The aim of the paper is to apply the method proposed by Denuit, Genest and Marceau (1999) for deriving stochastic upper and lower bounds on the present value of a sequence of cash flows, where the discounting is performed under a given stochastic return process. The convex approximation provided by Goovaerts, Dhaene and De Schepper (1999) and Goovaerts and Dhaene (1999) is then compared to these stochastic bounds. On the basis of several numerical examples, it will be seen that the convex approximation seems reasonable.Value; Functions;

Cossette, H

Denuit, M

Dhaene, Jan

Marceau, E

English

Research Papers in Economics

KATHOLIEKE 
UNIVERSITEIT 
LEUVEN 
DEPARTEMENT TOEGEPASTE 
ECONOMISCHE WETENSCHAPPEN 
RESEARCH  REPORT 0001 
STOCHASTIC APPROXIMATIONS OF 
PRESENT VALUE FUNCTIONS 
by 
H.  COSSETTE 
M. DENUIT 
J. DHAENE 
E.  MARCEAU 
0/2000/2376/01 STOCHASTIC APPROXIMATIONS OF 
PRESENT VALUE FUNCTIONS 
HELENE COSSETTE 
Ecole d'Actuariat 
Universite Laval 
Sainte-Foy,  Quebec, 
Canada, GIK 7P4 
Helene.Cossette@act.ulaval.ca 
MICHEL DENUIT 
Institut de Statistique 
Universite Catholique de Louvain 
Voie du Roman Pays, 20 
B-1348 Louvain-la-Neuve, Belgium 
denuit@stat.ucl.ac.be 
JAN DHAENE 
Departement Toegepaste Economische Wetenschappen 
Faculteit Economische 
and Toegepaste Economische Wetenschappen 
Katholieke Universiteit Leuven 
Minderbroedersstraat, 5 
B-3000 Leuven, Belgium 
Jan.Dhaene@econ.kuleuven.ac.be 
ETIENNE MARCEAU 
Ecole d'  Actuariat 
U  niversite Laval 
Sainte-Foy,  Quebec, 
Canada, GIK 7P4 
Etienne.Marceau@act.ulaval.ca 
January 2000 Abstract 
The aim of this paper is to apply the method proposed by Denuit, Genest 
and Marceau (1999)  for  deriving stochastic upper and lower  bounds on the 
present value of a sequence of cash flows,  where the discounting is  performed 
under a given stochastic return process.  The convex approximation provided 
by  Goovaerts,  Dhaene and De  Schepper  (1999)  and Goovaerts and Dhaene 
(1999)  is  then compared  to these  stochastic bounds.  On  the basis  of sev-
eral numerical examples, it will be seen that the convex approximation seems 
reasonable. 
Key words and phrases:  Dependence,  Stochastic Dominance,  Stochastic 
Annuities 1  Introduction 
Let vt be the present value at time 0 of an amount of at paid at time t.  The 
stochastic discounted value at time 0 of payments of amount at made at times 
t =  1,2, ... ,n is then given by 
(1.1) 
Consider for instance an insurance company facing payments of amount at at 
times t  =  1,2""  ,n; the present value of these n  deterministic payments is 
given by (1.1). 
The  Vi's  involved  in  (1.1)  are  obviously  correlated,  so  that  the  conve-
nient independence assumption for the summands in  Zn  is  not realistic.  As 
a consequence,  an exact expression for  the cumulative distribution function 
of Zn  requires  the knowledge  of the joint distribution of the random vector 
(Vi, V2 , ... ,V n ),  which  is  in general not available.  Goovaerts,  Dhaene and 
De Schepper (1999)  recently proposed to circumvent this problem by approx-
imating Zn  by means of a random variable Zn  dominating the original Zn  in 
the convex  sense.  If we  denote by  FI, F2, ... ,Fn the respective distribution 
functions of Vi, 1/2, ... ,  Vn  involved in (1.1), Zn  is given by 
where  U  is  a  unit uniform random variable and the  Fi-1,S are the quantile 
functions  associated to the Fi's.  We  obviously have that EZn  =  EZn  and it 
can be shown that the inequalities 
E max{ Zn - d,O}  ::;  E max{ Zn - d, O}  (1.2) 
