Random Integral Equation of the Volterra Type with Applications

In this paper we shall present some of the most general results which have been obtained to date concerning random integral equations of the Volterra type. Some results of Tsokos [4] are given for the random integral equation ( ) ; ∫ t x t w h t w k t w f x w d 0 ( ; ) ( ) ( , ; ) , ( ; ) = + τ τ τ τ (1.1) where t 0 ≥ and 1) w is appoint of Ω ; 2) h t w ( ; ) is the stochastic free term or free random variable defined for t 0 ≤ and w∈Ω ; 3) x t w ( ; ) is the unknown random variable for each t 0 ≥ 4) the stochastic kernel k t w ( , ; ) τ is defined for t 0 ≤ ≤ < ∞ τ and w∈Ω .

where t 0 ≥ and 1) w is appoint of Ω ; 2) h t w ( ; ) is the stochastic free term or free random variable defined for t 0 ≤ and w ∈ Ω ; 3) x t w ( ; ) is the unknown random variable for each t 0 ≥ 4) the stochastic kernel k t w ( , ; )

Introduction
In Section 2, we present some results of [4] for the random integral equation in Section 2, investigating the existence and uniqueness of random solution of Equation (1.1).The asymptotic behavior of the random solution and its stability properties are also considered.In Section 3 some applications of Equation (1.1) will be presented in the areas of telephone traffic theory and a generalization of the classical poincare'-Lyapunov theorem [3] Remark Below are some spaces which we use in our study a) ( ) This equation seems to be more general than any random Volterra integral equation which has been studied to date.The generality consists primarily in the choice of the stochastic kernel.In this Section we shall investigate the existence of a random solution and its uniqueness and asymptotic behavior and shall consider a number of special cases as corollaries of the main theorems.Finally, the stability properties of the random solution will be investigated.
To accomplish our objectives here we employ certain aspects of the methods of "admissibility theory" which has been utilized quite recently in the theory of deterministic integral equations by Corduneanu [4].

Existence and Uniqueness of a Random Solution
Let B and D be a pair of Banach spaces and T a linear operator.With respect to the study of this Section, we state the following Lemma which will be used in the main theorems.
Applying the condition of the theorem that

Some Special Cases
Now we shall derive some particular cases of theorem 3.2 choosing in a convenient manner the spaces B and D.
Consider that a g C space is a space of all continuous functions from 2 ( , , ) with the topology of uniform convergence on the interval [0, T] for any 0 T > and the norm of the stochastic kernel of the integral equation can be defined as follows: With 0 ( ) 0. P Ω = Theorem 4.1 Let us consider the random integral Equation (1.1) under the following conditions 1) there exist a number 0 A > and a continuous function ( ) 0 g t > such that 0 ( , ; ) ( ) , x t w dP w x t w g t Applying hypothesis (i) of the corollary, we have ( ) Utilizing the definition of norm as applied in theorem 4.5 to in equality (4.5) we have ( )

Asymptotic Stability of the Random Solution
With respect to the asymptotic behavior of the random solution of the stochastic integral Equation (1.1), we state and prove the following theorem, the objective of which is to investigate the possibility of the random solution being asymptotically exponentially stable.Theorem 5.1 Let us consider the stochastic integral Equation (1.1) under the following conditions 1) ≤ where ρ and β are positive numbers such that 0 β α < < .Then there exists a unique random solution of Equation ( 2 Using the definition of the norm on the g C space, inequality (5.4) can be written as In this Section we shall present some applications of the results of the previous Section.We shall first consider a generalization of the classical stability theorem of Poincaré and Lyapunov.We shall then study a stochastic integral equation arising in the theory of telephone traffic, arelated study of which was done by Fortet [1].

