KARATZAS, I., LEHOCSKY, J.P., SHREEVE, S.E. and XU, G.-L.. : Martingale and Duality methods for Utility Maximisation in an Incomplete Market. Preprint.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that the market is complete we mean that any candidate terminal wealth which is measurable with respect to the ltration generated by W alone can be represented as the sum of an initial wealth plus a gains from trade from an admissible strategy . The next result, adapted from Karatzas et al. [15], describes the form of the optimal target wealth random variable X T . 1. Then, for initial wealth x, the optimal target portfolio X T is given by X T I( Z T ) where is a Lagrange multiplier chosen to satisfy x e E [Z T I( Z T ) For w 0 de ne (w) e E [Z T ....

KARATZAS, I., LEHOCSKY, J.P., SHREVE, S.E. and XU, G.-L.. : Martingale and Duality methods for Utility Maximisation in an Incomplete Market. SIAM J. Control & Optimiasation, 29, 702-730, 1991.

....or explicit assumptions about utilities and preferences. The completeness property of a market can be lost in many ways; for example by the introduction of transaction costs, see [6] or when a small number of stocks have price processes driven by a larger number of Brownian motions (see [8]) The situation we choose to model is as follows: Consider an individual who is free to trade in an asset with price process P t . How much should the individual be prepared to pay now in order to receive a random payment Y at some pre determined time T in the future This problem is similar to ....

KARATZAS, I., LEHOCSKY, J.P., SHREEVE, S.E. and XU, G.-L.. : Martingale and Duality methods for Utility Maximisation in an Incomplete Market. Preprint.

....from below. 2 This notion turned out to be very useful for no arbitrage arguments (compare [HP 81] DS 94] and [DS 98] In the context of utility maximization for functions taking finite values only on R while being Gamma1 on R Gamma (typical example: U(x) ln(x) as analyzed, e.g. in [KLSX 91] and [KS 99] this concept also proved to be the appropriate one. Also in the present setting of utility functions taking finite values on all of R we may and do use this class of trading strategies to define the value function u(x) in (2) However in the present case this concept obviously ....

I. Karatzas, J.P. Lehoczky, S.E. Shreve, G.L. Xu, (1991), Martingale and duality methods for utility maximisation in an incomplete market. SIAM Journal of Control and Optimisation, Vol. 29, pp. 702--730.

.... which, under condition (1) is a smooth, convex function satisfying V (0) U(1) V (1) 1; V 0 (0) Gamma1; V 0 (1) 1: 12) We have the relation U 0 = GammaV 0 ) Gamma1 and we denote by I the inverse function (U 0 ) Gamma1 (which is equal to GammaV 0 ) compare [R70] [KLSX91], KS99] We also note the formula V (y) U(I(y) Gamma yI(y) 13) which will be used several times below. To give a concrete example: for U(x) Gammae Gammax we obtain V (y) y(ln(y) Gamma1) U 0 (x) e Gammax and V 0 (y) ln(y) A by now classical route to solve the ....

....be used several times below. To give a concrete example: for U(x) Gammae Gammax we obtain V (y) y(ln(y) Gamma1) U 0 (x) e Gammax and V 0 (y) ln(y) A by now classical route to solve the (primal) optimization problem (6) is to pass to the dual problem (see, e.g. B73] P86] [KLSX91]) v(y) inf Q2M a (S) E V y dQ dP : 14) Again the question arises about the appropriate domain over which the dual optimization problem is minimized. Morally speaking the proper set consists of the equivalent martingale probability measures for the process S: by this we mean ....

Karatzas, I., Lehoczky, J.P., Sethi, S.P., Xu, G.L. Martingale and duality methods for utility maximisation in an incomplete market. Journal of Control and Optimisation 29, (1991), pp. 702--730.

