Esko Valkeila

Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion

We study the possibility to control the moments of Wiener integrals of fractional Brownian motion with respect to the Lp- norm of the integrand. It turns out that when the self-similarity index , we can have only an upper inequality, and when we can have only a lower inequality.

Serum selenium and risk of cancer: a prospective follow-up of nine years

The association between serum selenium concentration and the risk of cancer was studied in 1110 men aged 55 to 74 years in two rural areas of Finland. The men were followed‐up prospectively for 9 years and there were 109 new cases of cancer, with the cases of the first follow‐up year excluded. The serum selenium concentrations were adjusted for age, area, smoking, serum cholesterol, and alcohol intake. The patients had a slightly lower adjusted mean serum selenium than the subjects without cancer at the end of the follow‐up (±standard error of mean) 53.9 ± 1.5 and 55.3 ± 0.5 μg/1, respectively. The relative risks of cancer were essentially the same when these were calculated in the tertiles of the serum selenium distribution. Thirty‐seven men had a history of cancer at baseline or had cancer diagnosed during the first follow‐up year and their adjusted mean serum selenium was 49.4 ± 2.6 μg/1, which was significantly lower (P<0.05) than that of the subjects without cancer during the follow‐up.

On some maximal inequalities for fractional Brownian motions

We prove some maximal inequalities for fractional Brownian motions. These extend the Burkholder-Davis-Gundy inequalities for fractional Brownian motions. The methods are based on the integral representations of fractional Brownian motions with respect to a certain Gaussian martingale in terms of beta kernels.

Hedging under Transaction Costs in Currency Markets: a Discrete-Time Model

We consider a discrete‐time model of a currency market with transaction costs and give a description of initial endowments that allow the investor to hedge a contingent claim in various currencies by a self‐financing portfolio.

Fractional Processes as Models in Stochastic Finance

We survey some new progress on the pricing models driven by fractional Brownian motion or mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with transaction costs. We end the paper by giving some approximation results and indicating some open problems related to the paper.

An Extension of the Lévy Characterization to Fractional Brownian Motion

Assume that X is a continuous square integrable process with zero mean, defined on some probability space (Ω, F, P). The classical characterization due to P. Levy says that X is a Brownian motion if and only if X and X 2 t ― t, t > 0, are martingales with respect to the intrinsic filtration F X . We extend this result to fractional Brownian motion.

Random variables as pathwise integrals with respect to fractional Brownian motion

We give both necessary and sufficient conditions for a random variable to be represented as a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand. We also show that any random variable is a value of such integral in an improper sense and that such integral can have any prescribed distribution. We discuss some applications of these results, in particular, to fractional Black–Scholes model of financial market.

A general poisson approximation theorem

We study the rate of convergence in a limit theorem due to Kabanov-Liptser-Shiryayev. We show how the probabilities P(N t= k) can be computed from the compensator, when it is deterministic.

Statistical inference with fractional Brownian motion

We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models.

A Prohorov bound for a Poisson process and an arbitrary counting process with some applications

An upper bound for the Prohorov distance between a Poission process and an arbitrary couting process in terms of their compensators is examined. We discuss the sharpness of this bound and apply it to the known weak convergense results of Brown [1] and Kabanov, Liptser and Shiryayev [2]. We also use our bound to investigate empirical and renewal counting processes.