Optimalna kontrola in njena uporaba v ekonomiji : magistrsko delo

Kelvišar, Katja (author), Perman, Mihael (mentor)

V magistrskem delu se ukvarjamo s problemom optimalne kontrole, ki ga nato uporabimo pri reševanju bančne krize. Najprej dobimo splošen pregled nad tipi problema optimalne kontrole. Vidimo, da obstajajo $4$ tipi modelov, ki so deterministični model v diskretnem in zveznem času ter stohastični model v diskretnem in zveznem času. Osredotočimo se na deterministični model v zveznem času in navedemo Pontryaginov princip. Dokažemo eksistenčni izrek z uporabo minimizirajočega zaporedja funkcij stanj. Nato s pomočjo dinamičnega programiranja razvijemo orodje za iskanje optimalne kontrole. Glavni rezultat je bila izpeljava enačbe dinamičnega programiranja. Spoznamo tudi pojem povratne kontrole in rešimo primer s pomočjo Pontryaginovega principa. V zadnjem poglavju se osredotočimo na modeliranje bančne krize kot problem optimalne kontrole s pomočjo modela SIR. Imamo tri različne primere širitve krize. Prva kriza se začne v banki na Portugalskem, druga v Španiji in tretja v Veliki Britaniji. Kriza na Portugalskem in Španiji se začne in konča dokaj hitro, medtem ko je v Veliki Britaniji proces počasnejši in zato tudi daljši, česar se bojimo. Nato vpeljemo kontrolno funkcijo, ki ima nekaj omejitev, s katero se kriza konča hitreje in ne razširi toliko. Te situacije sprogramiramo in rešimo s pomočjo Matlaba.

• Language: Slovenian
• Keywords: optimalna kontrola, problem optimalne kontrole, finančna kriza, bančna kriza, kontrolna funkcija, funkcija stanja, deterministični model v zveznem času, Pontryaginov princip, dinamično programiranje, povratna kontrola, SIR model.
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-108486
• UDC: 517.9
• COBISS.SI-ID: 18665305
• Publication date in RUL: 04.07.2019

KELVIŠAR, Katja, 2019, Optimalna kontrola in njena uporaba v ekonomiji : magistrsko delo [online]. Master’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=108486

Secondary language

• Language: English
• Title: Optimal control and its applications to economics

Abstract

In my master's thesis we deal with the optimal control problem, which we then apply to solving a bank crisis. Firstly, we get a general overview of types of optimal control problems. We learn that there are $4$ types of models, which are discrete time deterministic, continuous time deterministic, discrete time stochastic and continuous time stochastic models. We focus on the continuous time deterministic model and state Pontryagin principle. We prove the existence theorem by using a minimizing sequence of the state functions. Then we develop a tool for finding the optimal control via dynamic programming. The main result there is obtaining the equation of dynamic programming. We get familiar with the notion of feedback control and solve one example with the help of Pontryagin principle. Our last chapter focuses on modelling a bank crisis into optimal control problem by using the SIR model. We have three different examples of crisis propagation. The first crisis starts with a bank in Portugal, the second one with a bank in Spain and the third one starts in Great Britain. The crisis spreads quickly in Portugal and Spain but also ends relatively soon whereas in Great Britain it takes a lot more time for the crisis to spread but it lasts a lot longer, which is what we are afraid of. We then propose a control function with some restrains to end the crisis more quickly and prevent it from spreading. We model and solve these situations in Matlab.

• Keywords: optimal control, optimal control problem, finantial crisis, bank crisis, control function, state function, continuous time deterministic model, Pontryagin principle, dynamic programming, feedback control, SIR model.