....new measure we have that W becomes a Brownian Motion with drift 1 and so oe 1 a.s. The product L oe M oe becomes a uniformly integrable martingale The problem whether the product of two strictly positive strict local martingales could be a uniformly integrable martingale goes back to Karatzas, Lehoczky and Shreve (1991). Lepingle (1993) gave an example in discrete time. Independently Karatzas, Lehoczky and Shreve gave also such an example but the problem remained open whether such a situation could occur for continuous local martingales. The first example on the continuous case was given in Schachermayer (1993) ....

I. Karatzas, J.P. Lehoczky and S. Shreve (1991), Martingale and Duality Methods for Utility Maximisation in an Incomplete Market, SIAM J. Control Optim. 29, 702--730.

....and the stock to form a riskless hedge portfolio for the option. However this is not the case so there is no riskless hedge and the prices of options will depend on the risk preferences of investors. These preferences may be expressed via a utility function (see Hodges and Neuberger (1989) or Karatzas, Lehoczky, Shreve and Xu (1991)) or via a local risk minimisation criterion (Hofmann, Platen and Schweizer (1992) or Platen and Schweizer (1994) Substituting for doe we obtain dH = H 1 P dP P H 2 dt flae oe dP P Gamma dt # fl q 1 Gamma ae 2 dZ dt = H 1 P flaeH 2 oe dP P H 2 fl ....

KARATZAS, I., LEHOCZKY, J.P., SHREVE, S.E. and XU, G.-L.. (1991): Martingale and duality methods for utility maximisation in an incomplete market. SIAM Journal of Control and Optimisation, 29, 702-730.

....finite probability space model. Considerably more difficult is the case of incomplete financial models. It was studied in a discrete time, finite probability space model by He and Pearson [16] and in a continuous time diffusion model by He and Pearson [17] and by Karatzas, Lehoczky, Shreve and Xu [21]. The central idea here is to solve a dual variational problem and then to find the solution of the original problem by convex duality, similarly to the case of a complete model. In a discrete time, finite probability space model the solution of the dual problem is always a martingale measure. We ....

.... 1: To the best of our knowledge the notion of the asymptotic elasticity of a utility function has not been defined in the literature previously. We refer to Section 6 below for equivalent reformulations of this concept, related notions which have been investigated previously in the literature [21] and its intimate relation to the Delta 2 condition in the theory of Orlicz spaces. For the moment we only note that many popular utility functions like U(x) ln(x) or U(x) x ff ff , for ff 1, have asymptotic elasticity strictly less than one. On the other hand, a function U(x) ....

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. I. Karatzas and J.P. Lehoczky and S.E. Shreve and G.L. Xu, Martingale and duality methods for utility maximisation in an incomplete market, SIAM Journal of Control and Optimisation 29 (3) (1991), 702 -- 730.

....It is also equivalent to pricing options under the minimal martingale measure of Follmer and Schweizer (1990) Suppport for this idea is to be found in Hofmann et al. (1993) However other authors propose different criteria for determining b. For a utility based approach see for example Karatzas et al. (1991) and Duffie and Skiadas (1994) The implications for options pricing of a stochastic volatility model have been considered by many authors. Stein and Stein (1991) who assume no correlation between the pair of Brownian motions driving the asset price and the volatility, find implied volatility ....

KARATZAS, I., LEHOCZKY, J.P., SHREVE, S.E. and XU, G.-L.. (1991): Martingale and duality methods for utility maximisation in an incomplete market. SIAM Journal of Control and Optimisation, 29, 702-730.

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KARATZAS, I., LEHOCSKY, J.P., SHREEVE, S.E. and XU, G.-L.. : Martingale and Duality methods for Utility Maximisation in an Incomplete Market. Preprint.

No context found.

KARATZAS, I., LEHOCSKY, J.P., SHREVE, S.E. and XU, G.-L.. : Martingale and Duality methods for Utility Maximisation in an Incomplete Market. SIAM J. Control & Optimiasation, 29, 702-730, 1991.

No context found.

I. Karatzas, J.P. Lehoczky, S.E. Shreve, G.L. Xu (1991), Martingale and duality methods for utility maximisation in an incomplete market, SIAM J. of Control and Optimisation 29, 702-730.

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