hold for any d ;::::  0 (that is,  Zn  is  smaller than Zn  in the convex order). 
Since Zn  precedes Zn  in the convex sense, the approximation Zn  is  consid-
ered as  less favorable by all the risk-averse decision-makers,  and the method 
is thus conservative.  Moreover, the cumulative distribution function of Zn en-
joys  an explicit expression  and is  particularly easy to handle.  On the basis 
of numerical illustrations performed in a situation where the exact cumulative 
distribution function of Zn  can be obtained, Goovaerts  et  al.  (1999)  showed 
that the cumulative distribution functions  of Zn  and Zn  seem  to be rather 
close. 
The problem of estimating the distribution of Zn  has been studied, among 
others, by Beekman and F\lelling (1991), De  Schepper and Goovaerts (1992), 
Dufresne  (1990),  Frees  (1990),  Parker  (1994c,1997),  De  Schepper,  Teunen, 
Goovaerts (1994) and Vanneste, Goovaerts and Labie (1994).  This paper aims 
to carryon with Goovaerts  et  al.'s  (1999)  approach by providing lower  and 
upper bounds on  Zn  in  the stochastic  dominance  sense,  using  the method 
proposed in Denuit, Genest and Marceau (1999).  This approach also provides 
upper and lower  bounds on the quantiles of Zno  In risk management, these 
1 quantiles correspond to the Value at Risk at different probability levels.  Such 
bounds cannot be obtained with  the aid  of  the convex  approximation  Zn. 
Indeed, we see from  (1.2) that the stop-loss premium of Zn  is an upper bound 
of the stop-loss premium of Zn;  more generally, E¢(Zn) is an upper bound for 
E¢(Zn) for  any convex  function ¢.  However,  there is  in general no relation 
between P[Zn  :::;  z]  and P[Zn  :::;  z]  (since indicator functions are not convex). 
Another purpose of this work is  to provide several numerical illustrations 
which enhance the practical interest of our approach.  In these illustrations, we 
will examine the position of the cumulative distribution function corresponding 
to the convex approximation Zn  in the admissible region delimitated by the 
stochastic bounds on  Zn.  As  a  byproduct of our results,  the error in the 
approximation of Zn  by Zn  can be evaluated (in other words, we get an upper 
bound for the Kolmogorov distance between Zn  and Zn). 
2  Stochastic bounds on Zn 
In this section, we recall how to build two functions  Fmin  and Fmax  such that 
the inequalities 
(2.1) 
hold, as  well as 
(2.2) 
To this end, we use the following result due to Denuit et al.  (1999, Proposition 
2).  Let FI, F2,' .. ,Fn be the respective  cumulative distribution functions  of 
Yl, V2, .•• ,V n.  Then, the cumulative distribution function  FZn  of Zn  =  VI + 
V2 + ... + Vn  is constrained by (2.1)  with 
Fmin(t) =  sup  max {t  P[V; < Vi]  - (n - I), o} , 
(VI,V2, ... ,vn)E}J(t)  i=1 
and 
Fmax(t) =  inf  min {t  Fi(Vi), I} , 
(VI,V2, •.• ,vn)E}J(t)  i=1 
where 
Note that Fmax  is  a  bona fide  cumulative distribution function, whereas Fm'm 
is  the left-continuous version of some  cumulative distribution function.  The 
bounds  in  (2.1)  and  (2.2)  are  the  best-possible  bounds  on  Zn  and  Zn  in 
the sense of stochastic dominance when we  know  the distribution functions 
2 Fl , F2, • •. ,F n ,  but no  assumption is  made on the dependence structure be-
tween the Ws.  Equivalently, these bounds hold for  all sums  (1.1)  with given 
cumulative distribution functions for  Vi, \12, ... ,  Vn . 
Closed form expressions for the bounds (2.1) can in general not be obtained 
