Volume 27, Issue 3 -

Inspired by the practice of risk management in the financial industry, we introduce the notion of pivotal quantile estimates, and relate it to the theory of structural statistical models. This framework gives a mathematical foundation to unbiased estimation of exceedance probabilities. The application to Value at Risk calculations is illustrated by an operational risk example.

We consider a portfolio optimization problem in a Black-Scholes model with n stocks, in which an investor faces both fixed and proportional transaction costs. The performance of an investment strategy is measured by the average return of the corresponding portfolio over an infinite time horizon. At first, we derive a representation of the portfolio value process which only depends on the relative fractions of the total portfolio value that the investor holds in the different stocks. This representation allows us to consider these so-called risky fractions as the decision variables of the investor. We show a certain kind of stationarity (Harris recurrence) for a quite flexible class of strategies (constant boundary strategies). Then, using renewal theoretic methods, we are able to describe the asymptotic return by the behaviour of the risky fractions in a “typical” period between two trades. Our results generalize those of [irl1], who considered a financial market model with one bond and one stock, to a market with a finite number n >1 of stocks.

We consider the parametric estimation for the random design nonlinear regression model whose regression function has an unknown cusp location. The Fisher information of this location parameter is unbounded, that is caused by the non-differentiability of the likelihood function, so this is a non-regular estimation problem. In this paper, we verify the asymptotic properties of the Bayes estimator (BE), e.g. the consistency, the asymptotic distribution and the convergence of its moments, by the likelihood ratio process whose limit is expressed in terms of fractional Brownian motion. Further, we show that the BE is asymptotically efficient in a certain minimax sense.

In this paper, we examine the mean residual waiting time of record values from a sequence of identically independent random variables with a common continuous distribution F . Under the condition that the ( m +1)-st shock has not arrived by time t >0, we obtain a simplified expression for the mean residual waiting time of the ( n +1)-st shock. We investigate some monotonicity and aging properties for the mean residual waiting time of records. Further, it is shown that the underlying distribution function F can be recovered via the functional relationships between the mean residual waiting times of records.

We prove a maximal inequality for skew Brownian motion. This result generalizes similar inequalities for standard Brownian motion ([2]) and its modulus ([3, 4]). The proof relies on the solution to an optimal stopping problem.