Problems to understand a stochastic DGL equality

Question

currently I am reading a paper called "Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model" for self-study reasons. The paper can be found here: http://arxiv.org/abs/1110.6289

There, I found an equation I don't understand.

In Chapter 7 the authors are trying to calculate a wealth process.

The wealth process $X_t$ satisfies the following SDGL $$dX_t = (X_t - w(\bar{X_t}))\frac{dV_t}{V_t}, (1)$$ since in chaper 7 everthing is continuous.

The process $V_t$ satisfies the follwing equation $$dV_t=\pi_tdS_t := \sum{\pi_t^i}dS_t^i , (2) $$ for a preversibel $\pi_t$ and a d-dim semi-martingal $S_t$ with SDGL for every $i$ $$dS^i_t = S^i_t(\mu^i_t dt+\sum\sigma^{ij}_t dW^j_t) , $$ where $W_t = (W^1_t,\dots,W^d_t)$ is a d-dim Wiener process. Now the authors claim that by plugging $(2)$ in $(1)$ they obtain: $$dX_t = (X_t - w(\bar{X_t}))\sum\pi^i_t\frac{dS_t^i}{S^i_t} , (3) $$ where $\pi^i_t = (\frac{1}{1-\gamma(1-\alpha)}\theta'_i\sigma^{-1}_t)^i $. I think the exact form of $\pi_i$ shouldn't matter, only that it is preversible.

My question is: How do the authors came up with $(3)$? The way I'am doing it is: $$dX_t = (X_t - w(\bar{X_t}))\sum\pi^i_t\frac{dS_t^i}{V_t} .$$

Sadly, only $(3)$ seems to be consisting with the literature (one can find a similar solution for $(3)$ in "On Portfolio Optimization under "Drawdown" Constraints" http://www.math.columbia.edu/~ik/Drawdown.pdf) . What am I doing wrong?

Answer 1

I haven't read all the paper, just the section you mentioned. The previsible/predictable strategy $\pi_t$ represents the number of shares of the asset $S$ held at time $t$. The paper looks to use power utility in some way, and as is common in those types of problems, generally you want to think of $\tilde{\pi}_t$, which is the percentage of wealth at time $t$ held in the risky assets.

Towards the bottom of p. 21, you can see that they define $$ \tilde{\pi}_t := \pi_t S_t / V_t. $$

Using $\tilde{\pi}_t$ instead of $\pi_t$ solves your problem I believe. Also, equation (22) on p. 22 shows that the optimal $\tilde{\pi}$ has the form you describe above.