Essays in Finance Related Problems with PDE Approach

Abstract: This thesis is comprised of three scientific articles, all in PDE problems related to finance. The first two papers address problems concerning bonds and the third one treat s asymmetry problem in the mean field game theory.We investigate the free boundary for the problem of valuation of American type convertible bond with extra call feature, in paper one. In this case the free boundary behaviour is of interest since it shows the optimal conversion strategy. We study the regularity of the free boundary and also the solution to the pricing of this special financial derivative. The touching point of the free boundary and fixed boundary is discussed, as well.The second paper is devoted to the study of bonds with call feature and conversion. Further, it is shown that the price of a Bermudan type bond tends to that of an American one on condition that the conversion time goes to the whole life-span of the bond.Our main goal in the third paper is to investigate a symmetry case regarding the stationary problem in mean field game of Lasry-Lions. Starting with a priori unknown ring-'shaped region, taking the boundaries as level sets and admitting harmonic solutions, it is implied that the region has spherical symmetry. Moreover, therein we attempt to state the model to an higher dimensional case.