Nonlinear non-Gaussian state space model is considered for the analysis of nonstationary time series. Recursive formulas for state estimation such as predictor, filter and smoother can be derived for this model. These formulas can be realized by using Monte Carlo filter and smoother. These recursive computational methods facilitates very flexible nonlinear non-Gaussian modeling of nonstationary time series. As examples of this method, two types of modeling of volatility of financial time series are shown.

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This paper provides a survey of long memory models and their applications in economic and financial time series. Section 2 describes the definition of long memory models, particularly two popular models, ARFIMA and FGN. Section 3 describes unit roots tests and cointegration, two major topics of the recent decades in theoretical and applied econometrics. Section 4 discusses a generalization of unit roots tests and cointegration to long memory models. Section 5 describes ARCH models and SV models which have been paid much attention to analyze volatility in actual financial data. Section 6 considers a generalization of these volatility models to those with long memory properties. Finally Section 7 considers some problems to be solved in future.

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This paper compares some important differences between time-discrete approach and time-continuous approach in the no-arbitrage pricing theory. In particular, paying a central attention to the concept of no-arbitrage, from a practical viewpoint we discuss on the Markovian property of a diffusion model, the replicability of a contingent claim and the completeness of a model.

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We review a pricing theory of derivatives including exotic options. First, using stochastic calculus, we stute the general theory of pricing derivatives and constructing their replicating portfolios on the Black-Sholes model. Next, we apply this theory to some exotic options, and carry out the pricing and the construction of their replicating portfolios. Also, we remark a pricing theory on a discrete time model as an application of the random walk martingale representation theorem and the discrete Ito formula.

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Contrary to the classical static view of demand and supply, it is shown that open market price fluctuations have the properties of dynamic critical fluctuations between excess-demand and excess-supply phases. Self-similarity of price changes and power law distributions of price differences are naturally understood by introducing the concept of phase transition developed in statistical physics for materials. A set of dynamic stochastic equations for the market price is introduced, and the basic empirical properties, such as the bubbles and crashes, are explained with this formulation.

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Collections of participants in financial markets are represented as random partitions by the strategies or behavioral rules they employ. By suitable assignments of transition rates for entry, exit and type changes, the jump Markov process is defined which describe dynamics of the state of the market via the master (backward Chapman-kolmogorov) equation. This paper draws from the population genetics literature to discuss Ewens sampling formula, and Poisson-Dirichlet distribution of Kingman to discuss the circumstances in which a small number of large clusters of agents develop and they domimate the market. Price variations in such cases may exhibit power-laws. Power laws are known to govern price variations in financial markerts. Changes in volatilities of prices may also be due to switching of strategies by a small number of clusters of agents.

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[in Japanese]

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