Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance
Bibliographic record
Abstract
ABSTRACT Many nonlinear option pricing problems can be formulated as optimal control problems, leading to Hamilton–Jacobi–Bellman (HJB) or Hamilton– Jacobi–Bellman–Isaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods that ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB-type equations, we can guarantee convergence of a Newton-type (policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newton-type iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example, for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".