Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance

In this paper, on the basis of Newton’s method, we propose a fast quasilinearization numerical scheme, coupled with Rothe’s method, for fully nonlinear parabolic equations. General conditions that provide quadratic, uniform and monotone convergence of the quasilinearization method (QLM) of solving fully nonlinear ordinary differential equations that arise on each time level, are formulated and elaborated. The convergence of QLM and its rate are examined numerically, on a simple test example with an exact solution. The first few iterations already provide extremely accurate and stable numerical results. The second goal is to consider three applications of the proposed schemes in financial mathematics. Namely, numerical results for three nonlinear problems of optimal investment are presented and discussed. The numerical experiments of the last problem are based on the data from statistic information of the Bulgarian National Bank and Bratislava Interbank.