Hongwei Long

Nadaraya-Watson estimator for stochastic processes driven by stable Lévy motions

We discuss the nonparametric Nadaraya-Watson (N-W) estimator of the drift function for ergodic stochastic processes driven by α-stable noises and observed at discrete instants. Under geometrical mixing condition, we derive consistency and rate of convergence of the N-W estimator of the drift function. Furthermore, we obtain a central limit theorem for stable stochastic integrals. The central limit theorem has its own interest and is the crucial tool for the proofs. A simulation study illustrates the finite sample properties of the N-W estimator. AMS 2000 subject classification. 60G52, 62G20, 62M05, 65C30

Least squares estimators for discretely observed stochastic processes driven by small Lévy noises

We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Levy noises. We do not impose any moment condition on the driving Levy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient @e->0 and n->~ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to the jump part of the Levy process. Moreover, we briefly remark that our methodology can be easily extended to the more general case of semi-martingale noises.

A jump model for fads in asset prices under asymmetric information

This paper addresses how asymmetric information, fads and Levy jumps in the price of an asset affect the optimal portfolio strategies and maximum expected utilities of two distinct classes of rational investors in a financial market. We obtain the investors’ optimal portfolios and maximum expected logarithmic utilities and show that the optimal portfolio of each investor is more or less than its Merton optimal. Our approximation results suggest that jumps reduce the excess asymptotic utility of the informed investor relative to that of uninformed investor, and hence jump risk could be helpful for market efficiency as an indirect reducer of information asymmetry. Our study also suggests that investors should pay more attention to the overall variance of the asset pricing process when jumps exist in fads models. Moreover, if there are very little or too much fads, then the informed investor has no utility advantage in the long run.

A hybrid weighted interacting particle filter for multi-target tracking

A hybrid weighted interacting particle filter, the selectively resampling particle filter (SERP), is used to detect and track multiple ships maneuvering in a region of water. The ship trajectories exhibit nonlinear dynamics and interact in a nonlinear manner such that the ships do not collide. There is no prior knowledge on the number of ships in the region. The observations model a sensor tracking the ships from above the region, as in a low observable SAR or infrared problem. The SERP filter simulates particles to provide the approximated conditional distribution of the signal in the signal domain at a particular time, given the sequence of observations. After each observation, the hybrid filter uses selective resampling to move some particles with low weights to locations that have a higher likelihood of being correct, without resampling all particles or creating bias. Such a method is both easy to implement and highly computationally efficient. Quantitative results recording the capacity of the filter to determine the number of ships in the region and the location of each ship are presented. Thy hybrid filter is compared against an earlier particle filtering method.

A discontinuous mispricing model under asymmetric information

We study a discontinuous mispricing model of a risky asset under asymmetric information where jumps in the asset price and mispricing are modelled by Levy processes. By contracting the filtration of the informed investor, we obtain optimal portfolios and maximum expected utilities for the informed and uninformed investors. We also discuss their asymptotic properties, which can be estimated using the instantaneous centralized moments of return. We find that optimal and asymptotic utilities are increased due to jumps in mispricing for the uninformed investor but the informed investor still has excess utility, provided there is not too little or too much mispricing.

Impulse control with random reaction periods: A central bank intervention problem

a b s t r a c t We model an impulse control problem when the controllers action affects the state as well as the dynamics of the state process for a random amount of time. We apply our model to solve a central bank intervention problem in the foreign exchange market when the market observes and reacts to the banks interventions.

A stochastic grid filter for multi-target tracking

In this paper, we discuss multi-target tracking for a submarine model based on incomplete observations. The submarine model is a weakly interacting stochastic dynamic system with several submarines in the underlying region. Observations are obtained at discrete times from a number of sonobuoys equipped with hydrophones and consist of a nonlinear function of the current locations of submarines corrupted by additive noise. We use filtering methods to find the best estimation for the locations of the submarines. Our signal is a measure-valued process, resulting in filtering equations that can not be readily implemented. We develop Markov chain approximation approach to solve the filtering equation for our model. Our Markov chains are constructed by dividing the multi-target state space into cells, evolving particles in these cells, and employing a random time change approach. These approximations converge to the unnormalized conditional distribution of the signal process based on the back observations. Finally we present some simulation results by using the refining stochastic grid (REST) filter (developed from our Markov chain approximation method).

