Stochastic processes are a key element in various research fields. The thesis presents Itô’s processes, their properties and Itô’s formula. They are one of the most widesrpead stochastic processes with clear applications in finance. We define parametric Itô process and describe methods for parameter estimation from given sample. The described methods include Euler’s approximation, the Fokker-Planck equation and it’s connection to Kolmogorov’s forward equation and finally, Milstein’s method. Next, we define the problem of controlled stochastic processes, explain what stochastic control is and provide various examples of it. We introduce the solution using the dynamic programming principle, which is achieved using Hamilton-Jacobi-Bellman equations. Finally, we present the application to two examples. In the first example we solve the optimal protfolio management problem with different utility functions. In the second example we solve well-known linear regulator problem.
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