Mnrlwroricd Firinncr. zyxwvutsrq Vol. zyxwvutsr 5. No. 2 (April 1995). 121-131 ATTAINABLE CLAIMS IN A MARKOV MARKET' ALAIN BENSOUSSAN zyxwvu I.N.R.I.A., Rocquencourr, zyxwvu Lp Chesnay Cedex, France ROBERT J. ELLIUIT Department of Marhemarical Sciences, Universiv of Alberta, zyxw Edmonton, Alberta, Canada It is shown how, even when the market is incomplete, certain contingent claims are attainable: that is, they can zyxwvutsrqpo be represented zyxwvutsrq as stochastic integrals with respect to the process which describes the evolution of the asset prices. 1. INTRODUCTION In this section, following El Karoui and Quenez ( 1991), we give definitions of completeness of a financial market and attainability of a contingent claim. The objective of the paper is to show that in a market with Markov dynamics certain claims are attainable even if the market is incomplete. Such attainable claims have a unique, well-defined price and, using an explicit martingale representation theorem, we give the hedging policies. Consider, therefore,nriskyassetswhoSepricesattimer 2 Oare P'(r), P2(r), . . . , P"(r). We suppose the evolution of P'(t) is described by a stochastic differential equation of the form L j=l J Here W, = (w,', . . . , w:) is a d-dimensional Brownian motion on a probability space (52, zyxwvutsrqp F, Q) with respect to a filtration [Fl), r 2 0. There is also a riskless asset Po(t) such that (1.2) dPo(t) = Po(r)r(t) dt. It is usual to assume that the coefficients b'(.), dj(.), r(.) are predictable with respect to the filtration zyxwvutsr (A}: zyxwv a(.) denotes the matrix (c''(.)), and b(.) = (b'(,), . . ., b"(.)). We also suppose the coefficients are such that (1.1) and (1.2) have strong solutions. Write 1 for the vector (1, 1, . . . , 1) E R". To avoid arbitrage in the model, a second standard assumption is that there is an F,-predictable solution e(.) = (el(.), . . . , ed(.)) of the equation b(.) - r(-)1 = a(.)O(.). Manuscript received December 1992; final version received March 1994. 'This work was completed during a visit to INRIA in ApriMay 1992. The hospitality and support of MRIA is gratefully acknowledged. as is the support of NSERC grant A7964. @ 1995 Blackwell Publishers, 238 Main Street, Cambridge, MA 02142, USA, and 108 Cowley Road. Oxford, OX4 IJF. UK. 121