Completeness of security markets and solvability of linear backward stochastic differential equations

For a standard Black–Scholes type security market, completeness is equivalent to the solvability of a linear backward stochastic differential equation (BSDE, for short). An ideal case is that the interest rate is bounded, there exists a bounded risk premium process, and the volatility matrix has certain surjectivity. In this case the corresponding BSDE has bounded coefficients and it is solvable leading to the completeness of the market. However, in general, the risk premium process and/or the interest rate could be unbounded. Then the corresponding BSDE will have unbounded coefficients. For this case, do we still have completeness of the market? The purpose of this paper is to discuss the solvability of BSDEs with possibly unbounded coefficients, which will result in the completeness of the corresponding market.  2005 Elsevier Inc. All rights reserved