<s>
Multilevel	O
Monte	O
Carlo	O
(	O
MLMC	O
)	O
methods	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
are	O
algorithms	O
for	O
computing	O
expectations	O
that	O
arise	O
in	O
stochastic	O
simulations	O
.	O
</s>
<s>
Just	O
as	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
,	O
they	O
rely	O
on	O
repeated	O
random	O
sampling	O
,	O
but	O
these	O
samples	O
are	O
taken	O
on	O
different	O
levels	O
of	O
accuracy	O
.	O
</s>
<s>
MLMC	O
methods	O
can	O
greatly	O
reduce	O
the	O
computational	O
cost	O
of	O
standard	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
by	O
taking	O
most	O
samples	O
with	O
a	O
low	O
accuracy	O
and	O
corresponding	O
low	O
cost	O
,	O
and	O
only	O
very	O
few	O
samples	O
are	O
taken	O
at	O
high	O
accuracy	O
and	O
corresponding	O
high	O
cost	O
.	O
</s>
<s>
The	O
goal	O
of	O
a	O
multilevel	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
to	O
approximate	O
the	O
expected	O
value	O
of	O
the	O
random	O
variable	O
that	O
is	O
the	O
output	O
of	O
a	O
stochastic	O
simulation	O
.	O
</s>
<s>
Each	O
of	O
the	O
expectations	O
is	O
then	O
approximated	O
by	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
resulting	O
in	O
the	O
multilevel	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
MLMC	O
can	O
be	O
considered	O
as	O
a	O
recursive	O
control	B-Algorithm
variate	I-Algorithm
strategy	O
.	O
</s>
<s>
The	O
first	O
application	O
of	O
MLMC	O
is	O
attributed	O
to	O
Mike	O
Giles	O
,	O
in	O
the	O
context	O
of	O
stochastic	O
differential	O
equations	O
(	O
SDEs	O
)	O
for	O
option	B-Algorithm
pricing	I-Algorithm
,	O
however	O
,	O
earlier	O
traces	O
are	O
found	O
in	O
the	O
work	O
of	O
Heinrich	O
in	O
the	O
context	O
of	O
parametric	O
integration	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
the	O
random	O
variable	O
is	O
known	O
as	O
the	O
quantity	O
of	O
interest	O
,	O
and	O
the	O
sequence	O
of	O
approximations	O
corresponds	O
to	O
a	O
discretization	B-Algorithm
of	O
the	O
PDE	O
with	O
different	O
mesh	O
sizes	O
.	O
</s>
<s>
Recent	O
extensions	O
of	O
the	O
multilevel	B-Algorithm
Monte	I-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
include	O
multi-index	O
Monte	O
Carlo	O
,	O
where	O
more	O
than	O
one	O
direction	O
of	O
refinement	O
is	O
considered	O
,	O
and	O
the	O
combination	O
of	O
MLMC	O
with	O
the	O
Quasi-Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
