Affiliation:
1. Hitotsubashi University , Tokyo , Japan
Abstract
Abstract
The paper gives new results for the Milstein scheme of stochastic differential equations.
We show that
(i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields;
(ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense.
In particular, we prove
d
TV
(
X
T
ε
,
X
¯
T
ε
,
Mil
,
(
n
)
)
≤
C
ε
2
/
n
d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/n
and
d
TV
(
X
T
ε
,
X
¯
T
ε
,
EM
,
(
n
)
)
≤
C
ε
/
n
d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n
, where
d
TV
d_{\mathrm{TV}}
is the total variation distance,
X
ε
X^{\varepsilon}
is a solution of a stochastic differential equation with a small parameter 𝜀, and
X
¯
ε
,
Mil
,
(
n
)
\bar{X}^{\varepsilon,\mathrm{Mil},(n)}
and
X
¯
ε
,
EM
,
(
n
)
\bar{X}^{\varepsilon,\mathrm{EM},(n)}
are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively.
In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation.
Numerical examples demonstrate the validity of the method.
Funder
Japan Science and Technology Agency
Subject
Applied Mathematics,Statistics and Probability