Expected Log Growth with Path-Dependent Drift and Volatility
Summary
The document asks how to calculate expected log growth when the drift and volatility of a geometric Brownian motion depend on the stock’s past path. In the Markov case, where both coefficients depend only on the current price and time, it states that the conditional expectation of future accumulated growth satisfies a backward PDE with a zero terminal condition. The growth rate in that formulation is half the squared drift-to-volatility ratio; the document gives the constant-coefficient expression as an example.
For path-dependent coefficients, the usual state-based PDE no longer directly applies because the current price alone does not capture the relevant history. The author distinguishes conditioning on the current price from conditioning on the full observed price history, noting that the latter expectation is itself a random process. A stochastic PDE is suggested as a possible analogue, but no derivation or result is provided. The text frames an open mathematical question rather than presenting a finished method, and its conclusions are limited to the stated model setup.
Key ideas
- For Markov coefficients, the expected accumulated growth is described by a backward PDE with a zero terminal value.
- The growth contribution in the stated formulation is half the squared ratio of drift to volatility.
- When coefficients depend on the past path, the current price alone may not summarize the information needed for conditioning.
- Conditioning on the full price history produces a random conditional expectation rather than a function of time and current price alone.
- The document proposes a stochastic PDE as a possibility but does not derive one.
Tags
Full text
# optimal log growth under a path dependent GBM
# optimal log growth under a path dependent GBM
Consider an extension to the (one-dimensional) geometric Brownian motion model, $$dS_t = \mu(t,S_.)S_t dt + \sigma(t, S_.)S_t dB_t,$$ where $\mu$ and $\sigma$ are previsible path functionals, i.e. they depend on the path, denoted by $S_.$ of the stock price, at least up to the current time $t$. Then consider the function $$u(t,s) = \mathbb{E}\left[\int_t^T c(w,S_.) dw \big\lvert S_t=s\right]$$ where $$c(w,S_.)=\frac12 \left(\frac{\mu(w, S_.)}{\sigma(w, S_.)}\right)^2.$$
What is known:
When the coefficients depend only on the current price, i.e. (slight abuse of notation ahead) $$\mu(t, S.)=\mu(t, S_t) \text{ and } \sigma(t, S_.)=\sigma(t, S_t),$$ then it is known that $u(t,s)$ solves the PDE $$u_t + \mathscr{L} u +c(t,s) = 0,$$ with vanishing terminal condition $u(T,s)=0$ and where $\mathscr{L}$ is the infinitesimal generator of $S$, i.e. $$\mathscr{L} f(t,s) = \mu(t, s)s f_s + \frac12 \sigma(t, s)^2s^2 f_{ss}.$$
This actually gives us the expected log return in a portfolio that optimizes the terminal log wealth, and when $\mu(t,s)=\mu$, $\sigma(t,s)=\sigma$ are constant, we get the expected log-growth equal to $\frac12 \frac{\mu^2}{\sigma^2}(T-t).$
My question:
Is there a generalization to the above PDE that $u$ satisfies when the coefficients only depend on the current price, to the case when the coefficients are previsible path functionals?
Some thoughts: The standard approaches are no longer valid, I believe, as we lose Markov property. Bruno Dupire's paper on functional Ito calculus seems promising but I have yet to work out a meaningful application of it to this specific problem.
Please comment for clarifications, corrections, or questions.
Edit/Update 1/2/2023 Just realized a subtlety I overlooked. In the case I am concerned with, where the coefficients depend on the path up to time $t$, I am not actually trying to compute the conditional expectation conditional on $S_t=s$, $$\mathbb{E}\left(\int_t^T c(w, S_.)\big\vert S_t=s\right),$$ but rather the conditional expectation conditional on the entire history up to time $t$: $\mathscr{F}_t^S = \sigma(S_u: u\leq t)$ $$\mathbb{E}\left(\int_t^T c(w, S_.)\big\vert \mathscr{F}_t^S\right),$$ which complicates things greatly. First of all, in the former case the conditional expectation is a function of $(t,s)$. In the latter, the conditional expectation is a random process itself. An amateur guess is that then this latter expression should satisfy a stochastic PDE, analogous to the PDE that $u(t,s)$ solves in the Markov case.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.