The theory of functionally generated portfolios (FGPs) is an aspect of the continuous-time, continuous- path Stochastic Portfolio Theory of Robert Fernholz. FGPs have been formulated to yield a master equation - a description of their return relative to a passive (buy-and-hold) benchmark portfolio serving as the num\'eraire. This description has proven to be analytically very useful, as it is both pathwise and free of stochastic integrals. Historically, FGPs have been specified only as portfolios on the tradeable assets of a market, and the num\'eraire has been confined to be the market portfolio. Here we generalize the class of FGPs in several ways: (1) they may be specified over any strictly positive wealth processes resulting from investment in the tradeable assets, (2) the num\'eraire may be any strictly positive wealth process, (3 ) generating functions may be stochastically dynamic, adjusting to changing market conditions through an auxiliary continuous-path stochastic argument of finite variation. These generalizations do not forfeit the important tractability properties of the associated master equation. We show how these generalizations can be usefully applied to  statistical arbitrage, portfolio risk immunization, and the theory of mirror portfolios.
The theory of functionally generated portfolios (FGPs) is an aspect of the continuous-time, continuous-path Stochastic Portfolio Theory of Robert Fernholz. FGPs have been formulated to yield a master equation - a description of their return relative to a passive (buy-and-hold) benchmark portfolio serving as the num\'eraire. This description has proven to be analytically very useful, as it is both pathwise and free of stochastic integrals. Here we generalize the class of FGPs in several ways: (1) the num\'eraire may be any strictly positive wealth process, not necessarily the market portfolio or even a passive portfolio; (2 ) generating functions may be stochastically dynamic, adjusting to changing market conditions through an auxiliary continuous-path stochastic argument of finite variation. These generalizations do not forfeit the important tractability properties of the associated master equation. We show how these generalizations can be usefully applied to scenario analysis, statistical arbitrage, portfolio risk immunization, and the theory of mirror portfolios.
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Historically, FGPs have been specified only as portfolios on the tradeable assets of a market, and the num\'eraire has been confined to be the market portfolio.
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they may be specified over any strictly positive wealth processes resulting from investment in the tradeable assets, (2) the
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not necessarily the market portfolio or even a passive portfolio; (2
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scenario analysis,