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A new probabilistic approach to non local and fully non linear second order partial differential equations

We prove that weakly continuous solutions to martingale problems admit a canonical regular conditional probability distribution. This allows for the construction of time consistent convex dynamic procedures in a non dominated setting. Making use of the martingale problem approach for continuous diffusions and diffusions with Levy generator, we give an explicit construction of such procedures having furthermore a Feller property. These procedures lead to viscosity solution of fully non linear second order partial differential equations in case of continuous diffusions. In case of diffusions with Levy generator this provides a probabilistic approach for the resolution of non local fully non linear second order PDE.