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In shreve's book, we model the stock price dynamics as: $$S_i(t) = \alpha(t)S_i(t)dt +S_i(t)\sum ^d_{j=1}\sigma _{ij}(t)dW_j(t)$$ and the forward rate can be written as : $$df(t,T) = \gamma(t,T)dt + \sum ^d_{j=1}\beta _{ij}(t,T)dW_j(t)$$ I'm wondering are we assuming the risk factors $W_j$ to be the same for stock price and forward rate, and they could be viewed as the risk factors for the financial market. or they are different? And what's the relationship between the two risk neutral probability measures in two models?

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A priori, if those two $W_j$ notations are in the same paragraph, they identify the same Wiener processes, under the same probability measure. So yes, they are seen as the different risk factors in your financial market.

If you assume that your $W_j$'s are independent of one another, you can introduce correlation between your two variables (stock price and forward rate) by changing the volatility vectors, as long as you keep the Euclidean norm of those vectors (which is the total volatility of your variables) unchanged.

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