The 20th Finnish Summer School on Probability Theory

## The 20th Finnish Summer School on Probability Theory

20. todennäköisyyslaskennan kesäkoulu pidetään Lahden Mukkulassa 1-5.6.1998 (tiedotus ja ilmoittautumislomake suomeksi)

20 sommarskolan i sannolikhetsteori hålls på Mokulla gård i Lahtis den 1-5 juni 1998 (information och anmälningsblankett på svenska)

The 20th Finnish Summer School on Probability Theory will be arranged at Lahti from June 1 to June 5, 1998.

### Speakers: Robert S. Liptser, Laurens de Haan, Hans Gerber, Krishna Athreya

More information and lecture abstracts will be distributed in April - May. There is no participation fee, but we need to limit the number of participants to about 30.

## Course description

Krishna Athreya

#### Markov Chain Monte Carlo methods, random iterations and changes of measures

##### Two lectures on each topic:
1. Convergence criteria for MCMC (Markov chain Monte Carlo).

2. Iterations of IID random logistic maps.

3. Change of measures for Markov Chains and the LLOGL theorem for branching processes.

Hans Gerber

#### From Risk Theory to Finance:

1. Some results concerning the joint distribution of the time of ruin and the surpluses immediately before and at ruin

2. An actuarial approach to option pricing in continuous time

3. Pricing certain perpetual options

4. Applications of utility functions in risk theory and finance, including Merton's problem of finding the optimal portfolio

5. Dynamic solvency insurance and investment fund protection

Laurens de Haan sent us a course description:

#### Sample extremes and residual life times: theory and applications

The theory of extreme values (=sample extremes) has developed rapidly over the last decades. It is concerned with possible failure of a structure under the influence of shocks. This in contrasts to failure by gradual decay which is controlled by the central limit theory.

Extreme value theory is relevant in problems of corrosion, in assessing the occurrence of earthquakes, extreme wave heights -mostly in connection with the reliability of jack-up platforms, insurance etc. We have dealt with hydrological problems (what is the probability of a flood in Zuid Holland in 1999?) and with problems in finance (analyzing extreme risks in speculative markets, "value-at-risk"). The probabilistic theory of extreme values in one dimensional space (i.i.d.) attained its final form around 1940. Statistical methods have been in use from about 1950, but only around 1975 the present development towards semi parametric methods started. This development has greatly widened the scope of applications and has deepened the mathematical content.

The probabilistic theory in higher-dimensional spaces has been developed around 1975; the statistical applications started only ten years later. The probabilistic theory in infinite-dimensional space (dealing with i.i.d. stochastic processes) has been initiated in two papers in 1984 and 1990. The statistical theory is still in the development stage.

We shall review the probabilistic theory in one-dimensional and higher-dimensional space. This is connected with the analytic theory of regularly varying functions on the one hand and with the theory of point processes and empirical processes on the other hand. The statistical theory will be developed and applications will be discussed.

Robert S. Liptser

#### Asymptotically optimal filtering

There are a few filtering models, for which the "filtering equation" obeys a closed form like the Kalman filter, the conditionally Gaussian filter, the Wonham type filter, and the Kushner-Zakai filter. However in applications, realistic filtering models have more complicated structure than those for which the above-mentioned filters are at once applicable. In this chapter, we consider examples of filtering models for which a following approximation technique might be successful. Instead of the original model, a new model, where the underlying processes are replaced by simple ones, is applied which makes it possible to construct nearly optimal filters. To explain such an approach with more details, let us consider the filtering problem for a pair random processes $(X_t^\varepsilon,Y_t^\varepsilon)_{t\ge 0}$, where $X_t^\varepsilon$ represents an unobservable signal and $Y_t^\varepsilon$ is a corresponding observation, and where $\varepsilon$ is a small parameter. Suppose a probabilistic structure of $(X_t^\varepsilon,Y_t^\varepsilon)_ {t\ge 0}$ is too complicated to find the optimal (in the mean square sense) filtering estimate, but as $\varepsilon\to 0$ the pair $(X_t^\varepsilon,Y_t^\varepsilon)_{t\ge 0}$ converges (in some sense) to the limit pair $(\overline{X}_t,\overline{Y}_t)_{t\ge 0}$ which has a simpler description than the prelimit one, for example, it is a Markov diffusion process or, more specifically, a Gaussian diffusion. A natural procedure of creating a successful filter for prelimit model consists in finding the optimal filter for the limit model and then to use it for the prelimit observations.

The main problem in such approach is a verification of an asymptotic optimality for obtained filters. We give in two examples for which the asymptotic optimality can be checked effectively.

#### The 19th Finnish Summer School on Probability Theory, held on June 2 to June 6, 1997

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Updated 25.5.98 by Ann-Mari.Dahlstrom@abo.fi