Completeness of security markets and backward stochastic differential equations with unbounded coefficients

Abstract

For a standard Black–Scholes-type security market, completeness is equivalent to the solvability of a linear backward stochastic differential equation (BSDE). When the interest rate is bounded, there exists a bounded risk premium process, and the volatility matrix has certain surjectivity, then the BSDE will be solvable and the market will be complete. However, if the risk premium process and/or the interest rate is not bounded, one gets a BSDE with unbounded coefficients to solve. In this paper, we will discuss such a situation and will present some solvability results for the BSDE which will lead to the completeness of the market.