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A stochastic integral arising in discounting continuous cash flows and certain transformed characteristic functions
PERGAMON
Applied Mathematics Letters 13 (2000) 87-90
Applied Mathematics Letters www.elsevier.nl/locate/aml
A Stochastic Integral Arising in D i s c o u n t i n g C o n t i n u o u s Cash Flows and Certain Transformed Characteristic F u n c t i o n s T. A R T I K I S Department of Statistics and Insurance Science, University of Piraeus 80 Karaoli and Dimitriou Street, 185 34 Piraeus, Greece A. VOUDOURI Pedagogical Department, University of Athens 20 Hippocratous Street, 106 80 Athens, Greece
(Received March 1999; accepted April 1999) A b s t r a c t - - T r a n s f o r m e d characteristic functions are universally recognized as the most powerful tools for investigating distribution functions of complicated stochastic models. The paper is mainly devoted to the establishment of properties knd applications of a particular convolution model. More precisely, the paper derives the characteristic function of a convolution model based on a stochastic integral and provides applications of this model in discounting continuous cash flows. @ 2000 Elsevier Science Ltd. All rights reserved.
K e y w o r d s - - S t o c h a s t i c integral, Transformed characteristic function, Discounting.
1. I N T R O D U C T I O N Let { X ( t ) , t _~ 0} be a stochastic process with stationary, i n d e p e n d e n t i n c r e m e n t s a n d d e n o t e by "/(u) the characteristic f u n c t i o n of the i n c r e m e n t
X(t + 1) - X(t). It is k n o w n t h a t ./(u) is
infinitely divisible [1]. We assume t h a t
E[x(t)]-- la and V[X(t)] = ~2t, where - ~ < # < cxz a n d 0 < cre < oe. Moreover, we assume t h a t { X ( t ) , t > 0} is c o n t i n u o u s in p r o b a b i l i t y a n d t h a t its sample p a t h s are right c o n t i n u o u s a n d have left limits. T h e stochastic integral v =
c -~ dx(t),
f0 (:'
0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PIE S0893-9659(99)00214-1
(1.t)
Typeset by AAd,S-TEX
88
T. ARTIKIS AND A. VOUDOURI
where r > 0, exists in the sense of convergence in probability and is finite almost surely. Moreover, the distribution function of V is continuous and
5(u) =
exp
{/0
log 7
(ue-rt) dt ,
is its characteristic function [2]. The characteristic function given by
5(u) is easily
5(u) = exp { ~ jfo~ l°g tx (Y) dy}
(1.2) demonstrated to be (1.3)
The main purpose of the paper is to establish properties and applications in discounting continuous cash flows of the above stochastic integral.
2. C E R T A I N T R A N S F O R M A T I O N S OF C H A R A C T E R I S T I C F U N C T I O N S An important part of probability theory is devoted to the study of random variables. Many of the most important problems concerning random variables can be expressed in terms of distribution functions. The method of classical analysis provides an efficient approach to problems of this sort. It is frequently advisable to consider, instead of distribution functions, characteristic functions which are the Fourier transforms of distribution functions. It is now generally recognized that characteristic functions are the most powerful tools for investigating distribution functions. The uniqueness theorem, the convolution theorem, and the continuity theorem are the most important theorems which describe the connections between characteristic functions and distribution functions. These properties account for the significance of characteristic functions in probability theory [3]. During the last four decades there has been an increasing interest in transformations of characteristic functions, more precisely in operations which transform a given characteristic function into a new characteristic function. These transformations give some interesting information concerning the structure of characteristic functions. Naturally one investigates also the properties of the transformed characteristic function. The paper establishes a relationship between the characteristic function of the stochastic integral in (1.1) and the transformed characteristic functions defined below. Let ¢(u) be an arbitrary characteristic function. Then
to(u)=exp
-
e(x) d x d y
},
uET~
(2.1)
is the characteristic function of an infinitely divisible distribution having a finite second moment [4]. If ¢(u) belongs to a distribution function having a unique mode at the point 0 then Artikis [5,6] has proved that
u(u) = exp {- fo~'¢(y) y dy }
(2.2)
¢(y) dy j
(2.3)
and
w(u) =
exp { -
J0
are also characteristic functions of infinitely divisible distributions having a finite second moment. Sufficient conditions for embedding the characteristic functions t0(u), w(u), and 7(u) in several classes of infinitely divisible characteristic functions have also been established by Artikis [5-7].
