Merton Portfolio Optimization
1. Risky Asset Dynamics (GBM)
\[
dS_t = \mu S_t\,dt + \sigma S_t\,dW_t
\]
2. Wealth Dynamics
\[
dX_t = (rX_t + \pi_t(\mu - r))\,dt + \pi_t \sigma\, dW_t
\]
3. Objective Function
\[
\max_{\{\pi_t\}} \; \mathbb{E}[U(X_T)]
\]
4. Hamilton–Jacobi–Bellman Equation
\[
\frac{\partial V}{\partial t}
+ \sup_{\pi}
\left\{
(rx + \pi(\mu - r)) V_x
+ \frac{1}{2}\pi^2\sigma^2 V_{xx}
\right\}
= 0
\]
\[
V(T,x) = U(x)
\]
5. CRRA Utility
\[
U(x) = \frac{x^{1-\gamma}}{1-\gamma}
\]
6. Closed-Form Value Function
\[
V(t,x) = \frac{x^{1-\gamma}}{1-\gamma} e^{k(T-t)}
\]
\[
k = (1-\gamma)
\left(
r + \frac{(\mu - r)^2}{2\gamma\sigma^2}
\right)
\]
7. Optimal Policy
\[
\pi^*(x) = \frac{\mu - r}{\gamma \sigma^2}\, x
\]
\[
\pi^*(x) = -\frac{\mu - r}{\sigma^2}
\frac{V_x}{V_{xx}}
\]
8. State Space Discretization
\[
t \in [0, T], \quad N_t \text{ steps}
\]
\[
x \in [x_{\min}, x_{\max}], \quad N_x \text{ grid points}
\]
\[
x_{\min} > 0
\]
\[
\Delta t = \frac{T}{N_t}, \quad
\Delta x = \frac{x_{\max} - x_{\min}}{N_x - 1}
\]
9. Terminal Condition
\[
V(T, x) = \frac{x^{1-\gamma}}{1-\gamma}
\]
10. Boundary Conditions
\[
V(t, x) =
\frac{x^{1-\gamma}}{1-\gamma}
\exp\left( k (T - t) \right)
\]
11. Finite-Difference Approximations
\[
V_x \approx \frac{V_{i+1} - V_{i-1}}{2\Delta x}
\]
\[
V_{xx} \approx \frac{V_{i+1} - 2V_i + V_{i-1}}{\Delta x^2}
\]
\[
\frac{V_i^{n+1} - V_i^n}{\Delta t}
+ \mathcal{L} V_i^{n+1}
= 0
\]
12. Tridiagonal Linear System
\[
a_i V_{i-1}^{n+1}
+ b_i V_i^{n+1}
+ c_i V_{i+1}^{n+1}
= V_i^n
\]
\[
a_i v_{i-1} + b_i v_i + c_i v_{i+1} = d_i
\]
13. PINN Derivatives
\[
V_t,\; V_x,\; V_{xx}
\]