Costs Study through a Diffusion Process of Pensions Funds Held with an Outside Financing Effort

Pensions funds not auto financed and systematically maintained with an outside financing effort are considered in this work. Representing the unrestricted reserves value process of this kind of funds, a time homogeneous diffusion process with finite expected time till the ruin is proposed. It is also admitted a financial tool that regenerates the diffusion, at some level with positive value every time it hits a barrier at the origin. Then the financing effort may be modeled as a renewal-reward process if the regeneration level is kept constant. The perpetual maintenance cost expected values evaluation and of the finite time period maintenance cost are studied. An application of this approach, when the unrestricted reserves value process behaves as a generalized Brownian motion process, is presented.


Introduction
The protection cost present value expectation for a non-autonomous pension's fund is considered in this work.Two contexts are considered: • The protection effort is perpetual, • The protection effort happens for a finite time period.
It is admitted that the unrestricted fund reserves behavior may be modeled as a time homogeneous diffusion process.Then a regeneration scheme of the diffusion to include the effect of an external financing effort is used.
In Gerber and Parfumi [2] a similar work is presented.A Brownian motion process conditioned by a particular reflection scheme was considered there.With fewer constraints, but in different conditions, exact solutions were then obtained for both problems.
The work presented in Refait [1], on asset-liability management aspects, motivated the use of the Brownian motion application example in that domain.
Part of this work is considered in Ferreira [9].Other works on this subject are Figueira [3] and Figueira and Ferreira [4].

Pensions Fund Reserves Behavior Stochastic Model
Be Xሺtሻ, t 0 the reserves value process of a pensions fund given by an initial reserve amount a, a 0 added to the difference between the total amount of contributions received and the total amount of pensions paid both up to time t.It is assumed that Xሺtሻ is a time homogeneous diffusion process, with Xሺ0ሻ ൌ a, defined by drift and diffusion coefficients μሺxሻ and σ ଶ ሺxሻ, respectively.
Call S ୟ the first passage time of Xሺtሻ by 0, coming from a.The funds to be considered in this work are non-autonomous funds.So EሾS ୟ ሿ ൏ ∞, for any a 0 ሺ2.1ሻ, that is: funds where the pensions paid consume in finite expected time any initial positive reserve and the contributions received.Then other financing resources are needed in order that the fund survives.
The condition (2.1) may be fulfilled for a specific diffusion process using criteria based on the drift and diffusion coefficients.In this context, here the work presented in Bhattacharya and Waymire [13], pg.418-422, is followed.So accept that PሺS ୟ ൏ ∞ሻ ൌ 1 if the diffusion scale function is where x is a diffusion state space fixed arbitrary point, fulfilling qሺ∞ሻ ൌ ∞.
Then the condition (2.1) is equivalent to pሺ∞ሻ ൏ ∞, where It is admitted that whenever the exhaustion of the reserves happens an external source places instantaneously an amount θ, θ 0 of money in the fund so that it may keep on performing its function.
The reserves value process conditioned by this financing scheme is denoted by the modification X ෙ ሺtሻ of Xሺtሻ that restarts at the level θ whenever it hits 0. As Xሺtሻ was defined as a time homogeneous diffusion, X ෙ ሺtሻ is a regenerative process.Call T ଵ , T ଶ , T ଷ , … the sequence of random variables where T ୬ denotes the n ୲୦ X ෙ ሺtሻ passage time by 0. It is obvious that the sequence of time intervals between these hitting times D ଵ ൌ T ଵ , D ଶ ൌ T ଶ െ T ଵ , D ଷ ൌ T ଷ െ T ଶ , … is a sequence of independent random variables where D ଵ has the same probability distribution as S ୟ and D ଶ , D ଷ , … the same probability distribution as S .

