Abstract
In the paper expected utility from consumption over finite time horizon for discrete time markets with bid and ask prices and strictly concave utility function is considered. The notion of weak shadow price, i.e. an illiquid price, depending on the portfolio, under which the model without bid and ask price is equivalent to the model with bid and ask price is introduced. Existence and the form of weak shadow price is shown. Using weak shadow price usual (called in the paper strong) shadow price is then constructed.
Highlights
1 Introduction In this paper we study the problem of maximization of expected utility in the discrete time market with finite horizon and with transaction costs
We introduce the so called weak shadow price, i.e. a portfolio state dependent price process taking values between the bid and ask prices for which optimal value of expected utility in this frictionless
With the use of weak shadow price we construct shadow price, called in the paper strong shadow price, which is a sequence of random variables, playing the role of asset prices, taking values between bid and ask prices, depending on initial portfolio position, such that the optimal value of expected utility in the market with these asset prices is the same as in the market with transaction costs
Summary
In this paper we study the problem of maximization of expected utility in the discrete time market with finite horizon and with transaction costs. With the use of weak shadow price we construct shadow price, called in the paper strong shadow price, which is a sequence of random variables, playing the role of asset prices, taking values between bid and ask prices, depending on initial portfolio position, such that the optimal value of expected utility in the market with these asset prices is the same as in the market with transaction costs. Taking into account the fact that the random variables Sn and Sn, which represent the bid and ask prices of the stocks, are fully supported (see [11]), we are allowed to make an investment policy only in such a way that at time moment the amount on bank and stock accounts will be nonnegative almost surely This way we have short selling and short buying constraints. What is important we do not impose any additional conditions (besides of (1.1) and (A1)) for the processes S and S and we study the case with general strictly concave utility function
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