Overview
Excess portfolio growth is defined by Fernholz.
Definitions
The excess portfolio growth rate is defined by Fernholz to be
{% \alpha_{\pi}^* = \frac{1}{2}(\sum \pi_i(t) \sigma_{ii}(t) -\sum \pi_i(t)\pi_j(t) \sigma_{ij}(t) ) %}
With this definition, we can derive the following
{% \alpha_{\pi}(t) = \sum \pi_i(t) \alpha_i(t) + \alpha_{\pi}^* (t) %}
Portfolio Growth
Given the above defintions, Fernholz shows that the portfolio dynamics can be written as
{% d log Z(t) = \sum \pi_i(t) d log X_i(t) + \alpha_{\pi}^*(t) dt %}
Alternative Calculations
Given two portfolios {% \pi %} and {% \eta %}, the following holds
{% \alpha_{\pi}^*(t) = \frac{1}{2}(\sum \pi_i(t)\tau_{ii}^{\eta}(t) - \sum_{ij} \pi_i(t)\pi_j(t)\tau_{ij}^{\eta}(t)) %}
when we take {% \pi = \eta %}, we then have
{% \alpha_{\pi}^*(t) = \frac{1}{2} \sum_i \pi_i(t) \tau_{ii}^{\pi}(t) %}