In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

A rapidly growing literature has documented important improvements in financial return volatility measurement and forecasting via use of realized variation measures constructed from high-frequency returns coupled with simple modeling procedures. Building on recent theoretical results in Barndorff-Nielsen and Shephard (2004a, 2005) for related bi-power variation measures, the present paper provides a practical and robust framework for non-parametrically measuring the jump component in asset return volatility. In an application to the DM/$ exchange rate, the S&P500 market index, and the 30-year U.S. Treasury bond yield, we find that jumps are both highly prevalent and distinctly less persistent than the continuous sample path variation process. Moreover, many jumps appear directly associated with specific macroeconomic news announcements. Separating jump from non-jump movements in a simple but sophisticated volatility forecasting model, we find that almost all of the predictability in daily, weekly, and monthly return volatilities comes from the non-jump component. Our results thus set the stage for a number of interesting future econometric developments and important financial applications by separately modeling, forecasting, and pricing the continuous and jump components of the total return variation process.

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Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

Diffusion processes and their sample paths

Three expressions are provided for the first hitting time density of an Ornstein-Uhlenbeck process to reach a fixed level. The first hinges on an eigenvalue expansion involving zeros of the parabolic cylinder functions. The second is an integral representation involving some special functions whereas the third is given in terms of a functional of a 3-dimensional Bessel bridge. The expressions are used for approximating the density. 1Research supported by RiskLab, Switzerland, funded by Credit Suisse Group, Swiss Re and UBS AG. The third author was supported by a Steno grant from the Danish Natural Science Research Council.

Representations of the first hitting time density of an Ornstein-Uhlenbeck process

Assuming that the wealth process $X^u$ is generated self-financially from the given initial wealth by holding its fraction u in a risky stock (whose price follows a geometric Brownian motion with drift $\mu \in \mathbb {R}$ and volatility $\sigma >0$) and its remaining fraction $1 -u$ in a riskless bond (whose price compounds exponentially with interest rate $r \in \mathbb {R}$), and letting $\mathsf{P}_{t,x}$ denote a probability measure under which $X^u$ takes value x at time t, we study the dynamic version of the nonlinear mean-variance optimal control problem 


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where t runs from 0 to the given terminal time $T>0$, the supremum is taken over admissible controls u, and $c>0$ is a given constant. By employing the method of Lagrange multipliers we show that the nonlinear problem can be reduced to a family of linear problems. Solving the latter using a classic Hamilton-Jacobi-Bellman approach we find that the optimal dynamic control is given by 

$$\begin{aligned} u_*(t,x) = \frac{\delta }{2\; c\; \sigma }\; \frac{1}{x}\, e^{(\delta ^2-r)(T-t)} \end{aligned}$$

where $\delta = (\mu -r)/\sigma $. The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.

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Optimal mean-variance portfolio selection

Let (M t ) 0≤t≤1 be a continuous martingale with initial law M 0 μ 0, and terminal law M 1 μ 1 , and let S = sup 0≤t≤1 M t . In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of S. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.

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The minimum maximum of a continuous martingale with given initial and terminal laws

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

Discounted optimal stopping problems for the maximum process

At time 0 start to observe a Brownian path. Based upon the information, which is continuously updated through the observation of the path, a stopping time is determined such that the path is as close as possible to its unknown ultimate maximum over a finite time interval. The closeness is measured by a q-mean or by a probability distance. This can be formulated as an optimal stopping problem. The method of proof relies upon a representation of a conditional expectation of the gain process and the principle of smooth fit (at a single point).

Jesper Pedersen

Papers

Representations of the first hitting time density of an Ornstein-Uhlenbeck process

Optimal mean-variance portfolio selection

The minimum maximum of a continuous martingale with given initial and terminal laws

Discounted optimal stopping problems for the maximum process

Optimal prediction of the ultimate maximum of Brownian motion