Mathematics / research
Stochastic control and optimal stopping
Choose actions under uncertainty using dynamic programming, martingale verification, stopping rules, and viscosity solutions.
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Module 01 research
Stochastic control and optimal stopping
Choose actions under uncertainty using dynamic programming, martingale verification, stopping rules, and viscosity solutions.
- 01
Controlled diffusions and admissible controls
Specify the information, integrability, and well-posedness needed before optimizing a noisy state equation.
36 min - 02
The dynamic programming principle
Restart an optimization consistently and compute a two-stage policy with action-dependent transitions.
36 min - 03
Stochastic Hamilton–Jacobi–Bellman equations
Derive the second-order Bellman equation and solve a finite-horizon quadratic diffusion problem.
36 min - 04
Verification theorems and feedback controls
Turn a smooth HJB candidate into a certified lower bound and construct an admissible minimizing feedback.
36 min - 05
Linear–quadratic–Gaussian control
Separate estimation and control under explicit Gaussian observation assumptions and solve a conditional quadratic decision.
36 min - 06
Optimal stopping and Snell’s envelope
Construct the smallest dominating supermartingale and separate stopping rules from hindsight.
36 min - 07
Free-boundary problems and smooth fit
Solve a discounted Brownian stopping threshold and verify where derivative matching is legitimate.
36 min - 08
Viscosity solutions and comparison
Use smooth touching functions at nonsmooth values and state a precise comparison regime.
36 min - 09
The stochastic maximum principle
Fix a minimizing Hamiltonian convention and check an adjoint backward equation in an LQ diffusion.
36 min - 10
Risk-sensitive and robust stochastic control
Track exponential-cost signs, entropy tilting, and the conditions behind a robust Isaacs equation.
36 min