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This paper concerns with numerical resolution of an impulse control problem under state constraints arising from optimal portfolio selection under liquidity risk and price impact. We show that the value function could be obtained as the limit of an iterative procedure where each step is an optimal stopping problem and the reward function is related to the impulse operator. Given the dimension of our problem and the complexity of its solvency region, we use a numerical approximation algorithm based on quantization procedure instead of finite difference methods to calculate the value function, the transaction and no-transaction regions. We also focus on the convergence of our numerical scheme, in particular, we show that it satisfies monotonicity, stability and consistency properties. We further enrich our studies with some numerical results for the optimal transaction strategy. B Mohamed Mnif mohamed.mnif@enit.rnu.tn M'hamed Gaigi

In this paper we obtain time uniform propagation estimates for systems of interacting diffusion processes. Using a well defined metric function h , our result guarantees a time-uniform estimates for the convergence of a class of interacting stochastic differential equations towards their mean field equation, and this for a general model, satisfying various conditions ensuring that the decay associated to the internal dynamics term dominates the interaction and noise terms. Our result should have diverse applications, particularly in neuroscience, and allows for models more elaborate than the one of Wilson and Cowan, not requiring the internal dynamics to be of linear decay. An example is given at the end of this work as an illustration of the interest of this result. 1

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