Assume an investor has utility function $U(C_t)=\frac{C_t^\gamma}{\gamma}$. The investor wishes to consume some of their wealth at a rate $C_t$ per unit time, and invest in both risk-free bonds and a risky asset. Let $λ_t$ be the proportion of total wealth, $W_t$, invested in risky assets at time t and,
$$ dW_t = ((\lambda_t(\mu-r) + r )W_t-C_t)dt + \sigma\lambda_t W_t dX_t $$
where $dX_t$ is the infinitesimal increment of a standard Brownian motion.
Define the Bellman function,
$$ J(W, t)=\max_{\lambda_t, C_t} \mathbb{E}_t\left[\int_t^\infty U(C_\tau) d\tau \right] $$
Is the following the correct way to show that $J(W, t)=J(W,0)$?
Using change of varables $ \rho = \tau-t$, we have,
$$ \begin{align} J(W, t)&=\max_{\lambda_t, C_t} \mathbb{E}_t\left[\int_t^\infty U(C_\tau) d\tau \right]\\ &=\max_{\lambda_t, C_t} \mathbb{E}_t\left[\int_0^\infty U(C_{\rho+t}) d\rho \right]\\ &=\left[\int_0^t U(C_{\rho+t}) d\rho \right]+\max_{\lambda_t, C_t} \mathbb{E}_t\left[\int_t^\infty U(C_{\rho+t}) d\rho \right]\\ \end{align} $$
... and this somehow equals $J(w, 0)$?
I'm not sure how to make the argument flow through though.
Any help is appreciated!