract: While absence of arbitrage in frictionlessfinancial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an It^o process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an It^o process in a one-sided manner without reflection.

Robust Jump Regressions

Abstract: 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 (like L1) 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 ...

Optimal investment with intermediate consumption under no unbounded profit with bounded risk

Abstract: We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions. Copyright © 2017 Applied Probability Trust.

Shot-Noise Processes in Finance

Abstract: Shot-Noise processes constitute a useful tool in various areas, in particular in finance. They allow to model abrupt changes in a more flexible way than processes with jumps and hence are an ideal tool for modelling stock prices, credit portfolio risk, systemic risk, or electricity markets. Here we consider a general formulation of shot-noise processes, in particular time-inhomogeneous shot-noise processes. This flexible class allows to obtain the Fourier transforms in explicit form and is highly tractable. We prove that Markovianity is equivalent to exponential decay of the noise function. Moreover, we study the relation to semimartingales and equivalent measure changes which are essential for the financial application. In particular we derive a drift condition which guarantees absence of arbitrage. Examples include the minimal martingale measure and the Esscher measure.

Explicit Description of HARA Forward Utilities and Their Optimal Portfolios

Abstract: This paper deals with forward performances of HARA type. Precisely, for a market model in which stock price processes are modeled by a locally bounded $d$-dimensional semimartingale, we elaborate a complete and explicit characterization for this type of forward utilities. In particular, under some mild technical assumptions of integrability, we prove that the risk aversion process of a power-type forward utility is constant. Furthermore, the optimal portfolios for all HARA forward utilities are explicitly described. Our approach is based on the minimal Hellinger martingale densities that are obtained from the important statistical concept of Hellinger process. These martingale densities were introduced recently and appeared herein tailor-made for these forward utilities. After outlining our parametrization method for the HARA forward, we provide illustrations on discrete-time market models.

Testing for time-varying jump activity for pure jump semimartingales

Abstract: In this paper, we propose a test for deciding whether the jump activity index of a discretely observed Ito semimartingale of pure-jump type (i.e., one without a diffusion) varies over a fixed interval of time. The asymptotic setting is based on observations within a fixed time interval with mesh of the observation grid shrinking to zero. The test is derived for semimartingales whose “spot” jump compensator around zero is like that of a stable process, but importantly the stability index can vary over the time interval. The test is based on forming a sequence of local estimators of the jump activity over blocks of shrinking time span and contrasting their variability around a global activity estimator based on the whole data set. The local and global jump activity estimates are constructed from the real part of the empirical characteristic function of the increments of the process scaled by local power variations. We derive the asymptotic distribution of the test statistic under the null hypothesis of constant jump activity and show that the test has asymptotic power of one against fixed alternatives of processes with time-varying jump activity.

Continuous time analysis of fleeting discrete price moves

Abstract: This article proposes a novel model of financial prices where (i) prices are discrete; (ii) prices change in continuous time; (iii) a high proportion of price changes are reversed in a fraction of a second. Our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve. The resulting cadlag price process is a piecewise constant semimartingale with finite activity, finite variation, and no Brownian motion component. We use moment-based estimations to fit four high-frequency futures datasets and demonstrate the descriptive power of our proposed model. This model is able to describe the observed dynamics of price changes over three different orders of magnitude of time intervals. Supplementary materials for this article are available online.

Nonparametric tests for detecting breaks in the jump behaviour of a time-continuous process

Abstract: This paper is concerned with tests for changes in the jump behaviour of a time-continuous process. Based on results on weak convergence of a sequential empirical tail integral process, asymptotics of certain tests statistics for breaks in the jump measure of an It o semimartingale are constructed. Whenever limiting distributions depend in a complicated way on the unknown jump measure, empirical quantiles are obtained using a multiplier bootstrap scheme. An extensive simulation study shows a good performance of our tests in nite samples.

Some existence results for advanced backward stochastic differential equations with a jump time

Abstract: In this paper, we are interested by advanced backward stochastic differential equations (ABSDE), in a probability space equipped with a Brownian motion and a single jump process. The solution of the ABSDE is a triple (Y, Z, U) where Y is a semimartingale, Z is the diffusion coefficient and U the size of the jump. We allow the generator to depend on the future paths of the solution.

A note on central limit theorems for quadratic variation in case of endogenous observation times

Abstract: This paper is concerned with a central limit theorem for quadratic variation when observations come as exit times from a regular grid. We discuss the special case of a semimartingale with deterministic characteristics and finite activity jumps in detail and illustrate technical issues in more general situations.

On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries

Abstract: Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection (SDER) and the so-called submartingale problem. We consider a general formulation of the submartingale problem for (obliquely) reflected diffusions in domains with piecewise C2C2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding SDER. The main step involves showing existence of a weak solution to the SDER given a solution to the submartingale problem. This generalizes the classical construction, due to Stroock and Varadhan, of a weak solution to an (unconstrained) stochastic differential equation, but requires a completely different approach to deal with the geometry of the domain and directions of reflection and properly identify the local time on the boundary, in the presence of multi-valued directions of reflection at nonsmooth parts of the boundary. In particular, our proof entails the construction of classes of test functions that satisfy certain oblique derivative boundary conditions, which may be of independent interest. Other ingredients of the proof that are used to identify the constraining or local time process include certain random linear functionals, suitably constructed exponential martingales and a dual representation of the cone of directions of reflection. As a corollary of our result, under suitable assumptions, we also establish an equivalence between well-posedness of both the SDER and submartingale formulations and well-posedness of the constrained martingale problem, which is another framework for defining (semimartingale) reflected diffusions. Many of our intermediate results are also valid for reflected diffusions that are not necessarily semimartingales, and are used in a companion paper [Equivalence of stochastic equations and the submartingale problem for nonsemimartingale reflected diffusions. Preprint] to extend the equivalence result to a class of nonsemimartingale reflected diffusions.

Quasimartingales associated to Markov processes

Abstract: 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 dif- ference 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.

Stochastic invariance of closed sets for jump-diffusions with non-Lipschitz coefficients

Abstract: We provide necessary and sufficient first order geometric conditions for the stochastic invariance of a closed subset of $\mathbb{R} ^d$ with respect to a jump-diffusion under weak regularity assumptions on the coefficients. Our main result extends the recent characterization proved in Abi Jaber, Bouchard and Illand (2016) to jump-diffusions. We also derive an equivalent formulation in the semimartingale framework.

A weak law of large numbers for estimating the correlation in bivariate Brownian semistationary processes

Abstract: This article presents various weak laws of large numbers for the so-called realised covariation of a bivariate stationary stochastic process which is not a semimartingale. More precisely, we consider two cases: Bivariate moving average processes with stochastic correlation and bivariate Brownian semistationary processes with stochastic correlation. In both cases, we can show that the (possibly scaled) realised covariation converges to the integrated (possibly volatility modulated) stochastic correlation process.