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Non Markovian stochastic linear-quadratic control : Volterra equations, rough volatility and equations with delay

Abstract : The present thesis deals with non Markovian linear-quadratic stochastic control problems. It is divided into three parts. In the first part, we tackle stochastic Volterra control problems whose kernel can be expressed as Laplace transform. Such assumptions is inspired from the rewriting of fractional Brownian motion as infinite sum of Markovian processes. The optimal control and value functions are expressed in terms of Banach valued Riccati equation whose existence and uniqueness are proved. In the second part, we revisit the celebrated multivariate Markowitz portfolio selection problem combined with rough volatility. The optimal control and efficient frontier are derived in terms of explicit Hilbert valued Riccati operator. The completely explicit feature of our analysis enables us to implement an easy numerical scheme that we illustrate in the the case of portfolio allocation with 2 assets, one rough H=0.1 and one smooth H=0.45. Surprisingly our simulations were able to reproduce the buy rough sell smooth strategy exhibited in [GH20a], thus providing an endogenous explanation over this allocation. Finally, the last part deals with the delayed control of stochastic differential equations. We solve a simplified version by means of Riccati PDEs whose existence and uniqueness are derived, provided a condition combining the horizon, the delay, the drift and the volatility is satisfied. A deep learning method is used to solve the Riccati PDEs in the context of Markovitz portfolio selection with execution delay.
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Submitted on : Tuesday, July 5, 2022 - 3:09:16 PM
Last modification on : Friday, August 5, 2022 - 3:00:08 PM

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Enzo Miller. Non Markovian stochastic linear-quadratic control : Volterra equations, rough volatility and equations with delay. Optimization and Control [math.OC]. Université Paris Cité, 2021. English. ⟨NNT : 2021UNIP7114⟩. ⟨tel-03714427⟩

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