# SDE Parameter Estimation

Have a question about "How to estimate parameters for SDE with multiple Brownian Motions ?" Let's say $$X_t$$ follows the process: $$dX_t=\mu dt+\sigma_1 dW_t^1 + \sigma_2 dW_t^2$$

I think I've checked Sim.DiffProc for R and SDE Toolbox for MATLAB. Could someone lead me on this matter please ? Thanks for your kind attention.

Your question does not appear complete, that is, the rationale for using two Brownian motions is not clear. Note that \begin{align*} dX_t &= \mu dt + \sigma_1 dW^1_t + \sigma_2 dW^2_t \\ &=\mu dt + \sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}\frac{\sigma_1 dW^1_t + \sigma_2 dW^2_t}{\sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}}\\ &= \mu dt + \sigma dW_t, \end{align*} where $$\sigma = \sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}$$ and $$\Big\{W_t = \frac{\sigma_1 dW^1_t + \sigma_2 dW^2_t}{\sqrt{\sigma_1^2 + \sigma_2^2 + 2 \rho \sigma_1\sigma_2}}, \, t \ge0\Big\}$$ is a standard Brownian. That is, $$X_t$$ is completely described by a one factor model.