In this work we introduce correlated random walks on Z. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is the fractional Brownian motion. We have to use two radically different models for both cases 1 2 ≤ H < 1 and 0 < H < 1 2 . This result provides an algorithm for t...

Fractional Brownian motion has been suggested as a model for the movement of log share prices which would allow long-range dependence between returns on different days. While this is true, it also allows arbitrage opportunities, which we demonstrate both indirectly and by constructing such an arbitrage. Nonetheless, it is possible by looking at a process similar to the fractional Brownian motio...

In recent years fractional Brownian motion has been suggested to replace the classical Brownian motion as driving process in the modelling of many real world phenomena, including stock price modelling. In several papers seemingly contradictory results on the existence or absence of a riskless gain (arbitrage) in such stock models have been stated. This survey tries to clarify this issue by poin...

We give a simple criterion for a stochastic process Z := H+K ·W , where H and K are respectively continuous and left-continuous processes independent of the driving Brownian motion W , which ensures that Z has the conditional full support property, introduced by Guasoni, Rásonyi, and Schachermayer, in connection to pricing models with transaction costs. As an application of this result, we show...

Abstract We prove that the scaling limit of nearest-neighbour senile reinforced random walk is Brownian Motion when the time T spent on the first edge has finite mean. We show that under suitable conditions, when T has heavy tails the scaling limit is the so-called fractional kinetics process, a random time-change of Brownian motion. The proof uses the standard tools of time-change and invarian...

Since Merton (1969), the description of a contingent claim as a Brownian motion is commonly accepted. Thus an option price, a future price, a share price, a bond price, interest rates etc., can be modelled with a Brownian motion. In summary, any financial series which present value depends on only a few previous values, may be modelled with a continuous–time diffusion–type process. The general ...

and (1.6) For fixed u, p(t, u) is a continuous function of I for / different from zero. The above conditions on the transition function are suggested by applications to the Brownian motion and, in the discrete case, to sums of independent random variables belonging to the domain of attraction of symmetric stable laws. There are Markov processes besides the Brownian motion for which conditions (...