Stochaitic [ntegrM 3.
A
THEORETICAL
8!) RESULT
The purpose of this section is to establish a relationship between the characteristic function of the stochastic integral in (1.1) and the transformed characteristic functions in (2.2) and (2.3). THEOREM 1. Let {X(t), t >_ 0} be a stodmstic process with stationary independent increments and E(X(t)) = O, V(X(t)) = ~r2t, where 0 < cr'2 < ~o. and w(u) is the characteristic thnction of the increment X(t + 1) - X(t). We assume that {X(t), t > 0} is continuous in probability and that its sample paths are continuous and have left ]imits. Let Y and S be independent infinitely divisible random variables, and independent of {X(t), t > 0}, with characteristic timctions wl/"(v,), r > O, and
p(u) =
}
do(jj)9(h/ ,
exp
(3.1)
respectively, where ~(u) is the characteristic function of a distribution t)mction having a m~ique mode at the point O. Then wl/"(u) = exp
it" and only i£ y 'Z
j
{/,, -u
)
. 4,/ c:,(~:)
(3.2)
. o o
:"'
axu)
(3.3)
+ s,
)
where a denotes equality in distribution. PROOF. Only the sufficiency condition will be proved since the necessity condition can be proved by reversing the argument. Using characteristic functions in (3.3) we obtain the functional equation
wl/~ (u ) = exl) { ! ffoU h'g w (Y) d?/} eXp We consider the function ,~,(~) -
log w(~.) - - ,
.£" d)(y)y dy } .
.,,. ~ ~ .
(3.4)
(3.s)
It
It is easily seen that 9(tt) is well defined on ~ . follows that '~/,(u) satisfies the condition
Moreover, fl'om the fact that E[X(t)]
~':(o) = o.
(/ it
(3.6)
From (3.4) and (3.5) we obtain the functi(mal equation
Multiplying both sides of (3.7) by 'u and then differentiating we obtain tile differential (~quation .~'b,.) = -,<:,(,).
(3.s)
[ntegrating in (3.7) with due regard to the condition in (3.6) we get that
,~01/ From (3.5) and (3.9) we obtain
(3.9)
90
T. ARTIKIS AND A. VOUDOURI
4.
APPLICATIONS
Discounting models have often proved invaluable within the financial decision making process. Such models are becoming increasingly more complex in their stochastic formulation and more sophisticated in their statistical objectives. Recent articles attempt to make exact inferences regarding the distribution of several stochastic discounting models [8-13]. In this section, we provide an application of the convolution model Y in stochastic discounting. We suppose that X(t) represents the rate of payment at time t of a continuous cash flow with infinite duration and r represents the force of interest, then the stochastic integral V in (1.1) represents the present value, as viewed from time zero, of this continuous cash flow. Moreover, we suppos e that S represents a single cash flow at time 0, then the convolution model Y in (3.3) represents the overall cash flow at time 0. This convolution model can play an important role in the determination of the economic value of a firm as a going concern [14-16]. In general, any analytical determination of the distribution of Y is extremely difficult. The main contribution of the paper is the establishment of sufficient and necessary conditions for embedding the characteristic function of Y into a wide class of transformed characteristic functions. REFERENCES 1. E. Lukacs, Stochastic Convergence, 2 nd edition, Academic Press, New York, (1975). 2. I. Harrison, Ruin problems with compounding assets, Stochastic Processes AppI. 5, 67-79, (1977). 3. W. Feller, An Introduction to Probability Theory and its Applications 2, 2 nd edition, Wiley, New York, (1966). 4. E. Lukacs, On a transformation of characteristic functions,, Portugaliae Mathematica 16, 31-35, (1957). 5. T. Artikis, Convex densities and self-decomposability, Serdica 9, 326-329, (1983). 6. T. Artikis, Constructing infinitely divisible characteristic functions, Archivum Mathematicum 19, 57-61, (1983). 