# Cheap arbitrary high order methods for single integrand SDEs

Kristian Debrabant, Anne Kværnø

Research output: Contribution to journalJournal articleResearchpeer-review

### Abstract

For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order \$p_d\$ we obtain methods converging in the mean-square and weak sense with order \$\lfloor p_d/2\rfloor\$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.
Original language English Bit (Lisse) 57 1 153-168 0006-3835 https://doi.org/10.1007/s10543-016-0619-8 Published - 2017

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High-order Methods
Integrand
Arbitrary
B-series
Runge Kutta methods
Runge-Kutta Methods
Numerical Approximation
Mean Square
Calculus
Exact Solution

### Cite this

Debrabant, Kristian ; Kværnø, Anne. / Cheap arbitrary high order methods for single integrand SDEs. In: Bit (Lisse). 2017 ; Vol. 57, No. 1. pp. 153-168.
title = "Cheap arbitrary high order methods for single integrand SDEs",
abstract = "For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order \$p_d\$ we obtain methods converging in the mean-square and weak sense with order \$\lfloor p_d/2\rfloor\$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.",
author = "Kristian Debrabant and Anne Kv{\ae}rn{\o}",
year = "2017",
doi = "10.1007/s10543-016-0619-8",
language = "English",
volume = "57",
pages = "153--168",
journal = "BIT Numerical Mathematics",
issn = "0006-3835",
publisher = "Springer",
number = "1",

}

In: Bit (Lisse), Vol. 57, No. 1, 2017, p. 153-168.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Cheap arbitrary high order methods for single integrand SDEs

AU - Debrabant, Kristian

AU - Kværnø, Anne

PY - 2017

Y1 - 2017

N2 - For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order \$p_d\$ we obtain methods converging in the mean-square and weak sense with order \$\lfloor p_d/2\rfloor\$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.

AB - For a particular class of Stratonovich SDE problems, here denoted as single integrand SDEs, we prove that by applying a deterministic Runge-Kutta method of order \$p_d\$ we obtain methods converging in the mean-square and weak sense with order \$\lfloor p_d/2\rfloor\$. The reason is that the B-series of the exact solution and numerical approximation are, due to the single integrand and the usual rules of calculus holding for Stratonovich integration, similar to the ODE case. The only difference is that integration with respect to time is replaced by integration with respect to the measure induced by the single integrand SDE.

U2 - 10.1007/s10543-016-0619-8

DO - 10.1007/s10543-016-0619-8

M3 - Journal article

VL - 57

SP - 153

EP - 168

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 1

ER -