49 research outputs found
Strong convergence rates for Euler approximations to a class of stochastic path-dependent volatility models
We consider a class of stochastic path-dependent volatility models where the
stochastic volatility, whose square follows the Cox-Ingersoll-Ross model, is
multiplied by a (leverage) function of the spot price, its running maximum, and
time. We propose a Monte Carlo simulation scheme which combines a log-Euler
scheme for the spot process with the full truncation Euler scheme or the
backward Euler-Maruyama scheme for the squared stochastic volatility component.
Under some mild regularity assumptions and a condition on the Feller ratio, we
establish the strong convergence with order 1/2 (up to a logarithmic factor) of
the approximation process up to a critical time. The model studied in this
paper contains as special cases Heston-type stochastic-local volatility models,
the state-of-the-art in derivative pricing, and a relatively new class of
path-dependent volatility models. The present paper is the first to prove the
convergence of the popular Euler schemes with a positive rate, which is
moreover consistent with that for Lipschitz coefficients and hence optimal.Comment: 34 pages, 5 figure
Strong order 1/2 convergence of full truncation Euler approximations to the Cox-Ingersoll-Ross process
We study convergence properties of the full truncation Euler scheme for the
Cox-Ingersoll-Ross process in the regime where the boundary point zero is
inaccessible. Under some conditions on the model parameters (precisely, when
the Feller ratio is greater than three), we establish the strong order 1/2
convergence in of the scheme to the exact solution. This is consistent
with the optimal rate of strong convergence for Euler approximations of
stochastic differential equations with globally Lipschitz coefficients, despite
the fact that the diffusion coefficient in the Cox-Ingersoll-Ross model is not
Lipschitz.Comment: 16 pages, 1 figur
Recommended from our members
Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions
This Mini-Workshop is devoted to regularity and numerical analysis of stochastic ordinary and partial differential equations (SDEs for both). The standard assumption in the literature on SDEs is global Lipschitz continuity of the coefficient functions. However, many SDEs arising from applications fail to have globally Lipschitz continuous coefficients. Recent years have seen a prosper growth of the literature on regularity and numerical approximations for SDEs with non-globally Lipschitz coefficients. Some surprising results have been obtained – e.g., the Euler–Maruyama method diverges for a large class of SDEs with super-linearly growing coefficients, and the limiting equation of a spatial discretization of the stochastic Burgers equation depends on whether the discretization is symmetric or not. Several positive results have been obtained. However the regularity of numerous important SDEs and the closely related question of convergence and convergence rates of numerical approximations remain open. The aim of this workshop is to bring together the main contributers in this direction and to foster significant progress
On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient
We survey recent developments in the field of complexity of pathwise
approximation in -th mean of the solution of a stochastic differential
equation at the final time based on finitely many evaluations of the driving
Brownian motion. First, we briefly review the case of equations with globally
Lipschitz continuous coefficients, for which an error rate of at least in
terms of the number of evaluations of the driving Brownian motion is always
guaranteed by using the equidistant Euler-Maruyama scheme. Then we illustrate
that giving up the global Lipschitz continuity of the coefficients may lead to
a non-polynomial decay of the error for the Euler-Maruyama scheme or even to an
arbitrary slow decay of the smallest possible error that can be achieved on the
basis of finitely many evaluations of the driving Brownian motion. Finally, we
turn to recent positive results for equations with a drift coefficient that is
not globally Lipschitz continuous. Here we focus on scalar equations with a
Lipschitz continuous diffusion coefficient and a drift coefficient that
satisfies piecewise smoothness assumptions or has fractional Sobolev regularity
and we present corresponding complexity results
The order barrier for the -approximation of the log-Heston SDE at a single point
We study the -approximation of the log-Heston SDE at the terminal time
point by arbitrary methods that use an equidistant discretization of the
driving Brownian motion. We show that such methods can achieve at most order , where is the Feller index of the
underlying CIR process. As a consequence Euler-type schemes are optimal for
, since they have convergence order for
arbitrarily small in this regime
The weak convergence order of two Euler-type discretization schemes for the log-Heston model
We study the weak convergence order of two Euler-type discretizations of the
log-Heston Model where we use symmetrization and absorption, respectively, to
prevent the discretization of the underlying CIR process from becoming
negative. If the Feller index of the CIR process satisfies , we
establish weak convergence order one, while for , we obtain weak
convergence order for arbitrarily small. We
illustrate our theoretical findings by several numerical examples