Is the market price of risk deterministic or stochastic in the Heston model?

I am recently digging into the Heston model and I have noticed that every author refers to the market price of risk simply as $$\lambda$$, or sometimes it is more clearly specified to be bi-dimensional in a form such as $$\lambda=(\lambda_1, \lambda_2)^T$$ where $$\lambda_1=\frac{\mu-r}{\sqrt{v_t}}$$ and $$\lambda_2$$ is unknown as a result that volatility is not traded.

This notation seems confusing to me, and I wonder whether $$\lambda$$ should actually carry a subscript $$t$$, and therefore being written as $$\lambda_t$$, as a result of being dependant on $$\sqrt{v_t}$$. I mean, differently than what was true for the framework of B&S, even though $$r$$ and $$\mu$$ are constant, the market price of risk is now even stochastic, right?

• Yes. Besides there was no need to assume that $\mu$ is constant (BS, Heston or otherwise). – Antoine Conze Apr 7 at 9:27
• @AntoineConze I see. If $/mu$ was not constant it would still make $/lambda$ time-dependant although deterministic. In this case $/lambda$ is both time-dependent and stochastic – Mr Frog Apr 7 at 9:30
• $\mu$ being the drift under the historical measure, there is no reason to assume it is non stochastic. As long as it is measurable wrt the filtration generated by the model brownian motion(s), plus some technical conditions, Girsanov applies. – Antoine Conze Apr 7 at 10:01