# Existence and Asymptotic Behavior of an Optimal Barrier for an Optimal Consumption Problem in a Brownian Model with Absorption and Finite Time Horizon

Existence and Asymptotic Behavior of an Optimal Barrier for an Optimal Consumption Problem in a... We consider the following optimal control problem: Maximize $\mathbb{E}( \int_{(0,\tau)}e^{-\beta s}\,dC_{s}+e^{-\beta\tau}X_{\tau})$ , where X t = x + μt + σW t − C t , τ ≡inf{ t >0| X t =0}∧ T , T >0 is a fixed finite time horizon, W t is standard Brownian motion, μ , σ are constants, and C t describes accumulated consumption until time t . It is shown that the optimal strategy is given by a barrier strategy with time-dependent continuous barrier b ( t ). Moreover we determine the asymptotic behavior of b ( t ) for t → T . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Existence and Asymptotic Behavior of an Optimal Barrier for an Optimal Consumption Problem in a Brownian Model with Absorption and Finite Time Horizon

, Volume 69 (2) – Apr 1, 2014
39 pages

/lp/springer_journal/existence-and-asymptotic-behavior-of-an-optimal-barrier-for-an-optimal-aNU0UogIQw
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9223-3
Publisher site
See Article on Publisher Site

### Abstract

We consider the following optimal control problem: Maximize $\mathbb{E}( \int_{(0,\tau)}e^{-\beta s}\,dC_{s}+e^{-\beta\tau}X_{\tau})$ , where X t = x + μt + σW t − C t , τ ≡inf{ t >0| X t =0}∧ T , T >0 is a fixed finite time horizon, W t is standard Brownian motion, μ , σ are constants, and C t describes accumulated consumption until time t . It is shown that the optimal strategy is given by a barrier strategy with time-dependent continuous barrier b ( t ). Moreover we determine the asymptotic behavior of b ( t ) for t → T .

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2014

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