Existence and Uniqueness of an Optimum in the Infinate-Horizon Portfolio-cum-Saving Model with Semimartingale Investments
The model considered here is essentially that formulated in the authors previous paper Conditions for Optimality in the Infinite-Horizon Portfolio-cum Saving Problem with Semimartingale Investments, Stochastics 29 (1990) pp.133-171. In this model, the vector process representing returns to investments is a general semimartingale. Processes defining portfolio plans are here required only to be predictable and non-negative. Existence of an optimal portfolio-cum-saving plan id proved under slight conditions of integrability imposed on the welfare functi nal; the proofs rely on properties of weak precompactness of portfolio and utility sequences in suitable Lp spaces together with dominated and monotone convergence arguments. Conditions are also obtained for the uniqueness of the portfolio plan generating a given returns process and for the uniqueness of an optimal plan; here use is made of random measures associated with the jumps of a semimartingale.
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