Теоретические основы и методология имитационного и комплексного моделирования

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1 MONTE-CARLO STATISTICAL MODELLING METHOD USING FOR INVESTIGA- TION OF ECONOMIC AND SOCIAL SYSTEMS Vladmrs Jansons, Vtaljs Jurenoks, Konstantns Ddenko (Latva). THE COMMO SCHEME OF USI G OF TRADITIO AL METHOD OF STATIS- TICAL MODELLI G Usng tradtonal methods of statstcal modellng for nvestgaton of economc and socal systems t s possble to set the task of creatng an effcent procedure for generatng ncdental parameter values consttutng factors of a smulaton model, to effectvely use up-to-date nformaton technologes, to ensure contnuous control of the behavour of the specfc economc system that s beng researched. The tradtonal scheme of smulaton modellng s the generaton of a mass of ncdental parameter values featurng the changes of model factors (Fgure ). Identfcaton of factors to be generated Generaton of factors Modellng of system behavour Analyss of the modelled results Is t necessary to contnue modellng? yes no Applcaton of results Fgure. Algorthm of generaton of ncdental parameters The algorthm of generaton of ncdental contnuous value X (Fgure ), havng contnuous dstrbuton functon F, can be descrbed n the followng steps:. Let us generate, wthn an nterval (0.), an evenly dstrbuted ncdental parameter u ~U(0.).. Let us calculate X = F - (u). The value of F - (u) wll always be defnte, snce 0 < u <, but the area of defnng the functon F s the nterval [0,]. The fgure below presents the essence of the algorthm graphcally; here ncdental value may be assumed to be ether postve or negatve. Ths depends on the specfc value of parameter u. In the fgure, the value of parameter u produces a negatve ncdental value X, but parameter u yelds a postve ncdental value X (Fgure ). F u u 0 x x Fgure. Scheme of reverse transformaton ИММОД-03 33

2 The method of reverse transformaton may be also used f value X s dscrete. In ths case the dstrbuton s as follows: { X x} = å F ( x ) = P p ( x ), () where p(x ) s probablty p x ) = P{ X = } (. x x x It s admtted that ncdental parameter X may have only such values as x, x,..., for whch x < x <... Thus the algorthm of developng the values of ncdental parameter X wll have the followng consequences: Let us generate, wthn the nterval (0.), unform dstrbuted ncdental parameter u ~ U(0. ); Let us establsh the least postve round value I, for whch u < F(x ), and assume that X=x. Both optons of the method of the reverse transformaton for contnuous and dscrete values (at least formally) can be combned n one formula: { x : F( x U} X = P ) ³ mn, () whch s true also for mxed dstrbutons (.e., contanng both contnuous and dscrete components). In contrast to commonly used drect methods of generatng ncdental values (the method of the reverse transformaton composton and mploson), for mtatng the factors of the smulaton model t s recommended to use the so-called ndrect methods, namely, the acceptance-refusal method. Ths method may turn out to be sutable f due to certan reasons t s mpossble to apply drect methods or f these methods are neffcent.. STOCHASTIC MODELLI G OF I SURA CE I AGRICULTURE Let us to descrbe the usng of stochastc modellng method n nsurance. In order to ensure steady growth of agrcultural producton, especally n the prvate sector, t s necessary to ntroduce nsurance products wth regard to operaton of agrcultural enterprses (Fgure 3). A B Insurance fund Agrcultural sector's fund Insurance fund Agrcultural sector's fund Fgure 3. In unfavourable years (case A) the nsurance fund allows to stablze the agrcultural sector, n favourable years (case B) the nsurance fund can be ncreased usng funds from the agrcultural sector 34 ИММОД-03

3 Let us consder the modellng scheme of the agrcultural nsurance fund, whch later wll allow us to model the process of developng the model and to establsh the mnmum amount of the nsurance fund U (wthout a state subsdy). The mnmum fund amount U guarantees that wth a certanty g (probablty g) agrcultural losses wll be compensated. For modellng the nsurance fund, we wll use the smplest ndvdual rsk modellng scheme. Let us assume that the nsurance fund s satsfactory, gven the followng condtons: The number of regstered farms n the fund s constant; Rsks of ndvdual farms are ndependent; Payment of premums s effected at the begnnng of the perod; The loss dstrbuton functon s equal for all farms. Let us desgnate that: n number of agreements n the fund; j ordnal number of the farm; p probablty of settng n of the nsurance event; Y j possble losses of the farm j. Value Y j has probablty dstrbuton functon F(x); X j satsfed loss of the farm j. X j = Ind j *Y j. Ind j bnary ndex of the nsurance event of the farm j: By usng varable Ind, we can calculate total number of farms ncurrng losses: N = n å j= Ind j (3) Total amount of losses s: Z = X + X X n (4) Or by usng ndces of settng n of the events: Z = Ind Y + Ind Y Ind nyn = Ind j Y j (5) Fgure 4 shows that total losses are formed n n farms durng one tme perod. n å j= Fgure 4. Illustraton of process of loss formaton We are to compensate losses to farms wth a certanty g and are to ensure the requred operaton of the fund wth cash funds L. It means that the amount of the fund after compensatons ИММОД-03 35

