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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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Minimizer for Mean-Variance Portfolio Optimization [closed]

Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set $$ \{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ? $$
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