<s>
In	O
mathematics	O
,	O
the	O
Milstein	B-Algorithm
method	I-Algorithm
is	O
a	O
technique	O
for	O
the	O
approximate	O
numerical	B-General_Concept
solution	I-General_Concept
of	O
a	O
stochastic	O
differential	O
equation	O
.	O
</s>
<s>
with	O
initial	B-Algorithm
condition	I-Algorithm
,	O
where	O
stands	O
for	O
the	O
Wiener	O
process	O
,	O
and	O
suppose	O
that	O
we	O
wish	O
to	O
solve	O
this	O
SDE	O
on	O
some	O
interval	O
of	O
time	O
.	O
</s>
<s>
recursively	O
define	O
for	O
by	O
:	O
where	O
denotes	O
the	O
derivative	B-Algorithm
of	O
with	O
respect	O
to	O
and	O
:	O
are	O
independent	O
and	O
identically	O
distributed	O
normal	O
random	O
variables	O
with	O
expected	O
value	O
zero	O
and	O
variance	O
.	O
</s>
<s>
For	O
this	O
derivation	B-Algorithm
,	O
we	O
will	O
only	O
look	O
at	O
geometric	B-Language
Brownian	I-Language
motion	I-Language
(	O
GBM	O
)	O
,	O
the	O
stochastic	O
differential	O
equation	O
of	O
which	O
is	O
given	O
by	O
:	O
</s>
<s>
See	O
numerical	B-General_Concept
solution	I-General_Concept
is	O
presented	O
above	O
for	O
three	O
different	O
trajectories	O
.	O
</s>