7. T. Artikis, On the unimodality and self-decomposability of certain transformed distributions, Bull. Greek Math. Soc. 20, 3-9, (1979). 8. F. Delbaen and J. Haezendonck, Classical risk theory in an economic environment, Insurance- Mathematics and Economics 6, 85-116, (1987). 9. T. Artikis and A. Malliaris, Discounting certain random sums, Scand. Actuarial J. 99, 34-38, (1990). 10. T. Artikis and D.Jerwood, On the unimodality of present value distributions, Europ. J. Oper. Res. 52, 107-112, (1991). 11. T. Artikis, D. Jerwood and A. Voudouri, Analytical and simulation techniques for discounting binomial random sums, Mathl. Comput. Modelling 16 (9), 53-58, (1992). 12. T. Artikis, A.Voudouri and D. Jerwood, Analytical and computer simulation techniques for a stochastic model arising in discounting continuous uniform cash flows, Mathl. Comput. Modelling 18 (1), 9-16, (1993). 13. T. Artikis, D. Jerwood, J. Moshakis and A. Voudouri, A stochastic model for proactive risk management decisions, Mathl. Comput. Modelling 26 (7), 87-95, (1997). 14. T. Artikis, Properties and stochastic derivations of a characteristic function of class L, Demonstratio Mathematica 15, 207-213, (1982). 15. T. Artikis, On the distribution of a stochastic integral arising in the theory of finance, Studia Scient. Mathl. Hung. 23, 209-213, (1988). 16. E. Fama and M. Miller, The Theory of Finance, Dryden Press, IL, (1972).
Applied Mathematics Letters 13 (2000) 87-90
Applied Mathematics Letters www.elsevier.nl/locate/aml
A Stochastic Integral Arising in D i s c o u n t i n g C o n t i n u o u s Cash Flows and Certain Transformed Characteristic F u n c t i o n s T. A R T I K I S Department of Statistics and Insurance Science, University of Piraeus 80 Karaoli and Dimitriou Street, 185 34 Piraeus, Greece A. VOUDOURI Pedagogical Department, University of Athens 20 Hippocratous Street, 106 80 Athens, Greece
(Received March 1999; accepted April 1999) A b s t r a c t - - T r a n s f o r m e d characteristic functions are universally recognized as the most powerful tools for investigating distribution functions of complicated stochastic models. The paper is mainly devoted to the establishment of properties knd applications of a particular convolution model. More precisely, the paper derives the characteristic function of a convolution model based on a stochastic integral and provides applications of this model in discounting continuous cash flows. @ 2000 Elsevier Science Ltd. All rights reserved.
K e y w o r d s - - S t o c h a s t i c integral, Transformed characteristic function, Discounting.
1. I N T R O D U C T I O N Let { X ( t ) , t _~ 0} be a stochastic process with stationary, i n d e p e n d e n t i n c r e m e n t s a n d d e n o t e by "/(u) the characteristic f u n c t i o n of the i n c r e m e n t
X(t + 1) - X(t). It is k n o w n t h a t ./(u) is
infinitely divisible [1]. We assume t h a t
E[x(t)]-- la and V[X(t)] = ~2t, where - ~ < # < cxz a n d 0 < cre < oe. Moreover, we assume t h a t { X ( t ) , t > 0} is c o n t i n u o u s in p r o b a b i l i t y a n d t h a t its sample p a t h s are right c o n t i n u o u s a n d have left limits. T h e stochastic integral v =
c -~ dx(t),
f0 (:'
0893-9659/00/$ - see front matter @ 2000 Elsevier Science Ltd. All rights reserved. PIE S0893-9659(99)00214-1
(1.t)
Typeset by AAd,S-TEX
88
T. ARTIKIS AND A. VOUDOURI
where r > 0, exists in the sense of convergence in probability and is finite almost surely. Moreover, the distribution function of V is continuous and
5(u) =
exp
{/0
log 7
(ue-rt) dt ,
is its characteristic function [2]. The characteristic function given by
5(u) is easily
5(u) = exp { ~ jfo~ l°g tx (Y) dy}
(1.2) demonstrated to be (1.3)
The main purpose of the paper is to establish properties and applications in discounting continuous cash flows of the above stochastic integral.
