We consider a controlled diffusion process
dX_t = b(X_t, u_t)\,dt + \sigma(X_t, u_t)\,dW_twhere u_t \in \mathcal{U} is the control process. The value function
V(x) = \inf_{u \in \mathcal{U}} \mathbb{E}\left[\int_0^\infty e^{-\rho t} f(X_t, u_t)\,dt \,\Big|\, X_0 = x\right]satisfies the Hamilton-Jacobi-Bellman (HJB) equation:
\rho V(x) = \inf_{u \in \mathcal{U}} \left\{ f(x,u) + b(x,u) \cdot \nabla V(x) + \frac{1}{2} \mathrm{tr}\left[\sigma\sigma^T(x,u) D^2 V(x)\right] \right\}We establish existence and uniqueness of a viscosity solution V \in C(\mathbb{R}^n) under the following assumptions:
- The drift b and diffusion \sigma are Lipschitz in x, uniformly in u
- The running cost f satisfies a coercivity condition
- The discount rate \rho > 0 is strictly positive
The comparison principle for the HJB equation follows from the structure of the nonlinearity and is the key ingredient in the uniqueness proof.
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