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Density estimates for solutions to one dimensional backward SDE’s. (English) Zbl 1273.60066

The article shows that under sufficient conditions each component \((X_t,Y_t,Z_t)\) to a backward stochastic differential equation has a density. Moreover, there exist upper and lower Gaussian bounds for these densities. This result is especially derived for the components \(Y\) and \(Z\). For \(Y\) the authors first prove that the Malliavin derivatve of \(X_t\) is bounded and non-negative. Then, they use Nourdin-Viens formula for calculating the lower and upper bounds for the Malliavin derivatives of \(Y_t\). For the existence of the densitiy of \(Z\) it is first shown that die second-order Malliiavin derivatives of \(X\) and \(Y\) are non-negative. Then, the authors prove that \(DZ_t\) is positive a.s. and use the Bouleau-Hirsch Theorem. Finally, they derive lower and upper bounds for the density of \(Z\) by Gaussian estimates using again the Nourdin-Viens formula.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
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