We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocation problem, with the goal of minimizing the variance of the final wealth which results from trading over a fixed, finite horizon in a continuous-time, complete market, subject to the constraints that the expected final wealth equal a specified target value and the portfolio of the investor (defined by the dollar amount invested in each stock) take values in a given closed, convex set. The asset prices are modelled by Itô processes, for which the market parameters are random processes adapted to the information filtration available to the investor. We synthesize a dual optimization problem and establish a set of optimality relations, similar to the Euler-Lagrange and transversality relations of calculus of variations, giving necessary and sufficient conditions for the given optimization problem and its dual to each have a solution, with zero duality gap. We then solve these relations, to establish the existence of an optimal portfolio.

We apply conjugate duality to establish the existence of optimal portfolios in an asset-allocation problem, with the goal of minimizing the variance of the final wealth which results from trading over a fixed, finite horizon in a continuous-time, complete market, subject to the constraints that the expected final wealth equal a specified target value and the portfolio of the investor (defined by the dollar amount invested in each stock) take values in a given closed, convex set. The asset prices are modelled by Itô processes, for which the market parameters are random processes adapted to the information filtration available to the investor. We synthesize a dual optimization problem and establish a set of optimality relations, similar to the Euler-Lagrange and transversality relations of calculus of variations, giving necessary and sufficient conditions for the given optimization problem and its dual to each have a solution, with zero duality gap. We then solve these relations, to establish the existence of an optimal portfolio.

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We study a problem of stochastic control in mathematical finance, for which the asset prices are modeled by Itô processes. The market parameters exhibit "regime-switching" in the sense of being adapted to the joint filtration of the Brownian motion in the asset price models and a given finite-state Markov chain which models "regimes" of the market. The goal is to minimize a general quadratic loss function of the wealth at close of trade subject to the constraint that the vector of dollar amounts in each stock remains within a given closed convex set. We apply a conjugate duality approach, the essence of which is to establish existence of a solution to an associated dual problem and then use optimality relations to construct an optimal portfolio in terms of this solution. The optimality relations are also used to compute explicit optimal portfolios for various convex cone constraints when the market parameters are adapted specifically to the Markov chain.

No abstract