For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case?
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Risk neutral version of O-U process is apparently a simple GBM. See explanation here, http://web.mit.edu/wangj/www/pap/LoWang95.pdf specifically section II.
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$\begingroup$ II.B: "Despite the differences between the trending 0-U process and an arithmetic Brownian motion, Grundy (1991) points out that both data-generating processes yield the same risk-neutralized price process (equation (5)) hence the Black-Scholes formula still applies to options on stocks with log-price dynamics given by equation (6)." $\endgroup$ Commented Mar 14, 2020 at 23:21