160 research outputs found
Analysis of continuous strict local martingales via h-transforms
We study strict local martingales via h-transforms, a method which first
appeared in Delbaen-Schachermayer. We show that strict local martingales arise
whenever there is a consistent family of change of measures where the two
measures are not equivalent to one another. Several old and new strict local
martingales are identified. We treat examples of diffusions with various
boundary behavior, size-bias sampling of diffusion paths, and non-colliding
diffusions. A multidimensional generalization to conformal strict local
martingales is achieved through Kelvin transform. As curious examples of
non-standard behavior, we show by various examples that strict local
martingales do not behave uniformly when the function (x-K)^+ is applied to
them. Implications to the recent literature on financial bubbles are discussed.Comment: Significantly revised version. 28 page
Absolutely Continuous Compensators
We give sufficient conditions on the underlying filtration such that all
totally inaccessible stopping times have compensators which are absolutely
continuous. If a semimartingale, strong Markov process X has a representation
as a solution of a stochastic differential equation driven by a Wiener process,
Lebesgue measure, and a Poisson random measure, then all compensators of
totally inaccessible stopping times are absolutely continuous with respect to
the minimal filtration generated by X. However Cinlar and Jacod have shown that
all semimartingale strong Markov processes, up to a change of time and space,
have such a representation
No arbitrage without semimartingales
We show that with suitable restrictions on allowable trading strategies, one
has no arbitrage in settings where the traditional theory would admit arbitrage
possibilities. In particular, price processes that are not semimartingales are
possible in our setting, for example, fractional Brownian motion.Comment: Published in at http://dx.doi.org/10.1214/08-AAP554 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
From Discrete- to Continuous-Time Finance: Weak Convergence of the Financial Gain Process
Conditions suitable for applications in finance are given for the weak convergence (or convergence in probability) of stochastic integrals. For example, consider a sequence Sn of security price processes converging in distribution to S and a sequence 8 " of trading strategies converging in distribution to 0. We survey conditions under which the financial gain process J 0 " dSn converges in distribution to J 0 dS. Examples include convergence from discrete- to continuous-time settings and, in particular, generalizations of the convergence of binomial option replication models to the Black-Scholes model. Counterexamples are also provided. KEYWORDS: semimartingales, weak convergence, option valuation, Black-Scholes model 1
Stopping Times Occurring Simultaneously
Stopping times are used in applications to model random arrivals. A standard
assumption in many models is that they are conditionally independent, given an
underlying filtration. This is a widely useful assumption, but there are
circumstances where it seems to be unnecessarily strong. We use a modified Cox
construction along with the bivariate exponential introduced by Marshall and
Olkin (1967) to create a family of stopping times, which are not necessarily
conditionally independent, allowing for a positive probability for them to be
equal. We show that our initial construction only allows for positive
dependence between stopping times, but we also propose a joint distribution
that allows for negative dependence while preserving the property of non-zero
probability of equality. We indicate applications to modeling COVID-19
contagion (and epidemics in general), civil engineering, and to credit ris
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