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Quantitative Finance · Glossary

What is Hamilton–Jacobi–Bellman equation?

Also known as: Hamilton--Jacobi--Bellman equation

Definition 9.4 Quantitative Methods · Chapter 9 — Stochastic Control

For the controlled diffusion of Definition 9.1, the Hamilton–Jacobi–Bellman equation is

∂tV(t,x)+sup⁡u∈U{μ(t,x,u)∂xV+12σ2(t,x,u)∂xxV+f(t,x,u)}=0,V(T,x)=g(x).\partial_tV(t, x) + \sup_{u\in\mathcal U}\Bigl\{\mu(t, x, u)\partial_xV + \tfrac12\sigma^2(t, x, u)\partial_{xx}V + f(t, x, u)\Bigr\} = 0, \qquad V(T, x) = g(x).
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