OPTIMAL STOPPING AND POWER OPTIONS
Albert Shiryaev
(joint work with Alexander A. Novikov)
Abstract
We consider a class of the following
optimal stopping problems:
To find the value functions
and optimal stopping times for functions 
, 
,
where 
, the sequence
is i.i.d. one with 
.
The main result: if 
, then the
stopping time
is optimal, 
is the maximal root of the equation
, where 
is Appell's polynomial defined from
decomposition
with 
.
For case n=1 the corresponding result with 
was
obtained by D.A.Darling, T.Liggett, H.M.Taylor ("Optimal stopping
for partial sums", Ann. Math. Statist. 43 (1972),
1363-1368). We give a new proof which works for any ,
also for gain functions g(x) of type 
, 
.
For case n=1 
what explains the answer
for this case. For 
where 
are semiinvariants of the
random variable 
. Generally,
if 
and 
.
