Path-Dependent Geometric Brownian Motion and Continuous-Time Filters

Summary

The document asks about geometric Brownian motion models whose drift or volatility depends on the stock’s past path, rather than only its current value and time. It gives examples of drift built from an accumulated function of earlier prices or from the running minimum and maximum. It also suggests weighting historical prices and asks whether continuous-time versions of moving averages or exponentially weighted averages offer useful modeling choices.

The text is a request for references and established model choices, not a survey or a completed model specification. It provides no citations, empirical tests, or guidance on selecting functions for drift and volatility. Any proposed path functional would need to meet suitable mathematical conditions for the stochastic model to be well-defined, and the document does not address those conditions. Its value is in framing possible ways to encode price history in a continuous-time model and identifying the need for better-founded choices.

Key ideas

• A path-dependent GBM lets drift or volatility depend on the prior price trajectory.
• The document illustrates accumulated-price and running-extrema functionals for the drift.
• It asks about continuous-time counterparts to moving averages and exponentially weighted filters.
• No references, standard choices, or empirical evaluations are supplied.
• Model validity depends on conditions for the selected path functionals that are not discussed.

Tags

# References for path-dependent GBMs or continuous time analog of discrete time filters


# References for path-dependent GBMs or continuous time analog of discrete time filters












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?

Edit:

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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