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Weak error for stable driven stochastic differential equations: expansion of the densities. (English) Zbl 1235.60065
Consider the stochastic differential equation \[ X_t=x+\int_0^t b(X_s)\, ds+\int_0^tf(X_{s-})dZ_s,\quad 0\leq t\leq T \] for a symmetric stable process \(Z\), and the Euler approximation \(X_t^N\) for a time step \(h=T/N\). Under suitable assumptions, the laws of \(X_T\) and \(X_T^N\) have densities \(p(T,x,y)\) and \(p^N(T,x,y)\), and by means of a parametrix approach, an expansion of \(p-p^N\) in powers of \(h\) is obtained.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
65C30 Numerical solutions to stochastic differential and integral equations
60G52 Stable stochastic processes
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