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I thought this was an interesting example to add. It concerns a "ratio model" of habit (as opposed to a "difference" model of habit). See, for example, Abel (1990, American Economic Review). Let $$ x_t = \lambda \int_{-\infty}^t e^{-\lambda(t-s)} c_s ds. $$ (For context, $x_t$ is a log habit index that is given by a geometric average of past consumption, ...


$$ Z_t = f(S_t) := \left( \frac{S_t}{H} \right)^p $$ $$ dZ_t = \partial_x f(S_t) dS_t + \frac{1}{2} \partial^2_{xx} f(S_t) d\langle S \rangle_t = p\frac{S_t^{p-1}}{H^p} dS_t + \frac{1}{2} p(p-1) \frac{S_t^{p-2}}{H^p} S_t^2 \sigma^2 dt $$ Thus $$ dZ_t = Z_t \left( p r + \frac{1}{2} p(p-1) \sigma^2 \right)dt + p Z_t \sigma dW_t $$ so that $$ \frac{dZ_t}{Z_t} = ...

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