Boundary-Induced Apparent Risk Aversion in Nonergodic Multiplicative Growth

Finite multiplicative systems often cease to evolve when a lower continuation threshold is reached, whereas standard growth-optimal benchmarks assume uninterrupted continuation. We study a finite-horizon multiplicative process in which a fixed exposure is chosen ex ante and paths crossing an absorbing boundary are assigned a residual value. In the continuous-time limit the absorbed objective is available in closed form, and three results follow analytically. Costly absorption compresses exposure below the no-boundary Kelly fraction near the boundary; we prove that this holds strictly whenever the boundary is reachable within the horizon. The dependence on the horizon is non-monotone, and the optimum returns to a finite, cliff-independent long-horizon limit from below or from above according to an exact criterion, at which the threshold is the geometric mean of the residual value and the initial wealth, independently of drift and volatility. When interpreted through an unconstrained constant-relative-risk-aversion benchmark, this compression appears as elevated risk aversion. Exact lattice propagation confirms the analytical predictions without fitted parameters. Absorbing-boundary geometry can therefore generate state-dependent risk-averse-looking behavior without heterogeneous primitive preference parameters.