for distributions of the Vi's  and one must resort to numerical evaluation.  For 
more details, see Denuit et al.  (1999). 
Now,  assume  we  have at our disposal some  partial knowledge  of the de-
pendence existing between  the Ws,  namely that there exists a multivariate 
cumulative distribution function G satisfying 
G(Vl' V2, ... ,vn) :::;  P[Vi  :::;  Vb \12  :::; V2,' .. , Vn  :::;  vnl  for all Vb V2,"  . ,Vn E JR, 
(2.3) 
and a joint decumulative distribution function H such that 
P[Vi > Vb \12  > V2,' .. , Vn  > vnl  ~  H(Vb V2,' .. ,vn) for all Vb V2,' .. ,Vn E  JR. 
(2.4) 
From Denuit et al.  (1999, Proposition 5), the inequalities 
sup  G(Xl,X2,"',xn ):::;Fzn(t):::;1- sup  H(XbX2,"',Xn), 
(X1.X2.··· .Xn)eE(t)  (XloX2.··· .xn)eE(t) 
(2.5) 
hold for  all t  E  JR.  The bOlmds  in  (2.5)  are obviously  more  accurate than 
those in (2.1). 
In the literature, several notions of positive dependence have  been intro-
duced in order to express the fact that large values of one of the components 
of a random vector tend to be associated with large values of the others.  In 
our context, one intuitively feels that in most situations the Ws mainly "move 
together" (i.e.  a large value of Vi  is usually followed  by a large value of Vi+l)' 
For the numerical illustrations in this paper, we  will  assume that (2.3)  and 
(2.4)  are satisfied with 
and 
n 
G(VI, V2,' .. ,vn) = II  Fi(Vi) 
i=l 
n 
H(Vl' V2,' .. ,vn) = II  (1- Fi(Vi)). 
i=l 
In such a case,  the Vi's  are said to be Positively Orthant Dependent  (POD, 
in short).  POD comes thus down to assume that the probability that all the 
Vi's  assume  "small"  values  (i.e.  Vi:::;  Vi,  i  =  1,2, ... ,n)  is  larger than the 
corresponding probability under the assumption that the Ws  are mutually 
independent.  The interpretation for  H  is  similar by substituting  "large"  for 
"small".  For more details, see, e.g., Szekli  (1995, pp.  144-145). 
3 3  Applications 
3.1  Stochastic annuities 
Let 6. be the force of interest at time s and let yt denote the force of interest 
accumulation function at time t, i.e. 
The random present value at time 0 of a payment of 1 monetary unit at time 
t is  given by exp( -yt), t  ~  O. 
As noticed by Parker (1994b), there are mainly two possible approaches to 
model the interest randomness, namely the modeling of yt and the modeling 
of 6  •.  In the first approach, we could let yt be the sum of a deterministic drift 
of slope 6 and a perturbation modeled by a Wiener process, i.e. 
(3.1) 
where u is a non-negative constant and {Wt,  t E JR+} is a standardized Brow-
nian motion.  In such a case,  lit is  log-normally distributed with parameters 
-6t and u2t.  This corresponds to the approach adopted by Goovaerts  et  al. 
(1999) who considered a discounted cash flow  Zn  of the form 
n 
Zn = L  exp( -6i - Xi), 
i=l 
where the Xi's are assumed to be normally distributed with mean 0 and vari-
ance iu2 ,  and 6 is the expected force of interest.  The convex upper bound Zn 
on Zn obtained by Goovaerts et  al.  (1999) is 
n 
Zn = L exp { -6i - uViiP-1 (U) } , 
i=l 
(3.2) 
where  iP  is the cumulative distribution function of a standard normal distri-
bution and U is  a random variable uniformly distributed on the lmit interval 
[0, 1].  The survival function of Zn  then follows from 
P[Zn > x]  =  1 - FZn (x) =  iP(lIx), 
with IIx  the root of the equation 
n 
L  (}i exp( -6i - ViUllx )  =  X. 
i=l 
Let us  now  investigate  the  accuracy  of the bounds  (2.1)  and  (2.5)  on  the 
