arXiv subjects

Esko Valkeila

Publications and source records attributed to Esko Valkeila.

6 recordsLinked to original sources

arXiv2012

Random variables as pathwise integrals with respect to fractional Brownian motion

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

math.PR↗

arXiv2012

Spectral characterization of the quadratic variation of mixed Brownian fractional Brownian motion

Dzhaparidze and Spreij [5] showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. [2], where it is shown that the quadratic variation of the log-returns determines the hedging strategy.

math.PR↗

arXiv2011

Initial Enlargement in a Markov chain market model

Enlargement of filtrations is a classical topic in the general theory of stochastic processes. This theory has been applied to stochastic finance in order to analyze models with insider information. In this paper we study initial enlargement in a Markov chain market model, introduced by R. Norberg. In the enlargened filtration several things can happen: some of the jumps times can be accessible or predictable, but in the orginal filtration all the jumps times are totally inaccessible. But even if the jumps times change to accessible or predictable, the insider does not necessarily have arbitrage possibilities.

q-fin.TR↗

arXiv2011

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 $(Ω,\mathrm {F},\mathrm {P})$. The classical characterization due to P. Lévy says that $X$ is a Brownian motion if and only if $X$ and $X_t^2-t$, $t\ge0,$ are martingales with respect to the intrinsic filtration $\mathrm {F}^X$. We extend this result to fractional Brownian motion.

math.PR↗

arXiv2010

When does fractional Brownian motion not behave as a continuous function with bounded variation?

If we compose a smooth function g with fractional Brownian motion B with Hurst index H > 1/2, then the resulting change of variables formula [or It/^o- formula] has the same form as if fractional Brownian motion would be a continuous function with bounded variation. In this note we prove a new integral representation formula for the running maximum of a continuous function with bounded variation. Moreover we show that the analogue to fractional Brownian motion fails.

math.PR↗

arXiv2010

Fractional processes as models in stochastic finance

We survey some new progress on the pricing models driven by fractional Brownian motion \cb{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.

q-fin.PR↗