## Optional decomposition and Lagrange multipliers.(English)Zbl 0894.90016

Summary: Let $${\mathcal Q}$$ be the set of equivalent martingale measures for a given process $$S$$, and let $$X$$ be a process which is a local supermartingale with respect to any measure in $${\mathcal Q}$$. The optional decomposition theorem for $$X$$ states that there exists a predictable integrand $$\varphi$$ such that the difference $$X- \varphi\cdot S$$ is a decreasing process. In this paper, we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.

### MSC:

 91B28 Finance etc. (MSC2000) 60H05 Stochastic integrals
Full Text: