A new notion of transitive relative return rate and its applications using stochastic differential equations

We introduce a new notion of transitive relative return rate and present its applications based on the stochastic differential equations. First, we define the notion of a relative return rate and show how to construct the transitive relative return rate (TRRR) on it. Then, we state some propositions and theorems about relative return rate and TRRR and prove them. Moreover, we exhibit the theoretical framework of the generalization of TRRR for n ≥ 3 cases and prove it, as well. Furthermore, we illustrate our approach with real data applications of daily relative return rates for Borsa Istanbul-30 (BIST-30) and Intel Corporation indexes with respect to daily interest rate of Central Bank of the Republic of Turkey between June 18, 2003 and June 17, 2013. For this purpose, we perform simulations via Milstein method. We succeed to present usefulness of the relative return rate for the relevant real large data set using the numerical solution of the stochastic differential equations. The simulation results show that the proposed closely approximates the real data.