28 research outputs found
Strong convergence of an adaptive time-stepping Milstein method for SDEs with one-sided Lipschitz drift
We introduce explicit adaptive Milstein methods for stochastic differential
equations with one-sided Lipschitz drift and globally Lipschitz diffusion with
no commutativity condition. These methods rely on a class of path-bounded
timestepping strategies which work by reducing the stepsize as solutions
approach the boundary of a sphere, invoking a backstop method in the event that
the timestep becomes too small. We prove that such schemes are strongly
convergent of order one. This convergence order is inherited by an explicit
adaptive Euler-Maruyama scheme in the additive noise case. Moreover we show
that the probability of using the backstop method at any step can be made
arbitrarily small. We compare our method to other fixed-step Milstein variants
on a range of test problems.Comment: 20 pages, 2 figure
Strong Convergence of a GBM Based Tamed Integrator for SDEs and an Adaptive Implementation
We introduce a tamed exponential time integrator which exploits linear terms
in both the drift and diffusion for Stochastic Differential Equations (SDEs)
with a one sided globally Lipschitz drift term. Strong convergence of the
proposed scheme is proved, exploiting the boundedness of the geometric Brownian
motion (GBM) and we establish order 1 convergence for linear diffusion terms.
In our implementation we illustrate the efficiency of the proposed scheme
compared to existing fixed step methods and utilize it in an adaptive time
stepping scheme. Furthermore we extend the method to nonlinear diffusion terms
and show it remains competitive. The efficiency of these GBM based approaches
are illustrated by considering some well-known SDE models
First order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems
In this paper, we derive fully implementable first order time-stepping
schemes for McKean--Vlasov stochastic differential equations (McKean--Vlasov
SDEs), allowing for a drift term with super-linear growth in the state
component. We propose Milstein schemes for a time-discretised interacting
particle system associated with the McKean--Vlasov equation and prove strong
convergence of order 1 and moment stability, taming the drift if only a
one-sided Lipschitz condition holds. To derive our main results on strong
convergence rates, we make use of calculus on the space of probability measures
with finite second order moments. In addition, numerical examples are presented
which support our theoretical findings.Comment: 28 pages, 10 figure
Numerical contractivity preserving implicit balanced Milstein-type schemes for SDEs with non-global Lipschitz coefficients
Stability analysis, which was investigated in this paper, is one of the main issues related to numerical analysis for stochastic dynamical systems (SDS) and has the same important significance as the convergence one. To this end, we introduced the concept of -th moment stability for the -dimensional nonlinear stochastic differential equations (SDEs). Specifically, if and the -th moment stability constant \bar{K} < 0 , we speak of strict mean square contractivity. The present paper put the emphasis on systematic analysis of the numerical mean square contractivity of two kinds of implicit balanced Milstein-type schemes, e.g., the drift implicit balanced Milstein (DIBM) scheme and the semi-implicit balanced Milstein (SIBM) scheme (or double-implicit balanced Milstein scheme), for SDEs with non-global Lipschitz coefficients. The requirement in this paper allowed the drift coefficient to satisfy a one-sided Lipschitz condition, while the diffusion coefficient and the diffusion function are globally Lipschitz continuous, which includes the well-known stochastic Ginzburg Landau equation as an example. It was proved that both of the mentioned schemes can well preserve the numerical counterpart of the mean square contractivity of the underlying SDEs under appropriate conditions. These outcomes indicate under what conditions initial perturbations are under control and, thus, have no significant impact on numerical dynamic behavior during the numerical integration process. Finally, numerical experiments intuitively illustrated the theoretical results
Milstein schemes for delay McKean equations and interacting particle systems
In this paper, we derive fully implementable first order time-stepping
schemes for point delay McKean stochastic differential equations (McKean SDEs),
possibly with a drift term exhibiting super-linear growth in the state
component. Specifically, we propose different tamed Milstein schemes for a
time-discretised interacting particle system associated with the McKean
equation and prove strong convergence of order 1 and moment stability, making
use of techniques from calculus on the space of probability measures with
finite second order moments. In addition, we introduce a truncated tamed
Milstein scheme based on an antithetic multi-level Monte Carlo approach, which
leads to optimal complexity estimators for expected functionals without the
need to simulate L\'evy areas.Comment: 33 pages, 4 figure
Adaptive Milstein methods for stochastic differential equations
It was shown in [27] that the Euler-Maruyama (EM) method fails to converge with
equidistant timesteps in the strong sense to the solutions of stochastic differential
equations (SDEs) when either of the drift or diffusion coefficients is not globally
Lipschitz continuous. Higher-order methods or schemes that are developed based
on EM, e.g. Milstein method or EM with jumps, inherit the problem.
