I am trying to show that the solution of the following integral is as follows:
Define the stopping time: $C(a) = \inf(u \ge 0 : H(\pi(0) |\mu)-H(\pi(u) | \mu) > a)$
Where $H(\pi(t)|\mu(t))=\sum^{n}_{i=1} \pi_i \log \frac{\pi_i}{\mu_i}$ is the 'Relative Entropy'.
Introduce the vector field: $U_{\mu}(\pi)=\textbf{$\pi$}-\textbf{$\mu$}$ and associated flow: $$ \frac{d}{du} \pi (u ) = \pi'(u) = -U_\mu(\pi (u ) ) \quad \pi(0) = \pi $$
Then: $$ \pi(t+1) = \pi(t) -\int^{\min\{C(a),1\}}_0 U_{\mu(t+1)}(\pi(u)) du\\ =\pi(t)+s(\mu(t+1)-\mu(t)) $$
where $s:=\min\left\{\frac{\frac{a}{|\nabla H(\pi(t)|\mu(t+1))\bullet v|}}{\mu(t+1)-\pi(t)} ,1\right\}$
For clarity, the numerator is:
$$ \frac{a}{|\nabla H(\pi(t)|\mu(t+1))\bullet v|} $$ the dot product of the directional derivative of the relative entropy and $v$,
Where $v=\frac{\mu(t+1)-\pi(t)}{|\mu(t+1)-\pi(t)|}$
I'm confused by how they can solve the integral:
My Attempt: the flow is a first order ODE, solving for $\pi$ yields: $$ \pi(t) = e^{-t}(\pi(0)-\mu)+\mu $$ and the integral: $$ \int^{\min\{C(a),1\}}_0 U_{\mu(t+1)}(\pi(u)) du=\\ \bigg (\int_0^{\min\{C(a),1\}}\pi_1(u)-\mu_1(t+1)du,......,\int_0^{\min\{C(a),1\}}\pi_n(u)-\mu_n(t+1)du \bigg) $$ but solving these integrals with my solution to the ODE does not yield anything close to their result. Im confused about where the gradient vector of H comes in to play.. any hints appreciated.
Update: I think there may be a typo in the paper and the correct solution should be: $$ \pi(t)+s(\mu(t+1)-\pi(t)) $$ although I may be wrong. I also think that the solution above has been found by linear interpolation rather than directly solving the integral, I'm new to this method as well so any hints about this would be appreciated.
Edit: The function $s$ is defined from $\gamma_\pi^*$ which is the free energy (see page 15 http://arxiv.org/pdf/1308.5376v1.pdf). They are all functions of $t$ the time variable. so when one reads $$s (\mu(t+1)-\mu(t)) $$ this is meant as a scalar multiplications and not a function evaluation
As one begins with a particular portfolio position one desires to move in a profitable way in the set of portfolios. The movement of the investor's position evolves following an ODE, that means that we have to consider a flow an it's respective vector field.
hope this helps