distribution function  of Zn  in  the model  (3.1).  In  Figure  1,  one  sees  the 
4 ~~------~~~==~  /' 
'"  o 
'"  o 
.. 
o 
OJ  o 
/(" 
I  ! 
I  ' 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
"pp.bd~F'  I  ,-,-,  loworbd on Fa 
-- coovoxupperbd 
~L=====~===~==~-:~'!='-'-="-~"---'------,J 
10 
Figure 1:  Graph of the bounds (2.1)  and cumulative distribution function  of 
ZlO  for  (3.1) with 0 = 0.08 and (J" = 0.02. 
functions  F min  and Fmax involved in (2.1).  with in between the approximation 
FZn  of the unknown  FZn  for  n  =  10,  0 =  0.08  and  (J"  =  0.02.  Figure 3 is 
the analog for  n  =  20.  Comparing the cumulative  distribution function  of 
the convex approximation (3.2) with the stochastic bounds (2.1), we see from 
Figures  1 and 3 that (3.2)  lies  in  the very  middle  of the admissible  region 
bordered by F min and Fmax.  This indicates that (3.2)  could be reasonable.  In 
Figures 2 and 4,  we  further assume that the V;'s  are POD and we  computed 
the improved bounds furnished in  (2.1).  Only the lower bound got improved. 
As  it is  observed in Example 3 of Denuit  et  at.  (1999), both upper and lower 
bounds on the distribution of a sum of random variables got improved when 
the supports of the random variables are of the form  [ai, bJ  with -00 < ai < 
bi  < +00. If bi is equal to +00 as in Example 1 of Denuit et at.  (1999), only the 
lower bound will be improved with the assumption of POD. In our examples, 
the random variables are lognormally distributed with supports corresponding 
to  [0, +(0).  If,  as  in  Goovaerts and Dhaene  (1999),  Ot  is  defined  by  a  CIR 
model, then yt will be strictly positive, Vi = exp( - yt) will take values between 
o  and 1,  and therefore upper and lower bounds on the distribution of Zt will 
have been improved. 
A second approach to model interest randomness is  to model  Os.  For in-
stance, the force  of interest can be defined by the differential equation 
(3.3) 
5 ~~------~~/~==~ 
'"  o 
'"  o 
.. 
o 
OJ  o 
/r'· 
I  ! 
I  ! 
I  ! 
I  ! 
/  J 
I  " 
I 
I  .. 
I 
I  " 
/  Ii 
I 
I 
gL=====~==~~~~l=/~j~i  __ -,  ______  -.~ 
10 
Figure 2:  Graph of the bounds (2.5) and cumulative distribution  function  of 
ZlO  for  (3.1) with (j = 0.08 and (J" = 0.02. 
~r----~~/~============~ 
'"  o 
'"  o 
"!  o 
o 
o 
I  .,. 
f 
f 
f  I  : 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
I 
__  ....  /_._i  ........ , 
I 
"pp.bd~F.  I 
,.,.,  lov,oer bd on Fa 
-- CCflvmcupparbd 
~-----~-----~r------~ 
10  15  20 
Figure 3:  Graphs of the bounds (2.1) and cumulative distribution function  of 
Z20  for  (3.1) with (j = 0.08 and (J" = 0.02. 
6 ~r---~~~======~  /  .,. 
"!  o 
<D  o 
"  o 
I 
I 
/  : 
._._.  bwerbdcnFa  , /.,,'.,  1- _",dm" 
I  g  ::::  =';~:'~"POD 
:~==~~====!=;~1~1  ________ ~  __________ ~ 
10  15  20 
Figure 4:  Graphs of the bounds (2.5) and cumulative distribution function  of 
Z20  for  (3.1) with 0 =  0.08 and (J =  0.02. 
with  non-negative  constants  a  and  (J,  and with initial value  00  =  0  ~ 0; 
{Ot,  t  ~ O}  is  thus  an  Ornstein-Uhlenbeck  process.  The force  of interest 
accumulation function {yt,  t  ~  O}  is therefore a Gaussian process with mean 
function 
t f-+ /-It  =  ot + (00  _ 0) 1-exp(-at), 
a 
and auto  covariance (s, t)  f-+ Cov[Ys, yt]  == w(s, t), where 
(J2  (J2 
w(s, t)  =  2'  min(s, t) + -3  {-2 + 2exp( -as) + 2exp(-at) 
a  2a 
- exp( -a(t - s)) - exp( -a(t + s))}; 
see e.g.  Parker (1994a, Section 6).  Then, 
n 
Zn = L exp( -Yi), 
i=l 
where Yi  is  a Normal random variable with mean /-li  and variance w(i, i).  In 
