Further Results for General Financial Equilibrium Problems via Variational Inequalities

This paper is the sequel of the previous papers [1] and [2]. More precisely, we study the regularity of the solutions of the evolutionary variational inequality governing the general financial evolutionary problem. Specifically we obtain that such a solution is continuous and Lipschitz continuous with respect to time and we illustrate the achieved result through numerical examples. Moreover the numerical examples enables us to understand the behaviour of the financial equilibrium and the impact of the components of the model on the financial equilibrium.


Introduction
In the previous papers [1] and [2], a general equilibrium model of financial flows and prices is considered.The model is assumed evolving in time.The equilibrium conditions are considered in dynamic sense and the governing variational inequality formulation is presented.Precisely, the variational inequality we are working with is the following one:   =1 Find , , :

x t y t t r t x t x t x u t x t y t t r t h t y t y t t y t x t h t y t F t r t r t
is the set of feasible assets and liabilities for each sector i P 1, , i  given by : 0 , 0, , mn n A T ned as: , ,

t x y A t v t r t x x y t h t r t y t x t h t y t
, , 0, , : 0, 0, a.e. in 0, , , , a.e. in 0, , . in 0, ,

y r L T x t y t T x t s t y t l t T i m r t r t r t T
then variational inequality (1) becomes the problem: .
Variational inequality (1) proved to be a very useful ium of an economy evolving in time and in the previous papers [1] and [2] the authors provided an interesting and useful output as the deficit formula, a general balance la important formulas: 1) Deficit formula tool which enables us to study the financial equilibr w and the liability formula which could be of great importance for the theory of equilibrium problems evolving in time.For the reader's convenience we recall such , 1 , , , where     and     are the Lagrange variables associated to the price bounds: 0, 1, , .0, j j j j j j t r t r t j n t r t r t The meaning of      is that it represents the deficit per unit wher is the positive surplus per unit; 2) Balance law ; These suggested formulas could be of topical utility for the management of the world economy and to thi aim in [2] the authors give some suggestions for th achievement of the world financial equilibrium and for finding the necessary way to follow in order to reach an eme ec indin suggestions: Pro .The value of these propositions can be realized by taking into account that the increase in prices indicated in Pr over, in [2] an "Evaluation Index", that we deno e evaluation of an economy, given by oposition 1.4 has been forecasted many months before (June 2011) it happened in our days.More ted by   E t , has been introduced as a useful and simple tool for th where we set 1 We remark that if   E t is greater or equal to 1 the evaluation e financial equilibrium is positive (better if   E t is proximal to 1) whereas if  of th  E t is less than uation of the ial equilibrium is negative.The aim of this paper is to provide new theoretical and numerical results about solutions of financial equilibrium problems.In particular, we will prove a continuity result with respect to time of the solution, namely: .1 Let 0, , ly monotone map, namely there exists 0 be a strong Then variational inequality (1) admits a unique continuous solution.
Furthermore, we prove t e following Lips h chitz contin Theorem 1.2 L namely there exists 0 A be strongly monotone, ; Lipschitz continuous with respect to , namely there ex , , hat, for and Lipschitz continuous with respect to t , namely there where rium problems ty for ed, i m so ns can be obtained making use of a modified version of the algorithms (see for instances [3][4][5][6][7]) worth remarking that the Lipschitz continuity allo to calculate the error in approximating the solution.
der to clearly illust retical results, some significant examples are provided and, in such a way, the impact that the components of the model have on the equilibrium are highlighted.
It is worth mensioning that even in this case variat uilibr of the solution to the variat .

It is ws us
In or rate theo ional inequalities are able to express the time-dependent equilibrium conditions.Then, applying delicate tools of nonlinear analysis (see [8][9][10][11]), it is possible to prove existence results and qualitative analysis.For other economic problems where the time plays an important role we refer to the papers devoted to the Walrasian equilibrium problem [12][13][14][15], to the oligopolistic market equilibrium problem [16,17], to the weighted traffic eq ium problem [18,19], and to [20].The paper is organized as follows.In Section 2 we present the general financial model.In Section 3 we study the continuity results ional inequality which characterizes the financial model.In Section 4 we provide a Lipschitz continuity result for the solution.Finally, in Section 5 we propose a numerical examples from which we deduce that the solution, computed by means of the direct method (see [21]), is Lipschitz continuous.

The Model
We consider a financial economy consisting of m sec- tors, with a typical sector denoted by i , and of n in- struments, with a typical financial instrument denoted by j , in the time interval   .
a.e. in   0,T is that to each investor a minimal price j r for the assets held in the in- strument j is guaranteed, whereas each inv que estor is re sted to pay for the liabilities not less than the minimal price   . Analogously each investor cannot obtain for an asset a price greater than j r and as a liability the price cannot exceed the maximum price   The set of feasible instrument prices is . In order to de optimal termine for each sector the me liabisider, as usual, the sk ocess of optimization of e onomy, namely the desi o maximize the value of the asset holdings and to minimize the value of liabilities.Then, we introduce the tility function i re t u composition of instru nts held as assets and as lities, we con influence due to riaversion and the pr ach sector in the financial ec repr n re of the risk of the financial agent and  represents the value of the difference between the asset holdings and the value of liabilities.We suppose that the sector's utility function and In or determine the equilibrium prices, we estab-io r to lish the equilibrium condition which expresses the bration of the total assets, the total liabilities a on of financial transactions per unit de equilind the porti j F employed to cover the expenses of the financial in tions including possible dividends, as in [1].He he equilibrium condition for the price stitu nce, t j r of in the following: an In other words, the prices are determined taking into account the amount of the supply, the dem d of an instrument and the charges , namely if there is an actual supply excess of an instrument as assets and of the charges j F in the economy, then its price must be the floor price.If the price instrument is greater than of an   j r t , bu not at th , then the market of that instrum must clear.Finally, if there is an actual demand cess of an in nt as liabilities and of the charges t ent ex e ceiling strume j F in the ec y, then the price must be at the ceiling.Now, we can give different but equivalent equilibrium conditions, each of which is useful to illustrate particular features of the equilibrium.

