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 $L_2$ 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

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 $\theta, \eta \in [0, 1]$ 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 $\theta \in [\tfrac12, 1]$, 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 $(0, \infty)$. More specifically, the particular drift-diffusion implicit Milstein method ($\theta = \eta = 1$) is utilized to approximate the Heston $\tfrac32$-volatility model and the semi-implicit Milstein method ($\theta =1, \eta = 0$) 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