Stochastic Numerics for Mathematical Physics

Mil'shtein, G.N.; Tretyakov, Michael V.

doi:10.1007/978-3-662-10063-9

Cited by 575 publications

(899 citation statements)

References 0 publications

Supporting

Mentioning

877

Contrasting

Unclassified

Order By: Relevance

“…Assumption 3.2 is similar to the one used in the standard theory of numerical integration of SDEs in the weak sense (see, e.g. [18]). As we will see in Section 4, Assumptions 2.1-2.3 and 3.1-3.2 guarantee weak convergence of the numerical method (3.15) to the solution of the auxiliary system of SDEs (3.10) with order h q .…”

Section: T-discretizationmentioning

confidence: 99%

“…The Monte Carlo error can be decreased by variance reduction techniques (see, e.g. [9,10,18,20] and references therein). In this paper we deal with the numerical integration error and numerical algorithms which are effective with regard to (t, T )-discretization.…”

Section: Remark 32mentioning

confidence: 99%

“…In the analysis of R 2 the key is to show that convergence ofF (t 0 , f 0 ; t * , T * ) toF (t 0 , f 0 ; t * , T * ) is uniform in ∆, which is the reason why we cannot just apply here the standard results of weak convergence of numerical methods for SDEs (see, e.g. [18]). The convergence theorem is proved under the assumption that the pay-off function G(z) in (2.12) is sufficiently smooth.…”

Section: T-discretization Errormentioning

confidence: 99%

“…As it is known (see, e.g. [18]), this is a harder task, and, due to complexity of stochastic schemes, one usually restricts themselves to using weak methods of orders 1 or 2. As a result, in practice we will take p ≥ q.…”

Section: Remark 43 (Relationship Betweenmentioning

confidence: 99%

“…To get fully discrete methods (discrete in both T and t), we approximate the obtained finite dimensional system of SDEs in the weak and meansquare senses using the general theory of numerical integration of SDEs (see, e.g. [18]). 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, i.e., the number of forward rates we need to approximate at each time moment t is significantly less than it is usually required in the case when the time grids for t and T coincide.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Krivko¹,

Tretyakov²

2013

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

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 highly computationally 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.

show abstract

Section: T-discretizationmentioning

confidence: 99%

Section: Remark 32mentioning

confidence: 99%

Section: T-discretization Errormentioning

confidence: 99%

Section: Remark 43 (Relationship Betweenmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Krivko¹,

Tretyakov²

2013

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

show abstract

On the forward‐backward‐in‐time approach for Monte Carlo solution of Parker's transport equation: One‐dimensional case

et al. 2016

View full text Add to dashboard Cite

The cosmic rays propagation inside the heliosphere is well described by a transport equation introduced by Parker in 1965. To solve this equation, several approaches were followed in the past. Recently, a Monte Carlo approach became widely used in force of its advantages with respect to other numerical methods. In this approach the transport equation is associated to a fully equivalent set of stochastic differential equations (SDE). This set is used to describe the stochastic path of quasi‐particle from a source, e.g., the interstellar space, to a specific target, e.g., a detector at Earth. We present a comparison of forward‐in‐time and backward‐in‐time methods to solve the cosmic rays transport equation in the heliosphere. The Parker equation and the related set of SDE in the several formulations are treated in this paper. For the sake of clarity, this work is focused on the one‐dimensional solutions. Results were compared with an alternative numerical solution, namely, Crank‐Nicolson method, specifically developed for the case under study. The methods presented are fully consistent each others for energy greater than 400 MeV. The comparison between stochastic integrations and Crank‐Nicolson allows us to estimate the systematic uncertainties of Monte Carlo methods. The forward‐in‐time stochastic integrations method showed a systematic uncertainty <5%, while backward‐in‐time stochastic integrations method showed a systematic uncertainty <1% in the studied energy range.

show abstract

Stochastic Differential Equations

Lord

2015

Digital Encyclopedia of Applied Physics

View full text Add to dashboard Cite

We give an introduction to Brownian motion and illustrate that although sample paths are continuous, they are nowhere differentiable. This leads to a discussion on white and colored noise. In Section 3.3, we introduce stochastic integration and look at the difference between Itô and Stratonovich integrals. We examine Itô stochastic differential equations (SDEs) in Section 3.4, giving some motivating examples, and use the Itô formula to find some exact solutions. We then consider Stratonovich SDEs and show how to convert between Itô and Stratonovich cases. We discuss numerical approximation of both Itô and Stratonovich SDEs and discuss the multilevel Monte Carlo method for weak approximation in Section 3.6. We conclude in Section 3.7 by looking at the link between SDEs and partial differential equations (PDEs) and introduce both the Fokker–Planck and backward Fokker–Planck equations. Throughout, the aim is to develop intuition by first examining simple, one‐dimensional examples before giving more general formulations and results. Also included are some basic algorithms for numerical approximation.

show abstract

Stochastic Numerics for Mathematical Physics

Cited by 575 publications

References 0 publications

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

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

On the forward‐backward‐in‐time approach for Monte Carlo solution of Parker's transport equation: One‐dimensional case

Stochastic Differential Equations

Contact Info

Product

Resources

About