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U ovom radu smo konstruirali Itôv integral koji definiramo kao limes niza slučajnih varijabli. Dokazali smo analogon Newton-Leibnizove formule, tzv. Itôvu formulu. Također, dokazali smo martingalnost Itôvog integrala čime smo, uz spomenutu formulu, dobili alat za konstrukciju martingala te ispitivanje martingalnosti slučajnih varijabli. U nastavku rada, pokazali smo konformnu invarijantnost Brownovog gibanja. Odnosno, pokazali smo da preslikavanjem Brownovog gibanja po konformnoj funkciji dobivamo vremenski izmijenjeno Brownovo gibanje. Dodatno, u prvom poglavlju smo pokazali i invarijantnost Brownovog gibanja u odnosu na skaliranje (uz vremenski pomak). U zadnjem poglavlju smo dokazali Feynman-Kacovu formulu kojom smo eksplicitno izrazili rješenje jednadžbe provođenja. In this paper we have constructed Itô integral that is defined as a limit of certain sequence of random variables. We have proved Itô formula which is analogous to Newton–Leibniz formula. Also, we have proved that Itô integral has martingale property. Combining the last two observations, we get a powerful tool for constructing martingales and inspecting whether a random variable is a martingale. Later in the thesis we have observed conformal invariance of Brownian motion. It states that, under conformal mapping, Brownian motion is mapped to another time changed Brownian motion. However, conformal invariance is not the only invariance property of Brownian motion. In the first chapter we have shown that Brownian motion is also invariant under scaling (with a time change). In the final chapter we have proved Feynman–Kac formula which gives the explicit formula of the solution to the heat equation. |
