Consider a path-dependent GBM model for a stock price: $$dS_t = \mu(t, S_.)S_tdt + \sigma(t, S_.) S_t dB_t,$$ where $\mu, \sigma : [0,\infty)\times C_{[0,\infty)}\to \mathbb{R}$ are previsible path-functionals.

Does anyone know of any references of papers on this generalization and if there are any standard choices for $\mu$, $\sigma$ that are well studied?

Some examples: We might consider a drift in two forms: $$\mu(t, S_.) = \int_0^t f(S_u) du$$ or say $$\mu(t,S_.) = g(m_t, M_t),$$ where $m_t = \min_{0\leq u \leq t} S_u$ and $M_t = \max_{0\leq u \leq t} S_u$ and $f$ and $g$ are nice enough functions. These incorporate path-dependency in a sort of arbitrary way, but what might be some judicious choices? For example, $f(S_u)=e^{\lambda S_u}$ would "weight" recent prices heavier than past prices. Another way to phrase this might be: is there a survey of continuous time analogs of discrete time filters, like moving averages, EWMAs, etc?


I am not looking for online resources, I know of those and am searching among those, I was wondering if anybody already knew of specific papers about these topics. Like if you ask for functional Ito calculus, the classic to start with might be Bruno Dupire's paper on functional Ito calculus and he lists a variety of discrete time path-dependent functionals but does not go into much details about treating them in continuous time.



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