arXiv subjects

M. V. Tretyakov

Publications and source records attributed to M. V. Tretyakov.

24 records · Page 2Linked to original sources

arXiv2012

A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications

A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. Some numerical tests are presented.

math.NA↗

arXiv2012

Application of simplest random walk algorithms for pricing barrier options

We demonstrate effectiveness of the first-order algorithm from [Milstein, Tretyakov. Theory Prob. Appl. 47 (2002), 53-68] in application to barrier option pricing. The algorithm uses the weak Euler approximation far from barriers and a special construction motivated by linear interpolation of the price near barriers. It is easy to implement and is universal: it can be applied to various structures of the contracts including derivatives on multi-asset correlated underlyings and can deal with various type of barriers. In contrast to the Brownian bridge techniques currently commonly used for pricing barrier options, the algorithm tested here does not require knowledge of trigger probabilities nor their estimates. We illustrate this algorithm via pricing a barrier caplet, barrier trigger swap and barrier swaption.

q-fin.CP↗

arXiv2011

Numerical integration of Heath-Jarrow-Morton model of interest rates

We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense using the general theory of numerical integration of SDEs. The proposed numerical algorithms are computationally highly efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.

q-fin.CP↗

arXiv2010

Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantage of this approach is its simplicity and universality. It works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus and we study only smooth test functions. However we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.

math.PR↗

arXiv2010

Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation

We propose new semi-implicit numerical methods for the integration of the stochastic Landau-Lifshitz equation with built-in angular momentum conservation. The performance of the proposed integrators is tested on the 1D Heisenberg chain. For this system, our schemes show better stability properties and allow us to use considerably larger time steps than standard explicit methods. At the same time, these semi-implicit schemes are also of comparable accuracy to and computationally much cheaper than the standard midpoint implicit method. The results are of key importance for atomistic spin dynamics simulations and the study of spin dynamics beyond the macro spin approximation.

cond-mat.mes-hall↗

arXiv2009

Langevin Thermostat for Rigid Body Dynamics

We present a new method for isothermal rigid body simulations using the quaternion representation and Langevin dynamics. It can be combined with the traditional Langevin or gradient (Brownian) dynamics for the translational degrees of freedom to correctly sample the NVT distribution in a simulation of rigid molecules. We propose simple, quasi-symplectic second-order numerical integrators and test their performance on the TIP4P model of water. We also investigate the optimal choice of thermostat parameters.

physics.comp-ph↗