Stochastic evolution equations for nonlinear filtering of random fields in the presence of fractional Brownian sheet observation noise

The problem of nonlinear filtering of a random field observed in the presence of a noise, modeled by a persistent fractional Brownian sheet of Hurst index $(H_1,H_2)$ with $0.5<H_1,H_2<1$, is studied and a suitable version of the Bayes' formula for the optimal filter is obtained. Two types of spatial "fractional" analogues of the Duncan-Mortensen-Zakai equation are also derived: one tracks evolution of the unnormalized optimal filter along an arbitrary "monotone increasing" (in the sense of partial ordering in $\mathbb{R}^2$) one-dimensional curve in the plane, while the other describes dynamics of the filter along the paths that are truly two-dimensional. Although the paper deals with the two-dimensional parameter space, the presented approach and results extend to $d$-parameter random fields with arbitrary $d\geq 3$.Comment: 24 page

Interaction between stock indices via changepoint analysis

Stock market indices from several countries are modelled as discretely sampled diffusions whose parameters change at certain times. To estimate these times of parameter changes we employ both a sequential likelihood-ratio test and a non-parametric, spectral algorithm designed specifically for time series with multiple changepoints. Finally, we use point-process techniques to model relationships between changepoints of different financial time series. Copyright © 2006 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/55814/1/653_ftp.pd

Topics in Asset Pricing and Market Microstructure

This dissertation addresses various aspects of asset pricing theory in the following three contexts: the case of insider trading (of stocks) with uninformed biased traders, the case of trading of real options (specifically, of the option to sell a real indivisible asset), and the case of house pricing and construction of better house price indices. Chapter 1 examines the effects of insider trading on uninformed traders with bounded rationality in the context of a continuous-time Kyle-type model with a single perfectly informed risk-neutral agent (insider), a competitive risk-neutral market maker and a set of biased uninformed traders.Two cases of behavioral biases or bounded rationality on the part of the uninformed traders are considered. In the first case the uninformed traders' order flow has a non-zero covariation with a set of public signals (where positive covariation describes aggregate momentum strategies among the uninformed investors in reaction to news, while negative covariation indicates that the uninformed traders are predominantly contrarians). In the second case, the order flow from the uninformed traders has a strictly positive or a strictly negative covariance between its increments and is no longer Markov. The equilibrium strategy of the insider, taking into account such biases, is derived in both cases and the effects of the biases on the equilibrium price of the underlying asset are considered. The question of whether such biases benefit or harm the uninformed traders is answered.In Chapter 2 a class of mixed stochastic control/optimal stopping problems arising in the problem of finding the best time to sell an indivisible real asset, owned by a risk averse utility maximizing agent, is considered. The agent has power type utility based on the $\ell_{\alpha}$-type aggregator and has access to a frictionless financial market which can be used to partially hedge the risk associated with the real asset if correlations between the financial assets and the real asset value are nonzero. The solution to the problem of finding the optimal time to sell the real asset is characterized in terms of solution to a certain free boundary problem. The latter involves a nonlinear partial differential equation and includes, as special case with $\alpha=1$, the Hamilton-Jacobi-Bellman equation found in {Evans, Henderson, Hobson, 2008}. Comparisons with the case of exponential utility are also given.Due to lack of data, the U.S. primarily uses repeat-sales indices to measure real-estate returns, despite the serious shortcomings of these indices. Making use of a newly available data set that contains both time-varying characteristics for all properties in the U.S. and transaction details for those properties that traded, in Chapter 3 a new hedonic house-price index is developed that overcomes these shortcomings by allowing house prices and returns to depend on house characteristics and on local and national macroeconomic factors. The index is estimated using Markov Chain Monte Carlo (MCMC) linear filtering techniques and results in significant differences, in both the level and volatility of prices, between the new estimates and those from the Federal Housing Finance Board's weighted-repeat-sales (WRS) price index. This suggests that the new index is significantly superior to repeat-sales indices as a measure of U.S. real-estate returns for economic forecasting, mortgage valuation, and bank stress tests

Shrinkage estimation for convex polyhedral cones

Estimation of a multivariate normal mean is considered when the latter is known to belong to a convex polyhedron. It is shown that shrinking the maximum likelihood estimator towards an appropriate target can reduce mean squared error. The proof combines an unbiased estimator of a risk difference with some geometrical considerations. When applied to the monotone regression problem, the main result shows that shrinking the maximum likelihood estimator towards modifications that have been suggested to alleviate the spiking problem can reduce mean squared error.Degrees of freedom Maximum likelihood estimator Mean squared error Primal-dual bases Projections Stein's Identity Target estimator

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.Gaussian random field Multiparameter martingale Nonlinear filtering Fractional Wiener sheet Multiple stochastic integral

Product formula and independence criterion for multiple Huang-Cambanis integrals

Multiple stochastic integrals of Huang and Cambanis [1978. Stochastic and multiple Wiener integrals for Gaussian processes. Ann. Probab. 6, 585-614] with respect to a general Gaussian process , whose covariance function is of bounded variation on bounded subsets of , are considered. A product formula for the integrals is derived and a necessary and sufficient condition for independence of multiple Huang-Cambanis integrals is obtained. As an illustration, the results are applied to the special case of multiple integrals with respect to a persistent fractional Brownian motion.Gaussian process Multiple stochastic integral Product formula Independence Fractional Brownian motion