Definition 2.1 A vector of sector assets, liabilities and instrument prices
onom and equalities where , are Lagrange tions , 0 The same occurs for the liabilities and the of ( 6) is already illustrated.The functions and are Lagr ns associ meaning dynamic financial equili rium if and nly if it satisfies variational inequality (1).Mo ver, we recall the result about Lagrang , , 0, , be a olu o variational s tion t inequality (1).Then there exist

Continuity Results for Financial Equilibrium Solutions
In order to show the continuity result for the financial equilibrium problem, first of all, let us recall the wellknown property of set convergence due to K. Kuratowski (see [28]), that is a generalization of the classical Hausdorff definition of a metric for the space of closed sub-sets of a (compact) metric space.Let be a metric space and let   where x  converging to x in X, such that n x lies in n K for all n   , then the limit x belongs to K .Now, let us prove that the set of feasible vectors ce in Kur-satisfies the property of the set convergen atowski's sense. , , Let us verify that , .Taking into account that  min , there exist two index Then we can consider a sequence such that for where denotes the Hilbertian projection on , Then the first condition has been shown.For the second one, let Passing to the limit as in ( 15), ( 16) and ( 17), we obtain

Then
. The claim hieved.□For what follows, it is convenient to recall that variational inequality (1) can be rewritten in the equivalent parameterized form: where the constraint set ,t , is a closed and nonem convex pty subset of   .s been proved in [2] (see Section 6).
Taking into account the general continuity resu for solutions to parameter variational inequalities in reflexive Banach spaces (see [29], Theorem 4.1) and Proposition, we obtain Theorem 1.1 of Section 1.
T ma namely there e be a strongly monotone p, xists 0 .
Then variational inequality (1) a its a u tinuous solution.

Lipschitz Continuity Resu
The aim of this section is to provide a Lip tinuity result for the financial equilibrium so th eneral re ns to the parameterized variational inequality (18).More precisely, the followin result holds (see [30] , denotes the projection onto the set .Then, the unique solution , 1 2 t t  , the fo estim llowing ate holds: , sup For the sake of simplicity, we set Before applying vious result to o financial equilibrium p projec penden nstraint set the pre ur dynamic roblem, it is necessary to estimate the variation rate of tions onto time-de t co Making us of Proposition 1 in [3 e can show that e 0], w assuming  ,   itz co uous w describing the problem.It is useful to note that can be rewritten as the Cartesian product of th ng set: where T   be two Lipschitz continuous functions and let

 
, : 0, n r r T   be two Lipschitz continuous functions.Let z be an arbitrary point in mn   .Then it results to be where is the positive constant as in (3).As a consequence, it results .Moreover, let 1 where    Let us choose as the feasible set Let us consid sectors are the 1 er an economy with two sectors and two financial instruments, as shown in Figure 1, but this setting is not restrictive since we can consider a general economy by an iteration of this significan assume that the v variance matrices case, and ariance-co he two following: , 0 , a .e .in 0 0 4 3, a.e. in 0,1 , 0 7 8, a.e. in 0,1 .
The variational inequality which expresses the financial equilibrium conditions, becomes Following the direct method (see [21]) in order to find the solution, we can derive from the constraints of the convex set the values of some variables in terms of the others, mely we obtain, a.e. in variational inequality ( 22) can be expressed in the equivalent form: As the first step of the direct method suggests, let us search solutions obtained from the system We get the values of the variables ij x and in terms of For the sake of simplicity, we assume In the following, we study various examples of financial equilibrium for which the deficit (namely In particular, we assume     we obtain that they are negative and positive respectively, as it can be verified in Figure 4 where the graphs of the numerators are represented.
Let us remember that, by virtue of Theorem 2.2, the following relationships are satisfied 1, 2 j   , from (23), we obtain So, from (24), we get Whereas, for the instrument we have minimal price and the deficit is given by 2 j  , the economy has a positive average evaluation.The same situation happens if Now, we would like to calculate the Ev r this financial problem.More precisely, we aluation Index have fo As a consequence, the Evaluation Index is given by: In the interval Now let us consider the case where the values of is solution of the variational inequality in the interval   , we obtain So, from (24), we get In this situation, we can assert that, in the interval , t t , and the for the instrument , we have minimal price deficit is given by is solution of the variational inequality in the interval we obtain that they are both positive, as it can be verified in Figure 9 where the graphs of the numerators are represented.
We observe that, in our case so, from (23), it follows that for any Then, from (24), we obtain

Now, taking into account Proposition 4
Hence, applying Theorem 4.1, we get the following result.monotone (with const  ), Lipschitz continuous with respect to x (with constant  ),

2 F
.e.in 1 , means of va se component especially for what re ctor's assessment of the standard devi s for each instrument.We will see that the solutions f the examples are regular and we illustrate the impact f the s of the m F and and
As regards the calculation of the Evaluation Index, we have in   ,

τ 11 . Figure 2. Bound functions of tions of τ 21 . Figure 3. Bound func
 fulfil the constraints and