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Dynamic Corporate Transformation

Vol. II, Ch. 8 · Part 2. Dynamic Enterprise Optimization · Week 11

Hamilton–Jacobi–Bellman Enterprise Framework

Learning outcomes

After completing this chapter, the reader should be able to:

  1. Explain the relationship between optimal control and dynamic programming in continuous time.
  2. Derive the Hamilton–Jacobi–Bellman equation from the continuous-time Bellman principle.
  3. Construct continuous-time enterprise value functions.
  4. Interpret HJB equations economically and mathematically.
  5. Solve simple continuous-time enterprise optimization problems in closed form via value-function ans"atze.
  6. Distinguish necessary conditions (Pontryagin) from sufficient conditions (verification).
  7. Analyze enterprise feedback policies and closed-loop systems.
  8. Formulate continuous-time enterprise optimization models in HJB form.
  9. Implement numerical HJB approximations by monotone upwind schemes.
  10. Prepare enterprise models for stochastic optimization.

Reading guide

Work through the chapter in section order; the full development, proofs, and worked examples are in the book — this page indexes them and does not replace them.

  1. Motivation for Hamilton–Jacobi–Bellman Theory

    Motivation for Hamilton–Jacobi–Bellman Theory

  2. Continuous-Time Dynamic Programming

    Continuous-Time Dynamic Programming

  3. Enterprise Value Functions

    Enterprise Value Functions

  4. Derivation of the HJB Equation

    Derivation of the HJB Equation

  5. Interpretation of the HJB Equation

    Interpretation of the HJB Equation

  6. Feedback Enterprise Policies

    Feedback Enterprise Policies

  7. Verification Theorems

    Verification Theorems

  8. Numerical Approximation Methods

    Numerical Approximation Methods

  9. Enterprise Applications

    Enterprise Applications

  10. Computational Challenges

    Computational Challenges

  11. Preparation for Stochastic Enterprise Optimization

    Preparation for Stochastic Enterprise Optimization

  12. Chapter Summary

    Chapter Summary

  13. Worked Examples

    Worked Examples

  14. Exercises

    Exercises

  15. Notes and Sources

    Notes and Sources

AXIOM

This chapter is instrumented by:

Launch the module, load the chapter model, modify inputs, run the optimization, and compare against the worked examples in the book.

Exercises

12 exercises, grouped A concept checks · B mathematical · C computational · D enterprise applications. Starred (★) exercises are on the advanced track. Full solutions appear in the Instructor's Manual, Chapter 8.

A. Concept checks

  1. 8.1
    Explain, using Table (see book) and the asset- pricing reading of (see book), why a strict inequality ρV(x)>max⁡u{L+∇V⊤f}\rho V(\x) > \max_u\{L + \nabla V^\top f\} at some state signals overvaluation, and what governance action each inequality direction warrants.
  2. 8.2
    For each worked example, identify the verification items (V1)–(V3) and state which one would fail first under (a) an unbounded decision set, (b) a misdeclared terminal value.

B. Mathematical exercises

  1. 8.3
    Assemble the Lipschitz estimate for VV claimed in \S(see book) from (B1)–(B2), Gr"onwall, and the Bellman identity (see book).
  2. 8.4
    Complete the LQ transversality check of Example (see book): show e−ρTV(x(T))→0e^{-\rho T}V(x(T)) \to 0 along the closed loop and along every admissible comparison path with square-integrable controls, so verification applies.
  3. 8.5
    In Theorem (see book)'s proof, derive the mesh estimate for the piecewise-constant ε\varepsilon-greedy control: exhibit h(ε)h(\varepsilon) such that grid maximizers on steps of length hh attain the HJB maximum within ε\varepsilon along the trajectory.
  4. 8.6
    Show that the finite-horizon workforce Riccati solution P(0;T)P(0; T) of Chapter 6 converges as T→∞T \to \infty to the CARE solution of Example (see book), and quantify the gap at T=12T = 12.

C. Computational exercises

  1. 8.7
    Prove the stationary Bellman principle asserted after Lemma (see book): V(y)=sup⁡u{∫0se−ρtL dt+e−ρsV(x(s))}V(\mathbf{y}) = \sup_u\{\int_0^s e^{-\rho t}L\,dt + e^{-\rho s}V(\x(s))\} for all s>0s > 0, and derive (see book) from it for C1C^{1} VV.
  2. 8.8
    Prove the comparison refinement of Proposition (see book): a C1C^1 sub solution with terminal data ≤Φ\le \Phi lies below VV; conclude the maximal-subsolution characterization in Theorem (see book) and identify where smoothness was indispensable.

D. Enterprise applications

  1. 8.9
    Implement the upwind solver for Example (see book); reproduce the feedback law and steady state, then accelerate with policy iteration on the grid and report the sweep-count ratio.
  2. 8.10
    Reproduce the sustainability mesh ladder of Example (see book), add Richardson extrapolation, and compare the extrapolated value against 2.93752.9375; verify the policy-accuracy asymmetry on the coarsest mesh.
  3. 8.11
    Formulate Meridian's covenant-constrained deleveraging (Example 6.
  4. 8.12 ★
    Develop the viscosity-solution formulation for the digital instance with a hard capability floor c≥c‾c \ge \underline{c} (state constraint): state the constrained viscosity property on the boundary, verify it numerically for the upwind scheme, and prove convergence of the scheme using the Barles–Souganidis framework.

Downloads

All three companions consume the same seeded engine (26208), so their numbers agree by construction — the MFMF convention, carried forward.