- Source: Milstein method
In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published it in 1974.
Description
Consider the autonomous Itō stochastic differential equation:
d
X
t
=
a
(
X
t
)
d
t
+
b
(
X
t
)
d
W
t
{\displaystyle \mathrm {d} X_{t}=a(X_{t})\,\mathrm {d} t+b(X_{t})\,\mathrm {d} W_{t}}
with initial condition
X
0
=
x
0
{\displaystyle X_{0}=x_{0}}
, where
W
t
{\displaystyle W_{t}}
denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time
[
0
,
T
]
{\displaystyle [0,T]}
. Then the Milstein approximation to the true solution
X
{\displaystyle X}
is the Markov chain
Y
{\displaystyle Y}
defined as follows:
Partition the interval
[
0
,
T
]
{\displaystyle [0,T]}
into
N
{\displaystyle N}
equal subintervals of width
Δ
t
>
0
{\displaystyle \Delta t>0}
:
0
=
τ
0
<
τ
1
<
⋯
<
τ
N
=
T
with
τ
n
:=
n
Δ
t
and
Δ
t
=
T
N
{\displaystyle 0=\tau _{0}<\tau _{1}<\dots <\tau _{N}=T{\text{ with }}\tau _{n}:=n\Delta t{\text{ and }}\Delta t={\frac {T}{N}}}
Set
Y
0
=
x
0
;
{\displaystyle Y_{0}=x_{0};}
Recursively define
Y
n
{\displaystyle Y_{n}}
for
1
≤
n
≤
N
{\displaystyle 1\leq n\leq N}
by:
Y
n
+
1
=
Y
n
+
a
(
Y
n
)
Δ
t
+
b
(
Y
n
)
Δ
W
n
+
1
2
b
(
Y
n
)
b
′
(
Y
n
)
(
(
Δ
W
n
)
2
−
Δ
t
)
{\displaystyle Y_{n+1}=Y_{n}+a(Y_{n})\Delta t+b(Y_{n})\Delta W_{n}+{\frac {1}{2}}b(Y_{n})b'(Y_{n})\left((\Delta W_{n})^{2}-\Delta t\right)}
where
b
′
{\displaystyle b'}
denotes the derivative of
b
(
x
)
{\displaystyle b(x)}
with respect to
x
{\displaystyle x}
and:
Δ
W
n
=
W
τ
n
+
1
−
W
τ
n
{\displaystyle \Delta W_{n}=W_{\tau _{n+1}}-W_{\tau _{n}}}
are independent and identically distributed normal random variables with expected value zero and variance
Δ
t
{\displaystyle \Delta t}
. Then
Y
n
{\displaystyle Y_{n}}
will approximate
X
τ
n
{\displaystyle X_{\tau _{n}}}
for
0
≤
n
≤
N
{\displaystyle 0\leq n\leq N}
, and increasing
N
{\displaystyle N}
will yield a better approximation.
Note that when
b
′
(
Y
n
)
=
0
{\displaystyle b'(Y_{n})=0}
(i.e. the diffusion term does not depend on
X
t
{\displaystyle X_{t}}
) this method is equivalent to the Euler–Maruyama method.
The Milstein scheme has both weak and strong order of convergence
Δ
t
{\displaystyle \Delta t}
which is superior to the Euler–Maruyama method, which in turn has the same weak order of convergence
Δ
t
{\displaystyle \Delta t}
but inferior strong order of convergence
Δ
t
{\displaystyle {\sqrt {\Delta t}}}
.
Intuitive derivation
For this derivation, we will only look at geometric Brownian motion (GBM), the stochastic differential equation of which is given by:
d
X
t
=
μ
X
d
t
+
σ
X
d
W
t
{\displaystyle \mathrm {d} X_{t}=\mu X\mathrm {d} t+\sigma XdW_{t}}
with real constants
μ
{\displaystyle \mu }
and
σ
{\displaystyle \sigma }
. Using Itō's lemma we get:
d
ln
X
t
=
(
μ
−
1
2
σ
2
)
d
t
+
σ
d
W
t
{\displaystyle \mathrm {d} \ln X_{t}=\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)\mathrm {d} t+\sigma \mathrm {d} W_{t}}
Thus, the solution to the GBM SDE is:
X
t
+
Δ
t
=
X
t
exp
{
∫
t
t
+
Δ
t
(
μ
−
1
2
σ
2
)
d
t
+
∫
t
t
+
Δ
t
σ
d
W
u
}
≈
X
t
(
1
+
μ
Δ
t
−
1
2
σ
2
Δ
t
+
σ
Δ
W
t
+
1
2
σ
2
(
Δ
W
t
)
2
)
=
X
t
+
a
(
X
t
)
Δ
t
+
b
(
X
t
)
Δ
W
t
+
1
2
b
(
X
t
)
b
′
(
X
t
)
(
(
Δ
W
t
)
2
−
Δ
t
)
{\displaystyle {\begin{aligned}X_{t+\Delta t}&=X_{t}\exp \left\{\int _{t}^{t+\Delta t}\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)\mathrm {d} t+\int _{t}^{t+\Delta t}\sigma \mathrm {d} W_{u}\right\}\\&\approx X_{t}\left(1+\mu \Delta t-{\frac {1}{2}}\sigma ^{2}\Delta t+\sigma \Delta W_{t}+{\frac {1}{2}}\sigma ^{2}(\Delta W_{t})^{2}\right)\\&=X_{t}+a(X_{t})\Delta t+b(X_{t})\Delta W_{t}+{\frac {1}{2}}b(X_{t})b'(X_{t})((\Delta W_{t})^{2}-\Delta t)\end{aligned}}}
where
a
(
x
)
=
μ
x
,
b
(
x
)
=
σ
x
{\displaystyle a(x)=\mu x,~b(x)=\sigma x}
The numerical solution is presented in the graphic for three different trajectories.
= Computer implementation
=The following Python code implements the Milstein method and uses it to solve the SDE describing geometric Brownian motion defined by
{
d
Y
t
=
μ
Y
d
t
+
σ
Y
d
W
t
Y
0
=
Y
init
{\displaystyle {\begin{cases}dY_{t}=\mu Y\,{\mathrm {d} }t+\sigma Y\,{\mathrm {d} }W_{t}\\Y_{0}=Y_{\text{init}}\end{cases}}}
See also
Euler–Maruyama method
References
Further reading
Kloeden, P.E., & Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. ISBN 3-540-54062-8.{{cite book}}: CS1 maint: multiple names: authors list (link)
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