Notes on Stein-Sahi representations and some problems of non L2L^2 harmonic analysis

We discuss one natural class of kernels on pseudo-Riemannian symmetric spaces.Comment: 40p

Analysis of Fourier transform valuation formulas and applications

The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case, and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ

Stochastic Models Involving Second Order Lévy Motions

This thesis is based on five papers (A-E) treating estimation methods for unbounded densities, random fields generated by Lévy processes, behavior of Lévy processes at level crossings, and a Markov random field mixtures of multivariate Gaussian fields. In Paper A we propose an estimator of the location parameter for a density that is unbounded at the mode. The estimator maximizes a modified likelihood in which the singular term in the full likelihood is left out, whenever the parameter value approaches a neighborhood of the singularity location. The consistency and super-efficiency of this maximum leave-one-out likelihood estimator is shown through a direct argument. In Paper B we prove that the generalized Laplace distribution and the normal inverse Gaussian distribution are the only subclasses of the generalized hyperbolic distribution that are closed under convolution. In Paper C we propose a non-Gaussian Matérn random field models, generated through stochastic partial differential equations, with the class of generalized Hyperbolic processes as noise forcings. A maximum likelihood estimation technique based on the Monte Carlo Expectation Maximization algorithm is presented, and it is shown how to preform predictions at unobserved locations. In Paper D a novel class of models is introduced, denoted latent Gaussian random filed mixture models, which combines the Markov random field mixture model with the latent Gaussian random field models. The latent model, which is observed under a measurement noise, is defined as a mixture of several, possible multivariate, Gaussian random fields. Selection of which of the fields is observed at each location is modeled using a discrete Markov random field. Efficient estimation methods for the parameter of the models is developed using a stochastic gradient algorithm. In Paper E studies the behaviour of level crossing of non-Gaussian time series through a Slepian model. The approach is through developing a Slepian model for underlying random noise that drives the process which crosses the level. It is demonstrated how a moving average time series driven by Laplace noise can be analyzed through the Slepian noise approach. Methods for sampling the biased sampling distribution of the noise are based on an Gibbs sampler

Fourier Policy Gradients

We propose a new way of deriving policy gradient updates for reinforcement learning. Our technique, based on Fourier analysis, recasts integrals that arise with expected policy gradients as convolutions and turns them into multiplications. The obtained analytical solutions allow us to capture the low variance benefits of EPG in a broad range of settings. For the critic, we treat trigonometric and radial basis functions, two function families with the universal approximation property. The choice of policy can be almost arbitrary, including mixtures or hybrid continuous-discrete probability distributions. Moreover, we derive a general family of sample-based estimators for stochastic policy gradients, which unifies existing results on sample-based approximation. We believe that this technique has the potential to shape the next generation of policy gradient approaches, powered by analytical results