2. C E R T A I N T R A N S F O R M A T I O N S OF C H A R A C T E R I S T I C F U N C T I O N S An important part of probability theory is devoted to the study of random variables. Many of the most important problems concerning random variables can be expressed in terms of distribution functions. The method of classical analysis provides an efficient approach to problems of this sort. It is frequently advisable to consider, instead of distribution functions, characteristic functions which are the Fourier transforms of distribution functions. It is now generally recognized that characteristic functions are the most powerful tools for investigating distribution functions. The uniqueness theorem, the convolution theorem, and the continuity theorem are the most important theorems which describe the connections between characteristic functions and distribution functions. These properties account for the significance of characteristic functions in probability theory [3]. During the last four decades there has been an increasing interest in transformations of characteristic functions, more precisely in operations which transform a given characteristic function into a new characteristic function. These transformations give some interesting information concerning the structure of characteristic functions. Naturally one investigates also the properties of the transformed characteristic function. The paper establishes a relationship between the characteristic function of the stochastic integral in (1.1) and the transformed characteristic functions defined below. Let ¢(u) be an arbitrary characteristic function. Then
to(u)=exp
-
e(x) d x d y
},
uET~
(2.1)
is the characteristic function of an infinitely divisible distribution having a finite second moment [4]. If ¢(u) belongs to a distribution function having a unique mode at the point 0 then Artikis [5,6] has proved that
u(u) = exp {- fo~'¢(y) y dy }
(2.2)
¢(y) dy j
(2.3)
and
w(u) =
exp { -
J0
are also characteristic functions of infinitely divisible distributions having a finite second moment. Sufficient conditions for embedding the characteristic functions t0(u), w(u), and 7(u) in several classes of infinitely divisible characteristic functions have also been established by Artikis [5-7].
Stochaitic [ntegrM 3.
A
THEORETICAL
8!) RESULT
The purpose of this section is to establish a relationship between the characteristic function of the stochastic integral in (1.1) and the transformed characteristic functions in (2.2) and (2.3). THEOREM 1. Let {X(t), t >_ 0} be a stodmstic process with stationary independent increments and E(X(t)) = O, V(X(t)) = ~r2t, where 0 < cr'2 < ~o. and w(u) is the characteristic thnction of the increment X(t + 1) - X(t). We assume that {X(t), t > 0} is continuous in probability and that its sample paths are continuous and have left ]imits. Let Y and S be independent infinitely divisible random variables, and independent of {X(t), t > 0}, with characteristic timctions wl/"(v,), r > O, and
p(u) =
}
do(jj)9(h/ ,
exp
(3.1)
respectively, where ~(u) is the characteristic function of a distribution t)mction having a m~ique mode at the point O. Then wl/"(u) = exp
it" and only i£ y 'Z
j
{/,, -u
)
. 4,/ c:,(~:)
(3.2)
. o o
:"'
axu)
(3.3)
+ s,
)
where a denotes equality in distribution. PROOF. Only the sufficiency condition will be proved since the necessity condition can be proved by reversing the argument. Using characteristic functions in (3.3) we obtain the functional equation
wl/~ (u ) = exl) { ! ffoU h'g w (Y) d?/} eXp We consider the function ,~,(~) -
log w(~.) - - ,
.£" d)(y)y dy } .
