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GKW representation theorem under restricted information. An application to risk-minimization. (English) Zbl 1300.60061

The authors construct a version of the Galtchouk-Kunita-Watanabe representation for a random variable that is measurable w.r.t.the final \(\sigma\)-field \(\mathcal{F}_T\) of some stochastic basis \(\mathbb{F}\), but the decomposition is considered w.r.t.the restricted information, say, \(\mathbb{H}\). Moreover, an explicit characterization of the \(\mathbb{H}\)-adapted integrand process in the decomposition is provided. The concept of \(\mathbb{H}\)-predictable dual projection is introduced to get the main results. As a consequence, the decomposition allows one to construct a solution to special stochastic differential equations whose driver is equal to zero, in a partial information framework. The applications to risk-minimization under restricted information are considered.

MSC:

60G48 Generalizations of martingales
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
91G10 Portfolio theory
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