36 research outputs found
Continuous-time trading and the emergence of probability
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New procedures for testing whether stock price processes are martingales
We propose procedures for testing whether stock price processes are
martingales based on limit order type betting strategies. We first show that
the null hypothesis of martingale property of a stock price process can be
tested based on the capital process of a betting strategy. In particular with
high frequency Markov type strategies we find that martingale null hypotheses
are rejected for many stock price processes
Efficient discretisation of stochastic differential equations
The aim of this study is to find a generic method for generating a path of
the solution of a given stochastic differential equation which is more
efficient than the standard Euler-Maruyama scheme with Gaussian increments.
First we characterize the asymptotic distribution of pathwise error in the
Euler-Maruyama scheme with a general partition of time interval and then, show
that the error is reduced by a factor (d+2)/d when using a partition associated
with the hitting times of sphere for the driving d-dimensional Brownian motion.
This reduction ratio is the best possible in a symmetric class of partitions.
Next we show that a reduction which is close to the best possible is achieved
by using the hitting time of a moving sphere which is easier to implement
Local times for typical price paths and pathwise Tanaka formulas
Following a hedging based approach to model free financial mathematics, we
prove that it should be possible to make an arbitrarily large profit by
investing in those one-dimensional paths which do not possess local times. The
local time is constructed from discrete approximations, and it is shown that it
is -H\"older continuous for all . Additionally, we provide
various generalizations of F\"ollmer's pathwise It\^o formula
Rough paths in idealized financial markets
This paper considers possible price paths of a financial security in an
idealized market. Its main result is that the variation index of typical price
paths is at most 2, in this sense, typical price paths are not rougher than
typical paths of Brownian motion. We do not make any stochastic assumptions and
only assume that the price path is positive and right-continuous. The
qualification "typical" means that there is a trading strategy (constructed
explicitly in the proof) that risks only one monetary unit but brings infinite
capital when the variation index of the realized price path exceeds 2. The
paper also reviews some known results for continuous price paths and lists
several open problems.Comment: 21 pages, this version adds (in Appendix C) a reference to new
results in the foundations of game-theoretic probability based on Hardin and
Taylor's work on hat puzzle