Shigeyoshi Ogawa

An Improved Two-Step Regularization Scheme for Spot Volatility Estimation

We are concerned with the problem of parameter estimation in Finance, namely the estimation of the spot volatility in the presence of the so-called microstructure noise. In [16] a scheme based on the technique of multi-step regularization was presented. It was shown that this scheme can work in a real-time manner. However, the main drawback of this scheme is that it needs a lot of observation data. The aim of the present paper is to introduce an improvement of the scheme such that the modified estimator can work more efficiently and with a data set of smaller size. The technical aspects of implementation of the scheme and its performance on simulated data are analyzed. The proposed scheme is tested against other estimators, namely a realized volatility type estimator, the Fourier estimator and two kernel estimators.

Quasirandom Walk Methods

We investigate the simulation of diffusion by the random walk displacement of a set of particles. The method is a part of fractional step schemes when we consider problems involving more than one transport mechanism. We systematically replace pseudorandom numbers by quasirandom numbers in the random walk step. The application of quasirandom sequences is not straightforward, because of correlations, and a reordering technique must be used in every time step. We show that a significant improvement in both magnitude of error and convergence rate is achieved over standard random walk methods, for one- and two-dimensional problems.

On a stochastic Fourier transformation

Given a random function and an orthonormal basis in the , we are concerned with the basic properties of its stochastic Fourier coefficient (SFC) which is defined by the stochastic integral with respect to the Brownian motion, . More precisely we are concerned in this note with the problem of reconstructing the function from its SFCs. We will show as our main result the affirmative answer when the random function belongs to a certain restricted but wide enough subclass, which is the class of causal Wiener functionals.

On a real-time scheme for the estimation of volatility

We are interested in the numerical scheme for the estimation of the volatility of a given price process St , which in the Black-Sholes paradigm is supposed to follow the Itô type stochastic differential equation.

A quasi-random walk method for one-dimensional reaction-diffusion equations

Probabilistic methods are presented to solve one-dimensional nonlinear reaction-diffusion equations. Computational particles are used to approximate the spatial derivative of the solution. The random walk principle is used to model the diffusion term. We investigate the effect of replacing pseudo-random numbers by quasi-random numbers in the random walk steps. This cannot be implemented in a straightforward fashion, because of correlations. If the particles are reordered according to their position at each time step, this has the effect of breaking correlations. For simple demonstration problems, the error is found to be significantly less when quasi-random sequences are used than when a standard random walk calculation is performed using pseudo-random points.

On a Robustness of The Random Particle Method

We are concerned with a robustness of the so-called random particle methods that have been recognized as efficient tool for the numerical analysis of nonlinear diffusions. Among these, we take the random gradient method due to E.Puckette and we study the stability of this method against a slight perturbation in statistical quality of random numbers. 1. Random particle method Every Monte Carlo method is established on a basic assumption that an arbitrary amount of random numbers with prescribed distribution is always available. The efficiency of the method should more or less depend on the quality of random number generators. Hence it is needless to emphasize the importance of studying, with every specific Monte Carlo method, the robustness or sensitivity of the method. It is rather surprising therefore to find that, as far as the author knows, very few research has been done in this direction. From this viewpoint we take the random particle method due to E.Puckette and we are going to check the robustness of this method. The reason of investigating this method is simply because this is one of the well-known and successful stochastic methods and because the author has been interested in the stochastic simulation of nonlinear dif-, fusions. . v ^ ?i ^ -i

Stochastic Fourier Transformation

We have seen in the previous chapter that the stochastic Fourier transformation (SFT) and the stochastic Fourier coefficients (SFCs) serve as effective tools for the study of the noncausal SIE of Fredholm type. In this chapter we shall study basic properties of these SFT and SFC.

Appendices to Chapter 2

We are going to give here two notes that concern to the subject of Chap. 2, one is on the introduction of the Ito integral with respect to martingales and the other is on some elementary facts about inequalities concerning sub-martingales. Those are moved from Chap. 2 so as to keep the volume of chapter compact.

Introduction – Why the Causality?

We are concerned with an alternative theory of stochastic calculus, referred to nowadays with the adjective noncausal, that contrasts to the standard theory of Ito calculus. We would like to give in this chapter a historical sketch of the noncausal theory so as to convince ourselves of the necessity and importance of such a theory. The net lecture on noncausal theory begins from Chap. 3, while in Chap. 2 we give as a preliminary for the main subject short reviews on such basic materials from the causal theory of Ito calculus as the Ito integral, as well as the symmetric integrals and B-derivatives.

Preliminary – Causal Calculus

The theory of noncausal calculus is an alternative to the causal theory of Ito calculus but is not quite independent of it. As we will see in the main part of this book that starts from Chap. 3, our noncausal theory stands as a natural extension of the causal theory of Ito calculus, to be more precise, the causal theory based on the stochastic integral called symmetric integrals . We may emphasize that at this point our noncausal theory keeps a large part of its raisons d’etre.