First Passage Times Laplace Transforms
Call f ୟ ሺsሻ the probability density function of S ୟ ሺrelated to D ଵ ሻ .The corresponding probability distribution function is denoted by F ୟ ሺsሻ.The Laplace transform of S ୟ is denoted φ ୟ ሺλሻ.
Consequently, the density, distribution and transform of S ሺrelated to D ଶ , D ଷ , … ሻ will be denoted by f ሺsሻ, F ሺsሻ and φ ሺλሻ, respectively.

Perpetual Maintenance Cost Present Value
Consider the perpetual maintenance cost present value of the pension's fund given by the random variable Vሺr, a, θሻ ൌ ∑ θe ି୰ ஶ ୬ୀଵ , r>0, where r represents the appropriate discount rate.Note that Vሺr, a, θሻ is a random perpetuity.What matters is its expected value which is easy to calculate using Laplace transforms.Since the T ୬ Laplace transform is

Finite Time Period Maintenance Cost Present Value
Define the renewal process Nሺtሻ ൌ supሼn: T ୬ tሽ , generated by the extended sequence T ൌ 0, T ଵ , T ଶ , … .The present value of the pensions fund maintenance cost up to time t is represented by the stochastic process Wሺt; r, a, θሻ ൌ ∑ θe ି୰ , Wሺt; r, a, θሻ ൌ 0 if Nሺtሻ ൌ 0. To calculate the expected value function of the process evaluation: w ୰ ሺt; a, θሻ ൌ EሾWሺt; r, a, θሻሿ , begin to note that it may be expressed as a numerical series.In fact, evaluating the expected value function conditioned by Nሺtሻ ൌ n, it is obtained Repeating the expectation: where γሺt, ξሻ is the probability generating function of Nሺtሻ.
Denote now the T ୬ probability distribution function by G ୬ ሺsሻ and assume G ሺsሻ ൌ 1, for s 0. Recalling that PሺNሺtሻ ൌ nሻ ൌ G ୬ ሺtሻ െ G ୬ାଵ ሺtሻ, the above mentioned probability generating function is G ୬ ሺtሻ (5.2).Substituting (5.2) in (5.1), w ୰ ሺt; a, θሻ is expressed in the form of the series Call the w ୰ ሺt; a, θሻ ordinary Laplace transform ψሺλሻ .The probability distribution function G ୬ ሺsሻ, of T ୬ , ordinary Laplace transform is given and performing the Laplace transforms in both sides of (5.3)This imposes that the transform φ ሺλሻ is defined in a domain different from the one initially considered, that is a domain including a convenient subset of the negative real numbers.

The Brownian Motion Example
Suppose that the diffusion process Xሺtሻ , underlying the reserves value behavior of the pensions fund, is a generalized Brownian motion process, with drift μሺxሻ ൌ μ, μ ൏ 0 and diffusion coefficient σ ଶ ሺxሻ ൌ σ ଶ , σ 0. Observe that the setting satisfies the conditions that were assumed above in this work.Namely μ ൏ 0 implies condition (2.1).Everything else remaining as previously stated, it will be proceeded to present the consequences of this particularization.In general it will be added a ሺ * ሻ to the notation used before because it is intended to use these specific results later.To obtain the first passage time ୟ Laplace transform, remember (3.1), it must be solved the equation: motion, with no constraints in what concerns the drift coefficient, conditioned by a reflection scheme at the origin.
A way to reach an expression for the finite time period maintenance cost present value, is starting by the computation of k ୰ * ሺθሻ, solving (5.10).This means to determine a positive number k satisfying e ି ౨ e ି షಓ ൌ 1 or K ୰ K ି ൌ 0. This identity is verified for the value of k Note that the solution is independent of θ in these circumstances.A simplified solution, independent from a and θ , for c ୰ * ሺa, θሻ was also obtained.Using (5.11) the result is Merging these results, (6.4) and (6.5), as in (5.9) it is observable that the asymptotic approximation for this particularization reduces to w ୰ * ሺt; a, θሻ ൎ v ୰ * ሺa, θሻ െ π ୰ ሺtሻ, where the function π ୰ ሺtሻ is, considering (6.4) and (6.5),

