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We introduce some sparse grids interpolations used in Semi-Lagrangian schemesfor linear and fully non-linear diffusion Hamilton Jacobi Bellman equationsarising in stochastic control. We prove that the method introduced convergestoward the viscosity solution of the problem and we show that some potentiallyhigh order schemes can be efficiently implemented. Numerical test in dimension$2$ to $5$ are achieved and show that deterministic methods can be usedefficiently in stochastic control in moderate dimension.

Adaptive sparse grids for time dependent Hamilton-Jacobi-Bellman equations in stochastic control