Applied Numerical Mathematics 61 (2011) 160–169
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Applied Numerical Mathematics
www.elsevier.com/locate/apnum
Economical Runge–Kutta methods with strong global order one
for stochastic differential equations
F. Costabile, A. Napoli
∗
Department of Mathematics, University of Calabria, 87036 Rende (Cs), Italy
article info abstract
Article history:
Received 22 July 2009
Received in revised form 22 June 2010
Accepted 2 September 2010
Available online 6 September 2010
Keywords:
Stochastic differential equations
Stochastic Taylor expansion
Mean-square stability
Economical Runge–Kutta schemes for the numerical solution of Stratonovich stochastic
differential equations are proposed. The methods have strong global order 1. Numerical
stability is studied and some examples are presented to support the theoretical results.
© 2010 IMACS. Published by Elsevier B.V. All rights reserved.
1. Introduction
Stochastic differential equations (SDEs) are used for the description of many real-life phenomena in different fields,
including biology and physics, population dynamics, economics and finance. In fact if more realistic models are requested,
stochastic effects need to be taken into account.
Unfortunately, in many cases analytic solutions of SDEs are not available, thus numerical methods are needed to approx-
imate them.
In this paper we consider the scalar autonomous Stratonovich SDE [8]
dy(t ) = a
(
y(t )
)
dt + b
(
y(t )
)
◦ dW
t
, t
0
t T ,
y(t
0
) = y
0
, (1)
where W ={W
t
, 0 t T } denotes a standard Wiener process. The functions a and b are the drift and the diffusion
coefficients respectively, and we assume that they are defined and measurable in [t
0
, T ]× R and satisfy both Lipschitz and
linear growth bound conditions in y. These requirements ensure existence and uniqueness of solution of the SDE (1).
The integral formulation of (1) can be written as
y(t ) = y
0
+
t
t
0
a
(
y(s)
)
ds +
t
t
0
b
(
y(s)
)
◦ dW (s), (2)
where the first integral is a regular Riemann–Stieltjes integral and the second one is a Stratonovich stochastic integral with
respect to the Wiener process W (t ).
*
Corresponding author.
E-mail addresses: costabil@unical.it (F. Costabile), a.napoli@unical.it (A. Napoli).
0168-9274/$30.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.apnum.2010.09.001