We introduce an explicit adaptive Milstein method for SDEs with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together
satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the
boundary of a sphere, invoking a backstop method in the event that the timestep
becomes too small. We prove that such schemes are strongly L2 convergent of order
one. This order is inherited by an explicit adaptive EM scheme in the additive noise
case. Moreover, we show that the probability of using the backstop method at any
step can be made arbitrarily small. We compare our method to other fixed-step
Milstein variants on a range of test problems.
Secondly, we introduce a jump-adapted adaptive Milstein (JAAM) method for
SDEs driven by Poisson random measure. With the conditions of drift and diffusion
coefficients remaining the same as for the adaptive Milstein method, and the jump
coefficient is globally Lipschitz continuous. The corresponding time-stepping strategies that we propose are hence path-bounded and also jump-adapted. We prove
the L2 strong convergence of order one for JAAM and compare its computational
efficiency with jump-adapted and fixed-step methods on test models
Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean--Vlasov SDEs and associated particle systems
We study a class of McKean--Vlasov Stochastic Differential Equations
(MV-SDEs) with drifts and diffusions having super-linear growth in measure and
space -- the maps have general polynomial form but also satisfy a certain
monotonicity condition. The combination of the drift's super-linear growth in
measure (by way of a convolution) and the super-linear growth in space and
measure of the diffusion coefficient require novel technical elements in order
to obtain the main results. We establish wellposedness, propagation of chaos
(PoC), and under further assumptions on the model parameters we show an
exponential ergodicity property alongside the existence of an invariant
distribution. No differentiability or non-degeneracy conditions are required.
Further, we present a particle system based Euler-type split-step scheme
(SSM) for the simulation of this type of MV-SDEs. The scheme attains, in
stepsize, the strong error rate in the non-path-space root-mean-square
error metric and we demonstrate the property of mean-square contraction. Our
results are illustrated by numerical examples including: estimation of PoC
rates across dimensions, preservation of periodic phase-space, and the
observation that taming appears to be not a suitable method unless strong
dissipativity is present.Comment: 34 pages, 5 figure
Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients: applications to financial models
A novel class of implicit Milstein type methods is devised and analyzed in
the present work for stochastic differential equations (SDEs) with non-globally
Lipschitz drift and diffusion coefficients. By incorporating a pair of method
parameters into both the drift and diffusion parts,
the new schemes can be viewed as a kind of double implicit methods, which also
work for non-commutative noise driven SDEs. Within a general framework, we
offer upper mean-square error bounds for the proposed schemes, based on certain
error terms only getting involved with the exact solution processes. Such error
bounds help us to easily analyze mean-square convergence rates of the schemes,
without relying on a priori high-order moment estimates of numerical
approximations. Putting further globally polynomial growth condition, we
successfully recover the expected mean-square convergence rate of order one for
the considered schemes with , solving general SDEs in
various circumstances. As applications, some of the proposed schemes are also
applied to solve two scalar SDE models arising in mathematical finance and
evolving in the positive domain . More specifically, the
particular drift-diffusion implicit Milstein method () is
utilized to approximate the Heston -volatility model and the
semi-implicit Milstein method () is used to solve the
Ait-Sahalia interest rate model. With the aid of the previously obtained error
bounds, we reveal a mean-square convergence rate of order one of the positivity
preserving schemes for the first time under more relaxed conditions, compared
with existing relevant results for first order schemes in the literature.
Numerical examples are finally reported to confirm the previous findings.Comment: 36 pages, 3 figure