I'm looking into some classical utility maximization problems. In particular, I'm interested in looking at the wealth evolution where you invest your money across $n$ assets and each time period you have a deterministic cash inflow or outflow. I would like to know a reference request for the following:

Is there a research paper out there, where

  • Multidimension: The underlying market has $n$ assets to trade (multidimensional). I'm more than happy to look at a simple example first like GBM or a discrete setup, i.e. asset can be traded as $$ dS^i = S^i(\mu_i dt + \sigma_{i}dW_{i}) $$ with correlated Brownian Motions, $E[dW^idW^j]=\rho_{ij}dt$
  • Deterministic inflows / outflows: Each time period (or if continuous setup, continuously) you can contribute deterministic cash inflows or cash outflows. If it is a inflow this should be invested in the portfolio. If it is an outflow you basically reduce part of your holdings.
  • Terminal wealth distribution: I would be interested if it is possible to get a nice formula or computationally feasible to treat for the terminal wealth distribution. Even more, if there are formulas for quantities like, $P(W_T\le \alpha)$ where $W_T$ is your terminal wealth, this would be great. In general I'm interested in a terminal "utility" of the form of reaching a certain amount.

I've seen different papers covering one or two points of the above but not a single one covering all of them. This is a pure theoretical interest so I'm more than happy to see examples with simplifying assumptions as above or in a discrete case with iid returns too. Any hint would be appreciated


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