The HJB equation extends the deterministic Hamilton-Jacobi equation by adding the term (1/2)σ²∂²V/∂x². This second-order term arises from:
AThe uncertainty in the initial condition
BThe Itô correction — applying Itô's formula to V(X(t),t) produces a second-order term from the quadratic variation of X
CNumerical discretization error in the dynamic programming equation
DThe convexity of the cost functional J(u)
The HJB equation is derived by applying Itô's formula to the value function V(X(t),t) and requiring that the resulting process (after subtracting the running cost) be a supermartingale for all controls and a martingale for the optimal control. The (1/2)σ² ∂²V/∂x² term comes from Itô's formula — specifically from (dX)² = σ²dt. In the deterministic case (σ = 0), this term vanishes and the HJB equation reduces to the Hamilton-Jacobi equation of classical mechanics and optimal control.
Question 2 Multiple Choice
A verification theorem states: if a smooth function V solves the HJB equation with terminal condition V(x,T) = g(x), and the control u*(x,t) = argmin of the HJB minimization is admissible, then V is the value function and u* is optimal. Why is this called 'verification' rather than 'derivation'?
ABecause the HJB equation may not have a smooth solution, so assuming smoothness is a hypothesis that must be verified
BBecause the derivation of HJB involves non-rigorous infinitesimal arguments that need verification
CBoth — the HJB equation is derived heuristically, and its solution may not be smooth enough for the argument to work without additional verification
DBecause the optimal control might not exist even when V exists
The derivation of HJB uses the dynamic programming principle (an infinitesimal Bellman equation), which is heuristic. The verification theorem reverses the logic: start with a candidate V that solves HJB, then prove rigorously (via Itô's formula) that it equals the value function. The catch is that HJB may only have viscosity solutions (not C² smooth), in which case the classical verification theorem doesn't apply and the weaker theory of viscosity solutions is needed. In practice, many important problems have smooth solutions and the verification approach works directly.
Question 3 Short Answer
In the Merton portfolio problem, an investor with power utility U(x) = x^γ/γ (γ < 1) chooses what fraction π of wealth to invest in a risky asset. The optimal fraction turns out to be constant: π* = (μ-r)/((1-γ)σ²). Explain why this is remarkable.
Think about your answer, then reveal below.
Model answer: The optimal allocation is constant — it doesn't depend on wealth, time, or the state of the market. Despite the problem being a stochastic control problem with a continuous-time objective and random dynamics, the solution has the same structure as a static mean-variance optimization: invest a fixed fraction proportional to the Sharpe ratio (μ-r)/σ and inversely proportional to risk aversion (1-γ). This myopic property is special to power/log utility and GBM dynamics. With stochastic volatility, mean-reverting returns, or non-CRRA utility, the optimal fraction becomes state-dependent and the problem is genuinely dynamic.
The Merton fraction π* = (μ-r)/((1-γ)σ²) separates into the Sharpe ratio (μ-r)/σ² (how attractive the risky asset is) and 1/(1-γ) (how risk-tolerant the investor is). For γ → 0 (log utility), π* = (μ-r)/σ². The constancy of π* means the value function has a separable form V(w,t) = (w^γ/γ)f(t), which reduces the HJB PDE to an ODE for f(t) — the problem is exactly solvable.