114 research outputs found
SDEs Driven by SDE Solutions
We consider stochastic differential equations (SDEs) driven by Feller
processes which are themselves solutions of multivariate Levy driven SDEs. The
solutions of these 'iterated SDEs' are shown to be non-Markovian. However, the
process consisting of the driving process and the solution is Markov and even
Feller in the case of bounded coefficients. The generator as well as the
semimartingale characteristics of this process are calculated explicitly and
fine properties of the solution are derived via the stochastic symbol. A short
simulation study and an outlook in the direction of stochastic modeling round
out the paper.Comment: 16 pages, 9 figure
A Classification of Deterministic Hunt Processes with Some Applications
Deterministic processes form an important building block of several classes
of processes. We provide a method to classify deterministic Hunt processes.
Within this framework we characterize different subclasses (e.g. Feller) and
construct some (counter-)examples. In particular the existence of a Hunt
semimartingale (on R) which is not an It\^o process in the sense of Cinlar,
Jacod, Protter and Sharpe (1980) is proven.Comment: 18 pages, 9 figure
On the Semimartingale Nature of Feller Processes with Killing
Let U be an open set in R^d. We show that under a mild assumption on the
richness of the generator a Feller process in U with (predictable) killing is a
semimartingale. To this end we generalize the notion of semimartingales in a
natural way to those 'with killing'. Furthermore we calculate the
semimartingale characteristics of the Feller process explicitly and analyze
their connections to the symbol. Finally we derive a probabilistic formula to
calculate the symbol of the process directly
The Euler scheme for Feller processes
We consider the Euler scheme for stochastic differential equations with
jumps, whose intensity might be infinite and the jump structure may depend on
the position. This general type of SDE is explicitly given for Feller processes
and a general convergence condition is presented.
In particular the characteristic functions of the increments of the Euler
scheme are calculated in terms of the symbol of the Feller process in a closed
form. These increments are increments of L\'evy processes and thus the Euler
scheme can be used for simulation by applying standard techniques from L\'evy
processes
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