2,314 research outputs found
Convergence of the stochastic Euler scheme for locally Lipschitz coefficients
Stochastic differential equations are often simulated with the Monte Carlo
Euler method. Convergence of this method is well understood in the case of
globally Lipschitz continuous coefficients of the stochastic differential
equation. The important case of superlinearly growing coefficients, however,
has remained an open question. The main difficulty is that numerically weak
convergence fails to hold in many cases of superlinearly growing coefficients.
In this paper we overcome this difficulty and establish convergence of the
Monte Carlo Euler method for a large class of one-dimensional stochastic
differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in
the Foundations of Computational Mathematics 201
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Stochastic Differential Equations and Strict Local Martingales
In this thesis, we address two problems arising from the application of stochastic differential equations (SDEs). The first one pertains to the detection of asset bubbles, where the price process solves an SDE. We combine the strict local martingale model together with a statistical tool to instantaneously check the existence and severity of asset bubbles through the asset’s historical price process. Our approach assumes that the price process of interest is a CEV process. We relate the exponent parameter in the CEV process to an asset bubble by studying the future expectation and the running maximum of the CEV process. The detection of asset bubbles then boils down to the estimation of the exponent. With a dynamic linear regression model, inference on the exponent can be carried out using historical price data. Estimation of the volatility and calibration of the parameters in the dynamic linear regression model are also studied. When using SDEs in practice, for example, in the detection of asset bubbles, one often would like to simulate its paths using the Euler scheme to study the behavior of the solution. The second part of this thesis focuses on the convergence property of the Euler scheme under the assumption that the coefficients of the SDE are locally Lipschitz and that the solution has no finite explosion. We prove that if a numerical scheme converges uniformly on any compact time set (UCP) in probability with a certain rate under the globally Lipschitz condition, then when the globally Lipschitz condition is replaced with a locally Lipschitz one plus a no finite explosion condition, UCP convergence with the same rate holds. One contribution of this thesis is the proof of √n-weak convergence of the asymptotic normalized error process. The limit error process is also provided. We further study the boundedness for the second moment of the weak limit process and its running maximum under both the globally Lipschitz and the locally Lipschitz conditions. The convergence of the Euler scheme in the sense of approximating expectations of functionals is also studied under the locally Lipschitz conditio
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
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
Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient
We prove strong convergence of order for arbitrarily small
of the Euler-Maruyama method for multidimensional stochastic
differential equations (SDEs) with discontinuous drift and degenerate diffusion
coefficient. The proof is based on estimating the difference between the
Euler-Maruyama scheme and another numerical method, which is constructed by
applying the Euler-Maruyama scheme to a transformation of the SDE we aim to
solve
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