We propose a continuous-time version of the adaptive robust methodology introduced in Bielecki et al. ( 2019). An agent solves a stochastic control problem where the underlying uncertainty follows a jump-diffusion process and the agent does not know the drift parameters of the process. The agent considers a set of alternative measures to make the control problem robust to model misspecification, and employs a continuous-time estimator to learn the value of the unknown parameters to make the control problem adaptive to the arrival of new information. We use measurable selection theorems to prove the dynamic programming principle of the adaptive robust problem and show that the value function of the agent is characterised by a non-linear partial differential equation. As an example, we derive the optimal adaptive robust strategy for an agent who acquires a large amount of shares in an order driven market and illustrate the financial performance of the execution strategy.
We show the convergence of an online stochastic gradient descent estimator to obtain the drift parameter of a continuous-time jump-diffusion process. The stochastic gradient descent follows a stochastic path in the gradient direction of a function to find a minimum, which in our case determines the estimate of the unknown drift parameter. We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.
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