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The second order Mean Field Games system (MFGS) in a bounded domain with thelateral Cauchy data is considered. This means that both Dirichlet and Neumannboundary data for the solution the MFGS are given. Two H\"older stabilityestimates for two slightly diffeent cases are derived. These estimates indicatehow stable the solution of the MFGS is with respect to the possible noise inthe lateral Cauchy data. Our stability estimates imply uniqueness. The keymathematical apparatus is the apparatus of two new Carleman estimates.

On The Mean Field Games System With the Lateral Cauchy Data via Carleman Estimates