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I am currently working on to derive the following form of the stock price dynamics:

$$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$

where the parameter $\rho$ is the correlation between bond market returns and stock market returns, $\sigma_S$ is the volatility of the stock, and $\psi$ is the Sharpe ratio of the stock which we assume constant.

What I have done so far is, that I have considered a similar model to Ma (2009). It is a two-asset model with stochastic correlation under the standard assumptions regarding a martingale measure, and the assets $S_1, S_2$ are GBM with mean $r$ (the risk-free interest rate) and constant volatilities $\sigma_1 >0, \sigma_2 >0$, with respect to Brownian motions $W_1,W_2$ satisfying

$$dS_{1t} = rS_{1,t}dt + \sigma_1S_{1,t}dW_{1,t}$$ $$dS_{2t} = rS_{2,t}dt + \sigma_2S_{2,t}dW_{2,t}$$

where I let $\rho$ be the instantaneous correlation between $S_1$ and $S_2$ at time $t$, implying that $d[S_1,S_2] =S_{1,t}\sigma_1S_{2,t}\sigma_2\rho_t dt$ and $d[W_1,W_2]_t = \rho_t dt$.

So my question is: Is my approach to solving for the stock price dynamics incorrect? If so, how would the initial point look like, including Ito's Lemma?

And an another question is, why can the correlated part of the stock price dynamics be characterized as: $$\rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$

What is the intuition and the technicalities behind characterizing the correlation between stock and bonds returns as $\rho$ and $\sqrt{1-\rho^2}$? The joint distribution of these?

I would really appreciate some help.

Best regards,

Def

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