## Optimal investment-consumption models with constraints |

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a.e. Vt admissible controls asset assumption b-R vx(x bankruptcy Bellman equation bonds and stocks bounded Chapter compact set concave constraints continuous function contradiction control variables current wealth decreasing defined derivatives dS(t)Pt dynamic programming equation dynamic programming principle exists a constant exists a point F(ux finite following cases separately geometric Brownian motion horizon problem infinite horizon interval invested in bonds investor Jensen's inequality Lemma Let us assume linear Lipschitz functions locally Lipschitz Lu(z martingale max[-cu U(c maximum obtain optimal consumption optimal controls optimal policy optimal portfolio point x0 positive constant probability space Proof Proposition prove rate of return region rxux satisfies set of admissible short-selling stay nonnegative stochastic strictly increasing subsolution terminal wealth Theorem transaction costs transaction fees U(Ct)dt utility function ux(x value function variational inequality viscosity solution wealth Xt Wiener process yields zero