Generalization of Poincaré-Lyapunov Stability Theorem
As an example to illustrate our results, we shall generalized the classical stability theorem of Poincaré and Lyapunov (Tsokos [3]).That is consider the following random differential system ( ) where 1) ( ; ) x t w is the unknown 1 n × random vector; 2) ( ) A w is an n n × matrix whose elements are measurable functions; and 3) ( , ) f t x is, for t + ∈  and x ∈  , an 1 n × vector-valued function.Now we shall reduce the random differential system (6.1) to astochastic integral equation which will be a special case of the stochastic integral Equation (1.1).Myltiplying the random system (6.1) by , we have ( )

) ( , ( ; ) A w t A w t A w t e x t w A w e x t w e f t x t w
x t w e x t w A w e x t w dt dt Hence the stochastic differential system (6.1)reduces to the stochastic integral Equation (6.4), which is a special form of Equation (1.1).Now we state the following theorem.Theorem (6.1)Let us assume that the following conditions hold with respect to the stochastic integral Equation (6.4) 1) The matrix A(w) is stochastically stable, that is there exists an 0 are the characteristic roots of the matrix.2) ( , )  f t x is a continuous function from With ( , 0) 0 f t = and λ sufficiently small.Then there exists a unique random solution of the stochastic integral Equation (6.4) such that Proof To prove this result we want to prove that the pair of Banach spaces ( ) is admissible under conditions 1) and 2) with ( ) , and then apply theorem 4.1.Recall that the norm in the space ( ) x t w x t w dP w g t  Hence for ( ) ( ) , ( , , ) , ( , , ) , ( , , ) and the pair of Banach spaces ( ) is admissible.The rest of the proof is due to theorem 4.1.

A Problem in Telephone Traffic Theory
In this subsection we shall examine a stochastic integral equation arising in the study of telephone traffic.We shall describe the problem in detail and then apply corollary 4.3 to show existence of a unique random solution Consider a telephone exchange and suppose that calls arrive at the exchange at time instants , , t t  and as being independent of the state of the exchange, where the state of the exchange is the number of busy channels.
The number m of trunks or channels of the exchange is assumed to be finite and large, so that we approximate a continuous process.It is also assumed that any channel not being used may be utilized by an incoming call and that the holding time for a channel beings at the time instant that the call arrives at the exchange.
A conversation (or connection) is realized if a channel is not busy at the time a call arrives.If all channels are busy at the time t that a call arrives, then either the call is lost or a queueing problem develops.Only the first case will be considered here.Various problems have been studied in this situation.For example, the probability ( ) k P t that at time t, k of the m channels or busy has been examined in detail (Fortet [1]).We are concerned with the total number of "conversations" held (the number of busy channels) at time t which for each t is a random variable and may be described by as to chastic integral equation.Let ( ; )  x t w be the total number of conversations held at time t.
With respect to our aim here, we state and prove the following theorems.
[1]ma 3.1 Let T be a continuous operator from ( ) 2 , ( , , ) C C L P + Ω   into itself.If B and D are Banach spaces stronger than C C and the pair (B, D) is admissible with respect to T, then T is a continuous operator from B to D. Proof First we will prove that the operator T is closed form B to D. Let us consider the sequence ( ; ) n → ∞ .Now we must show that ( )( ; ) ( ; ) n Tx t w Tx t w → in C C .On the other hand ( ) ( ; ) ( ; ) n Tx t w y t w → in D, which implies that ( ) ( ; ) ( ; ) C .Therefore the operator T is closed.Then by the closed-graph theorem it follows that T is continuous operator from B to D.Remark since T is closed and continuous linear operator, it is also bounded (Yosida[1], p. 10).Then it Proof We must show that under condition (i) of the theorem the pair of Banach spaces( , ) + ∈  as long as ( ; ) h t w , λ and ( , 0) g C f t are small enough.g C , that is must be considered as random instants, so we denote the distribution function by A(t) on the time axis.For a call arriving at time t let the random variable ( ; ) H t w denote the holding time that is the length of time that a "conversation" is held for a call arriving at the exchange time t. are considered as being mutually independent for different times1 2 Suppose g(t) is any function satisfying condition 1) of corollary 4.3.
+ ∈  and w ∈ Ω .Hence we may write [ ] ∈ , a continuous bounded function for each w, is easily shown.Let w ∈ Ω and choose t + ∈  .
, then in order for condition 2) of corollary 4.3 to be satisfied, there must exist apposi- ≤, where g(t) satisfies the given conditions.