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Use column vectors for dollar holdings ${\theta }_t$ and put $b=\mu -r1$, $h={\sigma }^{-1}b$ and $V=\sigma {\sigma }^T$. The normalized state-price density isThe vector $h$ is the market price of risk. The density $e^{rt}{\zeta }_t$ defines an equivalent risk-neutral measure, under which $W_t^{\mathbb{Q}}=W_t+ht$ is Brownian motion and every stock has drift $r$. A terminal claim $H$ with finite state-price cost is priced at ${\zeta }_t^{-1}\mathbb{E}[{\zeta }_{T}H\mid {\mathcal{F}}_t]$. Nonsingularity of $\sigma$ and the natural Brownian market information make this a complete market; in particular the optimal nonnegative payoff below can be replicated.

Let $q=n^{-1}1$ and $v={\sigma }^{T}q$. Apply the Itô formula to each log price, whose return variance is $V_{ii}$:Summing and dividing by $n$ proves the geometric stock index formulaThus $J_t=J_0\exp (v^T{W}_t+a_{J}t)$, where $a_J=q^T\mu -\mathrm{tr}(V)/(2n)$. The factor $1/n$ applies to the whole bracket. The geometric average need not itself be a self-financing portfolio; its instantaneous drift is $a_J+|v{|}^2/2$.

If ${\theta }_t^i$ is the dollar amount invested in stock $i$, the bank account holds $w_t-1^T{\theta }_t$. The self-financing portfolio equation isEquivalently, ${\pi }_t={\theta }_t/w_t$ gives $d{w}_t/w_t=(r+{\pi }_t^{T}b)dt+{\pi }_t^T\sigma d{W}_t$. For share-holding notation replace ${\theta }_t^i$ by $S_t^i$ times the number of shares.

Static expected utility maximization uses the terminal budget $\mathbb{E}[{\zeta }_T{w}_T]\le w_0$. Since $J_T$ is an exogenous positive benchmark, differentiating $U(x/J_T)$ in $x$ gives $J_T^{-1}{U}^{\prime }(x/J_T)$. An interior Lagrange multiplier $\lambda >0$ for the budget therefore givesFor CRRA utility, write $p=1/R$. Since $U^{\prime }(y)=y^{-R}$, solving this marginal equation givesEvery positive lognormal moment is finite here, so the budget determines the constant uniquely. PutThenThis is also a sufficiency condition: concavity gives $U(X/J_T)-U(w_T^{\ast }/J_T)\le \lambda {\zeta }_T(X-w_T^{\ast })$, and taking expectations and using the budget inequality proves global optimality.

To find the whole optimal process, set $c=a_J-r-|h{|}^2/2$ andSince ${\zeta }_t{J}_t=J_0\exp ((v-h{)}^T{W}_t+ct)$, the moment-generating function of a normal distribution givesPricing the terminal payoff yields the explicit benchmark-relative power-utility portfolio wealthIts proportional diffusion coefficient is $\ell =ph+(1-p)v$. Since the wealth diffusion coefficient for dollar fractions is ${\sigma }^T\pi$, replication requiresUsing ${\sigma }^{-T}h=V^{-1}b$ gives the requested fixed proportions:The drift matches the self-financing portfolio equation as well. Indeed, $\ell =h+(1-p)(v-h)$, so the drift of $\log w^{\ast }$ from the explicit formula isConsequently an equivalent form of the solution isThe bank account fraction is $1-1^T{\pi }^{\ast }$, with negative values representing borrowing. The first stock term is ordinary power-utility risk exposure, while the second hedges the random benchmark. At the logarithmic limit $R=1$ the benchmark term disappears, consistently with $\log (w_T/J_T)=\log w_T-\log J_T$.