A stochastic control problem chooses an admissible control input using information available in a filtration to optimize an expected objective for a random dynamical system. Dynamic programming often characterizes its value function through a Hamilton-Jacobi-Bellman equation.
A controlled Markov process has transition probabilities or rates depending on its current state and the chosen control action. A Markov policy selects an action from the current state and possibly the time. Once such a policy is fixed, the induced state evolution is a Markov process. For general history-dependent controls, the state alone need not remain Markov. A dynamic programming operator optimizes over the available actions at each state.
A Markov decision process specifies states, admissible actions, action-dependent transition probabilities and rewards. A policy chooses actions from observed information; a time-homogeneous problem uses transition and reward rules independent of the date. A Bellman equation optimizes current reward plus the conditional value of subsequent decisions. For nonnegative rewards, finite-horizon values increase as the horizon grows, while identification of their limit with the infinite-horizon value requires an approximation argument for admissible policies.
Estimating the expected return under a specified control policy. It is distinct from improving or optimizing that policy. A Bellman expectation equation describes the fixed-policy value, whereas a Bellman optimality equation compares admissible actions.
Policy iteration alternates evaluation of a fixed decision policy with improvement of its actions using the resulting value function. A stable policy satisfies the Bellman equation when each chosen action attains the required optimum.
In singular stochastic control, a control is a finite-variation process rather than only a rate integrated against time. Its increments can have atoms or a part singular with respect to time. The Hamilton-Jacobi-Bellman equation becomes a variational inequality with a constraint on directional derivatives of the value function.
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Stochastic control is a branch of control theory that deals with decision-making in systems that are subject to randomness and uncertainty. Unlike deterministic control, where the system dynamics and external influences are predictable, stochastic control involves managing systems where future states are influenced by random variables. The key components of stochastic control include: 1. **State Space**: This describes all possible states the system can occupy. In stochastic control, the state can change randomly over time.