The Assets And Liability Behavior Representation
In this section it is presented an application of the results obtained above to an asset-liability management scheme of a pension's fund.Assume that the assets value process of a pensions fund may be represented by the geometric Brownian motion process Aሺtሻ ൌ be ୟାሺାஜሻ୲ାሺ୲ሻ with μ ൏ 0 and abρ μσ 0, where Bሺtሻ is a standard Brownian motion process.Suppose also that the fund liabilities value process performs such as the deterministic process Lሺtሻ ൌ be ୲ .
Consider now the stochastic process Yሺtሻ obtained by the elementary transformation of Aሺtሻ Yሺtሻ ൌ ln Aሺtሻ Lሺtሻ ൌ a μt σBሺtሻ.
This is a generalized Brownian motion process exactly as the one studied before, starting at a, with drift μ and diffusion coefficient σ ଶ .Note also that the passage time of the assets process Aሺtሻ by the mobile barrier T ୬ , the liabilities process, is the first passage time of Yሺtሻ by 0-with finite expected time under the condition, stated before, μ ൏ 0.
Consider also the pensions fund management scheme that raises the assets value by some positive constant θ ୬ , when the assets value falls equal to the liabilities process by the n ୲h time.This corresponds to consider modification A ഥ ሺtሻ of the process Aሺtሻ that restarts at times T ୬ when Aሺtሻ hits the barrier Lሺtሻ by the n ୲h time at the level LሺT ୬ ሻ θ ୬ .For purposes of later computations, it is a convenient choice the management policy where θ ୬ ൌ LሺT ୬ ሻ൫e െ 1൯, for some θ 0 ሺ7.1ሻ.
The corresponding modification Y ෩ ሺtሻ of Yሺtሻ will behave as a generalized Brownian motion process that restarts at the level ln ሺ ሻା ሺ ሻ ൌ θ when it hits 0 (at timesT ୬ ).
Proceeding this way, it is reproduced via Y ෩ ሺtሻ the situation observed before when the Brownian motion example was treated.In particular the Laplace transform in (6.1) is still valid.
Similarly to former proceedings, the results for the present case will be distinguished with the symbol ሺ#ሻ.It is considered the pensions fund perpetual maintenance cost present value, as a consequence of the proposed asset-liability management scheme, given by the random variable: V # ሺr, a, θሻ ൌ θ ୬ e ି୰ ஶ ୬ୀଵ ൌ b൫e െ 1൯e ିሺ୰ିሻ , r ρ ஶ ୬ୀଵ where r represents the appropriate discount interest rate.To obtain the above expression it was only made use of the Lሺtሻ definition and (7.1).Note that it is possible to express the expected value of the above random variable with the help of (6.2) as v ୰ # ሺa, θሻ ൌ b൫e െ 1൯ θ v ୰ି * ሺa, θሻ ൌ b൫e െ 1൯e ି ౨షಙ ୟ 1 െ e ି ౨షಙ ሺ7.2ሻ.
The results of section 6 with r replaced by r െ ρ may be combined as in (7.4) to obtain an asymptotic approximation.
Now, to obtain a common renewal equation from (5.7), it must be admitted the existence of a value k 0 such that න e ୩ୱ φ ሺrሻf ሺsሻds ൌ ஶ φ ሺrሻφ ሺെkሻ ൌ 1.

e
୩୲ Jሺtሻ ൌ e ୩୲ jሺtሻ න e ୩ሺ୲ିୱሻ Jሺt െ sሻe ୩ୱ ୲ φ ሺrሻf ሺsሻds from which, through the application of the key renewal theorem, it results ൌ φ ሺrሻφ ´ሺെkሻ , since e ୩୲ jሺtሻ is directly Riemann integrable.The integral in (5.8) may expressed in terms of transforms as