4 must be postve wth a certanty g ( P ( U - Z ³ 0) = g ). The degree of rsk (stablty) of the nsurance fund can be establshed by the varaton coeffcent: ( D( K var ( = s = (6) E( E( where s ( standard devaton from the amount Z (standard error); E( mathematcal expectaton of value Z, whch n practce s measured wth average value of Z; D( varaton of value Z. If the number of farms n the fund s bg, t s possble to use the central margnal theorem and to establsh value U. Let us consder the nequalty: whch has to be vald under probablty g: U - Z ³ 0, (7) P ( U - Z ³ 0) = g. (8) From the nequalty U - Z ³ 0 derves nequalty Z U, and after that nequalty Z- E( U-E(. Dvdng t by a postve value s(, we obtan: Z - E( U - E( Z ). (9) s ( s ( Z ) Z - E( Value S = normally dstrbuted wth E(S) = 0 and s(s) =. Then the s ( followng formula can be appled: P a ( S a ) = ò - e p x - From the equaton U - E( Z ) = a( g ), the requred nsurance fund amount U s s ( Z ) U = a ( g ) s ( + E( () The amount of the nsurance coverage n cereal sowngs nsurance depends on the average amount of crop receved by years, n whch no relevant losses took place. The calculatons show that very often varaton coeffcent K var fluctuates wthn the range from 0% to 50%, whch testfes to the fact that nsurance fund, s often not so stable and addtonal fnancng s requred from the state. In the age of up-to-date nformaton technologes the applcaton of the Monte-Carlo statstcal method s smple and frequently allows avodng from complcated theoretcal calculatons as well as allows obtanng suffcently accurate practcal results to take approprate decsons on nsurance parameters. dx (0) 36 ИММОД-03

5 CO CLUSIO The applcaton of statstcal modellng s connected wth the fact that frequently t s not possble to provde a defnte descrpton of the behavour of the economc and socal system beng nvestgated. When nvestgatng the dynamc behavour of the economc and socal system,.e. by makng defnte changes of parameters of the system under nvestgaton, we frequently observe the exstence of ncdental factors affectng the character of the behavour of the system. In addton, t should not be forgotten that the very character of the research also brngs ts ncdental elements nto the research process. The process of nvestgaton of economc and socal systems usng statstcal modellng, dynamc programmng and Monte Carlo method allows set of alternatve strateges supports stable functonng of economc and socal systems n the condtons of uncertanty. The theoretcal and practcal results obtaned as a result of ths research can be appled n practcal actvtes of companes for makng effectve decsons. REFERE CES. Glaz, J. Approxmatons for the Multvarate ormal Dstrbuton wth Applcatons n Fnance and Economcs. In: Appled Stochastc Models and Data Analyss. G. Govaert, J. Janssen and. Lmnos, eds., Volume, 00, Unverste de Technologe de Compegne, Compegne, France, pp Harrngton S.E., ehaus G.R. Rsk Management and Insurance. ew York, p. 3. Jurēnoks V., Jansons V. Stohaststc modellng and optmzaton of ndustral stock. Smulaton n Wder Europe. 9-th European conference on Modellng and Smulaton ECMS-005, Rga, Latva, June 4, 005. Рp Jansons, V., Ddenko, K., Jurenoks, V., Insurance as a tool for steady development of agrculture. VIII Internatonal scentfc conference, Management and Sustanable Development, Yundola, Bulgara, 4 6 March, 006, pp Jurenoks V., Jansons V., Ddenko K. Modellng of Stablty of Economc Systems Usng Benchmarkng and Dynamc Programmng X Internatonal Conference on Computer Modellng and Smulaton EUROSIM/UKSm, Cambrdge, Unted Kngdom, 3 Aprl, 008, pp Jansons, V., Jurenoks,V., Ddenko, K., 0. Investgaton of Fnancal Flows n Logstcs Usng Multvarate Statstcal Modellng Methods. The 3 RD Internatonal Conference on Harbor Martme Multmodal Logstcs Modellng & Smulaton HMS 0, September 4, Rome, Italy, pp Jansons, V., Jurenoks,V., Ddenko, K., 0. Safety Control Of Latvan State Road etwork Usng Statstcal Modellng Methods. The 4 th Internatonal Conference on Harbor Martme Multmodal Logstcs Modellng & Smulaton HMS 0, September 9, Venna, Austra, pp. 6. ИММОД-03 37