Advanced Material: Part II. Stochastic Optimal Control and the HJB Equation
Bibliographic record
Abstract
Learning Objectives Until now, we have assumed that there is no uncertainty other than the length of your lifetime. In real life, however, this is not the case. In this chapter we extend the results in the previous chapter by including uncertainties in both investment return and wage income. We start by presenting the standard materials on dynamic optimal asset allocation and consumption under the continuous time framework, originally developed by Robert Merton in a series of papers (1969, 1971) that led to him being awarded the Nobel Prize Memorial in Economics. We then conclude the chapter with a brief presentation of an advanced stochastic lifecycle model. Dynamic Asset Allocation and Optimal Consumption Let us for the time being forget about the lifecycle model(issues related to mortality risk and retirement) and instead consider a classical problem in finance. Suppose that you have inherited or accumulated a certain amount of money M t at time t . You have a choice between keeping your money in a safe bank account (cash) or investing it in the stock market, which promises a greater return on average. Of course the stock market is volatile and you may lose some or all of your investment (if you are one of the many who bought tech stocks before the great tech bubble burst, for example). So if you are risk averse and belong to those people who will lose sleep worrying about the possibility of losing your investment, clearly you face a dilemma: money in the savings account is safe but the return is low while the stock market might give you a greater return but you face the possibility oflosing your investment.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.050 | 0.014 |
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".