Stability Analysis in a Trend Depending Price Formation Model
Open AccessThis dissertation concerns the mathematical analysis of a mathematical model for priceformation. We take a large number of rational buyers and vendors in the market who aretrading the same good into consideration. Each buyer or vendor will choose his optimalstrategy to buy or sell the good. Since markets seldom stabilize, our model mimics the realmarket behavior.We introduce three models. All of them are a modification of the original J.-M.Lasryand P.L.Lions evolution model. In the first modified model, a random term is added tomimic the randomness of trading in the real market. This reflects markets with lowvolatility, where it might be difficulty to buy or sell the good at specific price. In the secondmodel, we use cumulative density function instead of density function. We give numericalsimulations on these two models in order to have a general picture on the solution. Thethird model is the one we mainly focus on; we add a term associated with the parameter Rto destabilize the original Larsy-Lions model and study oscillations and wave solutionsdepending on different values of R. We also study existence and uniqueness of the solution.In the numerical simulation, we use Crank-Nicolson scheme to discretize the space andwe also build a scheme for Dirac Delta function. Several plots are given to demonstrate theresults corresponding to the theoretical prediction.
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