892 research outputs found
Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations
It is shown that under a certain condition on a semimartingale and a
time-change, any stochastic integral driven by the time-changed semimartingale
is a time-changed stochastic integral driven by the original semimartingale. As
a direct consequence, a specialized form of the Ito formula is derived. When a
standard Brownian motion is the original semimartingale, classical Ito
stochastic differential equations driven by the Brownian motion with drift
extend to a larger class of stochastic differential equations involving a
time-change with continuous paths. A form of the general solution of linear
equations in this new class is established, followed by consideration of some
examples analogous to the classical equations. Through these examples, each
coefficient of the stochastic differential equations in the new class is given
meaning. The new feature is the coexistence of a usual drift term along with a
term related to the time-change.Comment: 27 pages; typos correcte
Projections, Pseudo-Stopping Times and the Immersion Property
Given two filtrations , we study under which
conditions the -optional projection and the -dual
optional projection coincide for the class of -optional processes
with integrable variation. It turns out that this property is equivalent to the
immersion property for and , that is every -local martingale is a -local martingale, which, equivalently, may
be characterised using the class of -pseudo-stopping times. We also
show that every -stopping time can be decomposed into the minimum of
two barrier hitting times
The strong weak convergence of the quasi-EA
In this paper, we investigate the convergence of a novel simulation scheme to the target diffusion process. This scheme, the Quasi-EA, is closely related to the Exact Algorithm (EA) for diffusion processes, as it is obtained by neglecting the rejection step in EA. We prove the existence of a myopic coupling between the Quasi-EA and the diffusion. Moreover, an upper bound for the coupling probability is given. Consequently we establish the convergence of the Quasi-EA to the diffusion with respect to the total variation distance
Anomalous Processes with General Waiting Times: Functionals and Multipoint Structure
Many transport processes in nature exhibit anomalous diffusive properties
with non-trivial scaling of the mean square displacement, e.g., diffusion of
cells or of biomolecules inside the cell nucleus, where typically a crossover
between different scaling regimes appears over time. Here, we investigate a
class of anomalous diffusion processes that is able to capture such complex
dynamics by virtue of a general waiting time distribution. We obtain a complete
characterization of such generalized anomalous processes, including their
functionals and multi-point structure, using a representation in terms of a
normal diffusive process plus a stochastic time change. In particular, we
derive analytical closed form expressions for the two-point correlation
functions, which can be readily compared with experimental data.Comment: Accepted in Phys. Rev. Let
Fractional smoothness and applications in finance
This overview article concerns the notion of fractional smoothness of random
variables of the form , where is a certain
diffusion process. We review the connection to the real interpolation theory,
give examples and applications of this concept. The applications in stochastic
finance mainly concern the analysis of discrete time hedging errors. We close
the review by indicating some further developments.Comment: Chapter of AMAMEF book. 20 pages
Power Utility Maximization in Discrete-Time and Continuous-Time Exponential Levy Models
Consider power utility maximization of terminal wealth in a 1-dimensional
continuous-time exponential Levy model with finite time horizon. We discretize
the model by restricting portfolio adjustments to an equidistant discrete time
grid. Under minimal assumptions we prove convergence of the optimal
discrete-time strategies to the continuous-time counterpart. In addition, we
provide and compare qualitative properties of the discrete-time and
continuous-time optimizers.Comment: 18 pages, to appear in Mathematical Methods of Operations Research.
The final publication is available at springerlink.co
A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise
We prove a functional non-central limit theorem for jump-diffusions with
periodic coefficients driven by strictly stable Levy-processes with stability
index bigger than one. The limit process turns out to be a strictly stable Levy
process with an averaged jump-measure. Unlike in the situation where the
diffusion is driven by Brownian motion, there is no drift related enhancement
of diffusivity.Comment: Accepted to Journal of Theoretical Probabilit
On the harmonic measure of stable processes
Using three hypergeometric identities, we evaluate the harmonic measure of a
finite interval and of its complementary for a strictly stable real L{\'e}vy
process. This gives a simple and unified proof of several results in the
literature, old and recent. We also provide a full description of the
corresponding Green functions. As a by-product, we compute the hitting
probabilities of points and describe the non-negative harmonic functions for
the stable process killed outside a finite interval
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