Grunnleggende konsepter
Designing explicit Milstein-type schemes for the Ait-Sahalia model with mean-square convergence.
Sammendrag
The article introduces novel explicit Milstein-type schemes for the Ait-Sahalia type model in mathematical finance. It addresses the challenges posed by superlinear growth and nonlinear drift, focusing on unconditionally positivity-preserving approximations. The proposed schemes achieve first-order strong convergence and mean-square error bounds without relying on high-order moment bounds. The theoretical analysis is supported by numerical experiments validating the findings.
Introduction
Stochastic differential equations (SDEs) applications.
Challenges in numerical approximation due to non-globally Lipschitz coefficients.
Preliminaries
Lemmas establishing moment bounds for exact solutions.
Monotonicity conditions for drift and diffusion terms.
Proposed Explicit Scheme
Introduction of a corrective mapping Φh.
Conditions ensuring well-posedness and positivity preservation.
Mean-Square Convergence Analysis
Error estimation through detailed calculations and inequalities.
Theorem proving expected order-one mean-square convergence.
Statistikk
E[|∆Wn|2] = h
E[(∆Wn)3] = 0
E[|∆Wn|4] = 3h^2