Kolmogorov equations in Hilbert spaces with application to essential self-adjointness of symmetric diffusion operators

The essential self-adjointness of differential operators over infinite dimensional spaces has been extensively studied. For historical comments and literature on this topic see the monograph by Berezanskii [3] and a recent paper by Albeverio, Kondratiev and Rόckner [2]. For some generalizations to certain Banach spaces we refer to Long [13]. Sometimes essential self-adjointness is also called strong uniqueness. There is another kind of uniqueness for symmetric diffusion operators, i.e. Markovian uniqueness, which means that one has uniqueness only within the class of selfadjoint operators which generate sub-Markovian semigroups. Obviously essential selfadjointness implies Markovian uniqueness. For details we refer to [2] and some references therein. In this paper, we aim to prove the essential self-adjointness of a certain class of perturbed Ornstein-Uhlenbeck operators associated to stochastic evolution equations (SEE) in a separable Hubert space, by using a general parabolic criterion of Berezanskii [3]. In [4], Berezanskii and Samoilenko established the essential selfadjointness of Ornstein-Uhlenbeck operators with a certain potential perturbation by using the finite-dimensional approximation approach. In [17], Shigekawa proved the essential self-adjointness of perturbed Ornstein-Uhlenbeck operators by using Malliavin calculus. Our method is completely different from Shigekawas. For our purpose, we need first to establish the existence and uniqueness of classical solutions to the Kolmogorov equations associated to the perturbed Ornstein-Uhlenbeck operators. The definition of classical solution will be given in Section 2 , following Cannarsa and Da Prato [5]. In [7], Da Prato proved the existence and regularity of classical solutions to Kolmogorov equations associated to Ornstein-Uhlenbeck operators. We consider the semilinear SEE :

Market-reaction-adjusted optimal central bank intervention policy in a forex market with jumps

Impulse control with random reaction periods (ICRRP) is used to derive a country’s optimal foreign exchange (forex) rate intervention policy when the forex market reacts to the interventions. This paper extends the previous work on ICRRP by incorporating a multi-dimensional jump diffusion process to model the state dynamics, and hence, enhance the viability of the extant model for applications. Furthermore, we employ a novel minimum cost operator that simplifies the computations of the optimal solutions. Finally, we demonstrate the efficacy of our framework by finding a market-reaction-adjusted optimal central bank intervention (CBI) policy for a country. Our numerical results suggests that market reactions and the jumps in the forex market are complements when the reactions increase the forex rate volatility; otherwise, they are substitutes.

Discrete-space particle filters for reflecting diffusions

We consider the low observable filtering problem of detecting and tracking a target buried in high amplitude synthetic spatial observation noise. Motivated by fish farming applications, we constrain our target to live in a rectangular region, undergoing reflections at the boundary of this region, and moving in a manner described by the unique solution to a Skorohod stochastic differential equation. Observations are taken at discrete times and consist of a nonlinear partial function of the current state corrupted by additive noise. We use the reference probability method to describe the solution to this filtering problem in terms of a discrete-time version of the Duncan-Mortensen-Zakai equation and then use Markov chain approximations to produce an implementable approximate solution. The approximations incorporate discretizations of both space and amplitude directly into the unnormalized conditional distribution of the signal given the back observations. These approximations converge to the actual filtering conditional distribution as the discretization mesh is refined. The algorithm to implement our filter is reduced to an algorithm to implement a specific time-inhomogeneous Markov chain, which can be done using a single Poisson process and independent sequences of Bernoulli trials. The inhomogeneity is due to the observations themselves. The discretization of amplitude results in particles representing a small mass of the conditional distribution at particular grid points in the signal domain. These particles diffuse, drift, give birth, and die within the region similarly to those of continuous-state particle filters. The particles include information from the observations through observation-dependent births and deaths. We discuss issues like mean time to localize the target and fidelity of filter estimates at various signal to noise ratios, and give visual demonstrations of filter performance.