Viability and martingale measures under partial information

We consider a financial market with a single risky asset whose price process S(t) is modeled by a jump diffusion, and where the agent only has access to a given partial information flow {Et}t≥0. Mathematically this means that the portfolio φ is required to be E predictable. We let AE denote the set of admissible portfolios. If U is a given utility function, we say that the market is (E , U) -viable if there exists a portfolio φ∗ ∈ AE (called an optimal portfolio) such that sup φ∈AE E[U(Xφ(T ))] = E[U(Xφ∗(T ))]. (0.1) We prove that, under some conditions, the following holds: The market is (E , U)-viable if and only if the measure Q∗ defined by dQ∗ = U (Xφ∗(T )) E[U ′(Xφ∗(T ))] dP on FT (0.2) is an equivalent local martingale measure (ELMM) with respect to E and with respect to the Et-conditioned price process S̃(t) := EQ∗ [S(t) | Et] ; t ∈ [0, T ]. (0.3) This is an extension to partial information of a classical result in mathematical finance. We also obtain a characterization of such partial information optimal portfolios in terms of backward stochastic differential equations (BSDEs), which is a result of independent interest. ∗INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, Le Chesnay Cedex, 78153, France, and Université Paris-Est, email: [email protected] †Center of Mathematics for Applications (CMA), Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N–0316 Oslo, Norway, email: [email protected] The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [228087]. ‡INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, Le Chesnay Cedex, 78153, France, and Université Paris-Est, email: [email protected]