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I guess for you to obtain the value for the real world probability measure P, the expected rate of return should be given...else the value of P should be given.


Note that, under measure $Q$, the dynamics is of the form \begin{align*} dS_t = S_t \big[(r+ \sigma \theta_t) dt + \sigma dW_t^Q \big]. \end{align*} Then, for $\Delta>0$ sufficiently small, \begin{align*} S_{t+\Delta} &= S_te^{\left(r-\frac{1}{2}\sigma^2\right)\Delta + \sigma \int_t^{t+\Delta} \theta_s ds + \sigma \left(W_{t+\Delta}^Q-W_t^Q\right)}\\ &...


in the new measure, the stock has drift $r + \sigma \theta$ so yes you just proceed with that drift as you say. If $\theta$ is time dependent, it gets more complicated.

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