As is celebrated, Pontryagin's greatest precept and Bellman's dynamic programming are the 2 vital and most typically used techniques in fixing stochastic optimum keep an eye on difficulties. * a fascinating phenomenon one could detect from the literature is that those techniques were built individually and independently. given that either equipment are used to enquire a similar difficulties, a typical query one will ask is the fol lowing: (Q) what's the courting betwccn the utmost principlc and dy namic programming in stochastic optimum controls? There did exist a few researches (prior to the Nineteen Eighties) at the dating among those . however, the consequences frequently werestated in heuristic phrases and proved below relatively restrictive assumptions, that have been now not happy more often than not. within the assertion of a Pontryagin-type greatest precept there's an adjoint equation, that's a standard differential equation (ODE) within the (finite-dimensional) deterministic case and a stochastic differential equation (SDE) within the stochastic case. The process along with the adjoint equa tion, the unique kingdom equation, and the utmost is often called an (extended) Hamiltonian procedure. however, in Bellman's dynamic programming, there's a partial differential equation (PDE), of first order within the (finite-dimensional) deterministic case and of moment or der within the stochastic case. this can be often called a Hamilton-Jacobi-Bellman (HJB) equation.
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