Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions
This paper deals with existence of smooth densities for solutions of Stochastic Differential Equations (SDEs) whose coefficients are smooth (i.e., admit bounded partial derivatives) and nondegenerate only on an open domain \(D\). More precisely, it is proved in Theorem 2.1 that under these conditions, a smooth density exists for the solution of the SDE. If in addition, the domain \(D\) is the complementary of a compact ball and under additional assumptions on the coefficients, asymptotics (meaning for large values of the state variable) upper bounds for the density are given in Theorem 2.2. In addition, the dependence between the bounding constants and the coefficients of the SDE is made explicit. These results find applications in Finance where for instance stochastic volatility models like the Heston model are defined \textit{via} such non-classical (non-Lipschitz) SDEs. These results are presented in Section 3. The proofs of the main results are presented in Section 2 and make use of the Malliavin calculus and of a Fourier transform argument.
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