such a case, the convex upper bound Zn follows from Goovaerts et al.  (1999): 
Zn = texp  {-/-li - VW(i,i)<I>-l(U)}, 
i=l 
7 ~r-----~~~~====~  /~  .:. 
'"  ,; 
'"  ,; 
.. 
,; 
OJ 
,; 
/  I  .,. 
I 
I 
I 
I 
J 
I 
J 
J 
J 
J 
J 
J 
J 
J 
J 
J 
J 
J 
J 
J 
I 
/  . 
/  , 
I 
"",.bdooF.  I  ,.,.,  lowerbdonFa 
-- conva.xupp"rbd 
~~====~==:-~~=-.-;.-=,-.-='-='_'_~i  ______ -r  __________  ~ 
10  15  20 
Figure 5:  Graphs of the bounds (2.1) and cumulative distribution function of 
ZlO  for  (3.3) with 0 = 0.06, 00 = 0.08,  0: = 0.3 and (J = 0.01. 
where U is a random variable uniformly distributed on the unit interval [0,1]. 
In Figure 5, you can see the bounds on the cumulative distribution function of 
ZlO in the model (3.3) with 0 = 0.06, 00  = 0.08, 0: = 0.3 and (J = 0.0l, together 
with the cumulative distribution function of ZlO'  Figure 7 is  the analog for 
n = 20.  The comments inspired from Figures 1 and 3 still apply.  In Figures 
6 and 8,  we  assumed that the Vi's  were  POD. Again, the improvement with 
POD is  moderate. 
3.2  Life insurance 
Consider a temporary life annuity issued to an individual aged x with curtate-
future-lifetime K  and denote P[k < K  ::::;  k + 1] = klq",  and P[K > n] = nP",' 
We assume that K is independent ofthe random discount factors Vi, 112, V3, .... 
The net single premium relating to this contract is  given by 
with 
o  if K =  0, 
ZK if K  = 1, ... ,n - 1, 
Zn if K  ~  n, 
8 qr-------~~~==~ 
!~/"-
CD 
ci 
co 
ci 
'"  ci 
I  ! 
I  i 
I  i 
I  i 
! I 
I  i 
I  E  !  ~Ir--::-.  -~-'-~-~- .. ----l, 
I  :! I  ,-...  ~  bwerlxtllllFawilhPOO 
I  Ii 
/  II 
)  I  Ii 
~ ~======;=====~~~==  ,)='_=._=l~.J_~ ____  ~ 
10 
Figure 6:  Graphs of the bounds (2.5) and cumulative distribution function of 
ZlO  for  (3.3) with 0 = 0.06,  00 = 0.08, a = 0.3 and (J" = 0.01. 
~~--~==~~====~  ((/ 
CD 
ci 
co 
ci 
... 
ci 
"!  o 
I I  Ir-::-·:-=-:..-~  ..  -::-~--'I 
/ /  I 
/  , 
~~====::==:_:~:_._;._.=_._:._:'M'_~! ____  ~  __________ ~ 
10  15  20 
Figure 7:  Graphs of the bounds (2.1) and cumulative distribution function of 
Z20  for  (3.3) with 0 = 0.06,  00 = 0.08, a = 0.3 and (J" = 0.01. 
9 10  15  20 
Figure 8:  Graphs of the bounds (2.5) and cumulative distribution function of 
Z20  for  (3.3) with 0 = 0.06,  00 = 0.08,  ex = 0.3 and (J" = 0.01. 
where Z is defined as in (1.1).  By conditioning on K, the net single premium 
relating to such a contract is 
n-l 
ax;;j = L E[Zklklqx + E[ZnlnPx. 
k=l 
The cumulative distribution function of aO -I is also obtained by conditioning 
Xjn 
onK: 
n-l 
P[a:;;:;j" ::;  yl = qx + L P[Zk ::;  ylklqx + P[Zn ::; ylnPx' 
k=l 
No explicit expression exists for P[a:;;j ::;  yl,  but we  use the approach devel-
oped above allows us to find stochastic dominance bounds on a:;;j'  In Figure 
9, we depicted the graph of the bounds on P[a:;;j ::;  yl  for an individual aged 
45 in the model (3.1)  with 0 = 0.08 and  (J"  =  0.02.  Figure 10 is the analog in 
model (3.3) with 0 =  0.06,  00 = 0.08, ex = 0.3 and (J" =  0.01.  For these numeri-
cal illustrations, we used the standard mortality table (Makeham model) given 
in Bowers et al.  (1996).  The bounds in Figures 9 and 10 give a good idea of the 
danger inherent to the stochastic interest rate combined with the stochastic 
mortality.  Let us mention that the convex approximation of Goovaerts et al. 
(1999)  also applies in this situation. 
10 '"  ci 
I  upper bdonPt(B10<y) 
bwerlx:lllllPr(Bl0q) 
10 
Figure 9:  Bounds on P[a:;Wf  :::;  y]  for x  =  45  and (3.1)  with 0 =  0.08 and 
