11,986 research outputs found
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
Euler Scheme and Tempered Distributuions
Given a smooth R^d-valued diffusion, we study how fast the Euler scheme with
time step 1/n converges in law. To be precise, we look for which class of test
functions f the approximate expectation E[f(X^{n,x}_1)] converges with speed
1/n to E[f(X^x_1)]. If X is uniformly elliptic, we show that this class
contains all tempered distributions, and all measurable functions with
exponential growth. We give applications to option pricing and hedging, proving
numerical convergence rates for prices, deltas and gammas.Comment: 26 page
Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments
We prove a global asymptotic equivalence of experiments in the sense of Le
Cam's theory. The experiments are a continuously observed diffusion with
nonparametric drift and its Euler scheme. We focus on diffusions with
nonconstant-known diffusion coefficient. The asymptotic equivalence is proved
by constructing explicit equivalence mappings based on random time changes. The
equivalence of the discretized observation of the diffusion and the
corresponding Euler scheme experiment is then derived. The impact of these
equivalence results is that it justifies the use of the Euler scheme instead of
the discretized diffusion process for inference purposes.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1216 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions
For a stochastic differential equation(SDE) driven by a fractional Brownian
motion(fBm) with Hurst parameter , it is known that the existing
(naive) Euler scheme has the rate of convergence . Since the limit
of the SDE corresponds to a Stratonovich SDE driven
by standard Brownian motion, and the naive Euler scheme is the extension of the
classical Euler scheme for It\^{o} SDEs for , the convergence
rate of the naive Euler scheme deteriorates for . In
this paper we introduce a new (modified Euler) approximation scheme which is
closer to the classical Euler scheme for Stratonovich SDEs for ,
and it has the rate of convergence , where
when , when
and if . Furthermore, we study the
asymptotic behavior of the fluctuations of the error. More precisely, if
is the solution of a SDE driven by a fBm and if
is its approximation obtained by the new modified Euler
scheme, then we prove that converges stably to the solution
of a linear SDE driven by a matrix-valued Brownian motion, when
. In the case , we show the
convergence of , and the limiting process is identified as the
solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate
of weak convergence is also deduced for this scheme. We also apply our approach
to the naive Euler scheme.Comment: Published at http://dx.doi.org/10.1214/15-AAP1114 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
On the one hand, the explicit Euler scheme fails to converge strongly to the
exact solution of a stochastic differential equation (SDE) with a superlinearly
growing and globally one-sided Lipschitz continuous drift coefficient. On the
other hand, the implicit Euler scheme is known to converge strongly to the
exact solution of such an SDE. Implementations of the implicit Euler scheme,
however, require additional computational effort. In this article we therefore
propose an explicit and easily implementable numerical method for such an SDE
and show that this method converges strongly with the standard order one-half
to the exact solution of the SDE. Simulations reveal that this explicit
strongly convergent numerical scheme is considerably faster than the implicit
Euler scheme.Comment: Published in at http://dx.doi.org/10.1214/11-AAP803 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
An adaptive scheme for the approximation of dissipative systems
We propose a new scheme for the long time approximation of a diffusion when
the drift vector field is not globally Lipschitz. Under this assumption,
regular explicit Euler scheme --with constant or decreasing step-- may explode
and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is
explicit and we prove that any weak limit of the weighted empirical measures of
this scheme is a stationary distribution of the stochastic differential
equation. Several examples are presented including gradient dissipative systems
and Hamiltonian dissipative systems
Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications
As a first step towards the numerical analysis of the stochastic primitive
equations of the atmosphere and oceans, we study their time discretization by
an implicit Euler scheme. From deterministic viewpoint the 3D Primitive
Equations are studied with physically realistic boundary conditions. From
probabilistic viewpoint we consider a wide class of nonlinear, state dependent,
white noise forcings. The proof of convergence of the Euler scheme covers the
equations for the oceans, atmosphere, coupled oceanic-atmospheric system and
other geophysical equations. We obtain the existence of solutions weak in PDE
and probabilistic sense, a result which is new by itself to the best of our
knowledge
Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme
In the present paper, we prove that the Wasserstein distance on the space of
continuous sample-paths equipped with the supremum norm between the laws of a
uniformly elliptic one-dimensional diffusion process and its Euler
discretization with steps is smaller than where
is an arbitrary positive constant. This rate is intermediate
between the strong error estimation in obtained when coupling the
stochastic differential equation and the Euler scheme with the same Brownian
motion and the weak error estimation obtained when comparing the
expectations of the same function of the diffusion and of the Euler scheme at
the terminal time . We also check that the supremum over of the
Wasserstein distance on the space of probability measures on the real line
between the laws of the diffusion at time and the Euler scheme at time
behaves like .Comment: Published in at http://dx.doi.org/10.1214/13-AAP941 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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