281 research outputs found
An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients
We consider the approximation of stochastic differential equations (SDEs)
with non-Lipschitz drift or diffusion coefficients. We present a modified
explicit Euler-Maruyama discretisation scheme that allows us to prove strong
convergence, with a rate. Under some regularity and integrability conditions,
we obtain the optimal strong error rate. We apply this scheme to SDEs widely
used in the mathematical finance literature, including the
Cox-Ingersoll-Ross~(CIR), the 3/2 and the Ait-Sahalia models, as well as a
family of mean-reverting processes with locally smooth coefficients. We
numerically illustrate the strong convergence of the scheme and demonstrate its
efficiency in a multilevel Monte Carlo setting.Comment: 36 pages, 17 figures, 2 table
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
Complexity of randomized algorithms for underdamped Langevin dynamics
We establish an information complexity lower bound of randomized algorithms
for simulating underdamped Langevin dynamics. More specifically, we prove that
the worst strong error is of order , for
solving a family of -dimensional underdamped Langevin dynamics, by any
randomized algorithm with only queries to , the driving Brownian
motion and its weighted integration, respectively. The lower bound we establish
matches the upper bound for the randomized midpoint method recently proposed by
Shen and Lee [NIPS 2019], in terms of both parameters and .Comment: 27 pages; some revision (e.g., Sec 2.1), and new supplementary
materials in Appendice
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
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