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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
Note on q-extensions of Euler numbers and polynomials of higher order
In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order
twisted -extension of Euler polynomials and numbers, by using -adic
q-deformed fermionic integral on . By applying their generating
functions, they derived the complete sums of products of the twisted
-extension of Euler polynomials and numbers, see[13, 14]. In this paper
we cosider the new -extension of Euler numbers and polynomials to be
different which is treated by Ozden-Simsek-Cangul. From our -Euler numbers
and polynomials we derive some interesting identities and we construct
-Euler zeta functions which interpolate the new -Euler numbers and
polynomials at a negative integer. Furthermore we study Barnes' type -Euler
zeta functions. Finally we will derive the new formula for " sums products of
-Euler numbers and polynomials" by using fermionic -adic -integral on
.Comment: 11 page
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