a =  0.02. 
co 
ci 
~ 
I 
~ 
I 
I  cq 
o 
-.t; 
o  ~ 
- upperbdonPI(BIO<~) 
-- bwerbdooPrtBIO.:y) 
o 
ci  ~------r------.------r------.------~ 
Figure 10:  Bounds  on  P[a:;Wf  :::;  y]  for x 
00  =  0.08, a =  0.3 and a =  0.01. 
11 
10 
45  and (3.3)  with 0  0.06, Acknowledgements 
Partial funding in support of this work was  provided by the Natural Sciences 
and Engineering Research Council of Canada and the "Chaire en Assurance 
l'Industrielle-Alliance" . 
References 
[1]  Beekman,  J.A.,  and C.P.  Fuelling  (1991).  Extra randomness in  certain 
annuity models.  Insurance:  Mathematics & Economics 10, 275-287. 
[2]  Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, O.A. and C.J. Nesbitt 
(1996).  Actuarial Mathematics. Society of Actuaries, Itasca, Illinois. 
[3]  Denuit, M., Genest, C. and E. Marceau (1999). Stochastic bounds on sums 
of dependent risks.  Insurance:  Mathematics & Economics 25, 85-104. 
[4]  De  Schepper,  A.,  and  M.J.  Goovaerts  (1992).  Some  further results  on 
annuities certain with random interest.  Insurance:  Mathematics & Eco-
nomics 11, 283-290. 
[5]  De Schepper, A.,  Teunen, M.,  and M.J. Goovaerts (1994). An analytical 
inversion  of  a  Laplace transform related to annuity certain.  Insurance: 
Mathematics & Economics 14, 33-37. 
[6]  Dufresne, D.  (1990). The distribution of a perpetuity, with applications to 
risk theory and pension funding.  Scandinavian Actuarial Journal 33-79. 
[7]  Frees, E.W.  (1990).  Stochastic life  contingencies with solvency consider-
ations.  Transactions of the Society of Actuaries XLII, 91-148. 
[8]  Goovaerts,  M.J.,  and  Dhaene,  J.  (1999).  Supermodular  ordering  and 
stochastic annuities. Insurance:  Mathematics & Economics 24, 281-290. 
[9]  Goovaerts,  M.J.,  Dhaene,  J.  and  A.  De  Schepper  (1999).  Stochastic 
bounds for  present value function.  Mimeo. 
[10]  Parker, G.  (1994a). Limiting distribution of the present value of a port-
folio.  ASTIN Bulletin 24,47-60. 
[11]  Parker, G.  (1994b).  Two  stochastic approaches for  discounting actuarial 
functions.  ASTIN Bulletin 24, 167-181. 
[12]  Parker, G.  (1994c).  Stochastic analysis of a portfolio of endowment poli-
cies.  Scandinavian Actuarial Journal,  119-130. 
[13]  Szekli, R.  (1995).  Stochastic Ordering and Dependence in Applied Proba-
bility.  Lecture Notes in Statistics 97. Springer Verlag.  Berlin. 
12 [14]  Vanneste, M.,  Goovaerts, M.J., and E.  Labie (1994). The distribution of 
annuities. Insurance:  Mathematics f3  Economics 15, 37-48. 
13 

Actuarial Mathematics. Society of Actuaries,

An analytical inversion of a Laplace transform related to annuity certain.

Extra randomness in certain annuity models.

Limiting distribution of the present value of a portfolio.

Some further results on annuities certain with random interest.

Stochastic analysis of a portfolio of endowment policies.

Stochastic bounds for present value function.

Stochastic bounds on sums of dependent risks.

Stochastic life contingencies with solvency considerations.

Stochastic Ordering and Dependence in Applied Probability.

Supermodular ordering and stochastic annuities.

The distribution of a perpetuity, with applications to risk theory and pension funding.

Two stochastic approaches for discounting actuarial functions.

Stochastic approximations of present value functions.

https://lirias.kuleuven.be/bitstream/123456789/118589/1/OR_0001.pdf

Stochastic approximations of present value functions.

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