.,,. ~ ~ .
(3.4)
(3.s)
It
It is easily seen that 9(tt) is well defined on ~ . follows that '~/,(u) satisfies the condition
Moreover, fl'om the fact that E[X(t)]
~':(o) = o.
(/ it
(3.6)
From (3.4) and (3.5) we obtain the functi(mal equation
Multiplying both sides of (3.7) by 'u and then differentiating we obtain tile differential (~quation .~'b,.) = -,<:,(,).
(3.s)
[ntegrating in (3.7) with due regard to the condition in (3.6) we get that
,~01/ From (3.5) and (3.9) we obtain
(3.9)
90
T. ARTIKIS AND A. VOUDOURI
4.
APPLICATIONS
Discounting models have often proved invaluable within the financial decision making process. Such models are becoming increasingly more complex in their stochastic formulation and more sophisticated in their statistical objectives. Recent articles attempt to make exact inferences regarding the distribution of several stochastic discounting models [8-13]. In this section, we provide an application of the convolution model Y in stochastic discounting. We suppose that X(t) represents the rate of payment at time t of a continuous cash flow with infinite duration and r represents the force of interest, then the stochastic integral V in (1.1) represents the present value, as viewed from time zero, of this continuous cash flow. Moreover, we suppos e that S represents a single cash flow at time 0, then the convolution model Y in (3.3) represents the overall cash flow at time 0. This convolution model can play an important role in the determination of the economic value of a firm as a going concern [14-16]. In general, any analytical determination of the distribution of Y is extremely difficult. The main contribution of the paper is the establishment of sufficient and necessary conditions for embedding the characteristic function of Y into a wide class of transformed characteristic functions. REFERENCES 1. E. Lukacs, Stochastic Convergence, 2 nd edition, Academic Press, New York, (1975). 2. I. Harrison, Ruin problems with compounding assets, Stochastic Processes AppI. 5, 67-79, (1977). 3. W. Feller, An Introduction to Probability Theory and its Applications 2, 2 nd edition, Wiley, New York, (1966). 4. E. Lukacs, On a transformation of characteristic functions,, Portugaliae Mathematica 16, 31-35, (1957). 5. T. Artikis, Convex densities and self-decomposability, Serdica 9, 326-329, (1983). 6. T. Artikis, Constructing infinitely divisible characteristic functions, Archivum Mathematicum 19, 57-61, (1983). 7. T. Artikis, On the unimodality and self-decomposability of certain transformed distributions, Bull. Greek Math. Soc. 20, 3-9, (1979). 8. F. Delbaen and J. Haezendonck, Classical risk theory in an economic environment, Insurance- Mathematics and Economics 6, 85-116, (1987). 9. T. Artikis and A. Malliaris, Discounting certain random sums, Scand. Actuarial J. 99, 34-38, (1990). 10. T. Artikis and D.Jerwood, On the unimodality of present value distributions, Europ. J. Oper. Res. 52, 107-112, (1991). 11. T. Artikis, D. Jerwood and A. Voudouri, Analytical and simulation techniques for discounting binomial random sums, Mathl. Comput. Modelling 16 (9), 53-58, (1992). 12. T. Artikis, A.Voudouri and D. Jerwood, Analytical and computer simulation techniques for a stochastic model arising in discounting continuous uniform cash flows, Mathl. Comput. Modelling 18 (1), 9-16, (1993). 13. T. Artikis, D. Jerwood, J. Moshakis and A. Voudouri, A stochastic model for proactive risk management decisions, Mathl. Comput. Modelling 26 (7), 87-95, (1997). 14. T. Artikis, Properties and stochastic derivations of a characteristic function of class L, Demonstratio Mathematica 15, 207-213, (1982). 15. T. Artikis, On the distribution of a stochastic integral arising in the theory of finance, Studia Scient. Mathl. Hung. 23, 209-213, (1988). 16. E. Fama and M. Miller, The Theory of Finance, Dryden Press, IL, (1972).