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5The no arbitrage pricing of Guaranteed Minimum Withdrawal Benefits (GMWB) contracts 6 results in a singular stochastic control problem which can be formulated as a Hamilton Jacobi

The piezoelectric properties of (K 0.5 Na 0.5 )NbO 3 (KNN) are normally enhanced by chemical substitutions or doping to form solid solutions. In this study, we report that the piezoelectric properties of KNN and thermal stability of piezoelectric coefficient d 33 can be both enhanced by forming the composite of KNN:ZnO. The d 33 of KNN:0.2ZnO can be improved to 110 pC/N by introducing the ZnO nanoparticles, which is better than the pure KNN (d 33 = 85 pC/N). The Curie temperature (T C = 407°C) remains well comparable to the pure KNN (T C = 408°C). Furthermore, the thermal stability of both remanent polarization (P r ) and piezoelectric parameter (d 33 ) is improved. The enhanced thermal stability could be related to the induced built-in electric field or the enhanced sinterability by the addition of ZnO. The present results may help to optimize the piezoelectric properties of lead-free materials by forming composite.