289 research outputs found
Convergence of numerical methods for stochastic differential equations in mathematical finance
Many stochastic differential equations that occur in financial modelling do
not satisfy the standard assumptions made in convergence proofs of numerical
schemes that are given in textbooks, i.e., their coefficients and the
corresponding derivatives appearing in the proofs are not uniformly bounded and
hence, in particular, not globally Lipschitz. Specific examples are the Heston
and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia
model with rational coefficient functions. Simple examples show that, for
example, the Euler-Maruyama scheme may not converge either in the strong or
weak sense when the standard assumptions do not hold. Nevertheless, new
convergence results have been obtained recently for many such models in
financial mathematics. These are reviewed here. Although weak convergence is of
traditional importance in financial mathematics with its emphasis on
expectations of functionals of the solutions, strong convergence plays a
crucial role in Multi Level Monte Carlo methods, so it and also pathwise
convergence will be considered along with methods which preserve the positivity
of the solutions.Comment: Review Pape
A Milstein scheme for SPDEs
This article studies an infinite dimensional analog of Milstein's scheme for
finite dimensional stochastic ordinary differential equations (SODEs). The
Milstein scheme is known to be impressively efficient for SODEs which fulfill a
certain commutativity type condition. This article introduces the infinite
dimensional analog of this commutativity type condition and observes that a
certain class of semilinear stochastic partial differential equation (SPDEs)
with multiplicative trace class noise naturally fulfills the resulting infinite
dimensional commutativity condition. In particular, a suitable infinite
dimensional analog of Milstein's algorithm can be simulated efficiently for
such SPDEs and requires less computational operations and random variables than
previously considered algorithms for simulating such SPDEs. The analysis is
supported by numerical results for a stochastic heat equation and stochastic
reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some
numerical simulations are remove
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
Simulation-based assessment of the stationary tail distribution of a stochastic differential equation
A commonly used approach to analyzing stochastic differential equations (SDEs) relies on performing
Monte Carlo simulation with a discrete-time counterpart. In this paper we study the impact of such a
time-discretization when assessing the stationary tail distribution. For a family of semi-implicit Euler
discretization schemes with time-step h > 0, we quantify the relative error due to the discretization, as a
function of h and the exceedance level x. By studying the existence of certain (polynomial and exponential)
moments, using a sequence of prototypical examples, we demonstrate that this error may tend to 0 or ¥.
The results show that the original shape of the tail can be heavily affected by the discretization. The cases
studied indicate that one has to be very careful when estimating the stationary tail distribution using Euler
discretization schemes
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
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
Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises
In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, cà dlà g, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived inL 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler-Maruyama approximation. Finally, simulations complete the pape
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