Say I have a portfolio, $X_t$, using a leverage of $f$, such that the dynamics are given by \begin{equation} dX_t = \mu f X_t dt + \sigma f X_t dW_t \end{equation} I want to optimize the expected utility after some time $T$, $E[U(V_T)]$, and find the optimal leverage $f$. With the utility function $U(x)=\frac{x^\gamma}{\gamma}$ this is fairly easy. The SDE can be solved and the expected utility is maximized with $f^* = \frac{\mu}{\sigma^2 (1-\gamma)}$. With $\gamma=0$ and log-utility this is just the Kelly criterion.
But what if I also have to pay a constant cost $C$ such that the dynamics are \begin{equation} dX_t = (\mu f X_t - C) dt + \sigma f X_t dW_t, \quad X_t > 0 \end{equation} and $dX_t = 0$ when $X_t=0$ (i.e., I go bankrupt). The utility function would need to be altered to account for the non-zero probability of bankruptcy, so $U(x)=\frac{(x + b)^\gamma}{\gamma}$ with some $b>0$ so that the utility is bounded at bankruptcy.
Is there any way I can formulate the problem such that I can get an expression for the optimal $f^*$ that maximizes the expected utility $E[U(V_T)]$ when costs are included?