Tommi Sottinen

Fractional Brownian motion, random walks and binary market models

Abstract. We prove a Donsker type approximation theorem for the fractional Brownian motion in the case

Pricing by hedging and no-arbitrage beyond semimartingales

H>1/2.

Fractional Processes as Models in Stochastic Finance

Using this approximation we construct an elementary market model that converges weakly to the fractional analogue of the Black–Scholes model. We show that there exist arbitrage opportunities in this model. One such opportunity is constructed explicitly.

Application of Girsanov Theorem to Particle Filtering of Discretely Observed Continuous - Time Non-Linear Systems

We show that pricing a big class of relevant options by hedging and no-arbitrage can be extended beyond semimartingale models. To this end we construct a subclass of self-financing portfolios that contains hedges for these options, but does not contain arbitrage opportunities, even if the stock price process is a non-semimartingale of some special type. Moreover, we show that the option prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. We end the paper by giving no-arbitrage results even with stopping times for our model class.

Path Space Large Deviations of a Large Buffer with Gaussian Input Traffic

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.

Stochastic Analysis of Gaussian Processes via Fredholm Representation

Summary. This article considers the application of particle filterin g to continuousdiscrete optimal filtering problems, where the system model is a stochastic differential equation, and noisy measurements of the system are obtained at discrete instances of time. It is shown how the Girsanov theorem can be used for evaluating the likelihood ratios needed in importance sampling. It is also shown how the methodology can be applied to a class of models, where the driving noise process is lower in the dimensionality than the state and thus the laws of state and noise are not absolutely continuous. Rao-Blackwellization of conditionally Gaussian models and unknown static parameter models is also considered.

Yukawa Potential, Panharmonic Measure and Brownian Motion

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having different Hurst indices.

Walk On Spheres Algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence

We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.

Lipschitz conditions for _{}(Ω)-processes and applications to weakly self-similar processes with stationary increments

In [25] a Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz PDEs was developed by using the so-called Duffin correspondence. In this paper we investigate the foundations behind the algorithm for the case of the Yukawa PDE. We study the panharmonic measure that is a generalization of the harmonic measure for the Yukawa PDE. We show that there are natural stochastic definitions for the panharmonic measure in terms of the Brownian motion and that the harmonic and the panharmonic measures are all mutually equivalent. Furthermore, we calculate their Radon--Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm.

CONDITIONAL-MEAN HEDGING UNDER TRANSACTION COSTS IN GAUSSIAN MODELS

We show that a constant-potential time-independent Schrödinger equation with Dirichlet boundary data can be reformulated as a Laplace equation with Dirichlet boundary data. With this reformulation, which we call the Duffin correspondence, we provide a classical Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the boundary value problem. We compare the obtained Duffin WOS algorithm with existing modified WOS algorithms.