I am writing my Bachelor's Thesis on the fast Ninomiya-Victoir calibration of the Double Mean Reverting model and have a question to its Stratonovich formulation. I am new to mathoverflow and a novice in financial mathematics, so please freely point out my obvious mistakes.
The Double Mean Reverting Model is given by
$$dS_t=\sqrt{v_t} S_t dB_t^1$$
$$dv_t=\kappa_1(v_t'-v_t)dt + \xi_1 v_t^{\alpha_1}(\rho_{1,2}dB_t^1+\sqrt{1-\rho_{1,2}^2}dB_t^2)$$
$$dv_t'=\kappa_2(\theta-v_t')dt + \xi_2 v_t'^{\alpha_1}(\rho_{1,3}dB_t^1+\rho_{2,3}dB_t^2+\sqrt{1-\rho_{1,3}^2-\rho_{2,3}^2}dB_t^3)$$
where the $B^i$ are independent Brownian motion. To be able the use the Ninomiya-Victoir scheme it is needed to reexpress this Itô-SDE's in Stratonovich form and for that I need to compute the quadratic variations
$$d[\sqrt{v_t}S,B^1]_t = \{\frac{1}2{}\rho_{1,2}v_t^{\alpha_1-\frac{1}{2}}+v_t\}S_tdt$$
$$d[\xi_1v_t^{\alpha_1},(\rho_{1,2}B^1+\sqrt{1-\rho_{1,2}^2}B^2)]_t=\xi_1^2\alpha_1v_t^{2\alpha_1-1}dt$$
$$d[\xi_2v_t'^{\alpha_2},(\rho_{1,3}B^1+\rho_{2,3}B^2+\sqrt{1-\rho_{1,3}^2-\rho_{2,3}^2}B^3)]_t=\xi_2^2\alpha_2v_t'^{2\alpha_2-1}dt$$
This is where my problem lies. I know that this are the solutions since it is done in the the paper "Fast Ninomiya-Victoir calibration of the Double-Mean-Reverting Model" by Bayer et al.
However I cannot comprehend the steps needed to calculate the quadratic variation. Could you maybe give me a hint how to calculate such terms or show me how it is done? Thank you in advance
