Limits of Stochastic Optimal Control Models
This page catalogs the models obtained as limits of a more canonical model: a parameter tends to a boundary value and the richer model reduces to a simpler one with its own exact solution (for example the risk-neutral market maker as \(\gamma \to 0\)). It records only the limit relationships. The dynamics, HJB, and (where one exists) closed form of every model are stated once in Exact Solutions in Stochastic Optimal Control or SOC Models Without Exact Solutions and are not restated here.
The companion page Symmetries in Stochastic Optimal Control catalogs the same limits from the symmetry viewpoint; Rigor Reference records which limit is rigorous versus an approximation; and the solver index maps each model to its implementation.
Limit models
| Parent model | Limit | Resulting model | Exact solution |
|---|---|---|---|
| CARA market making (Avellaneda-Stoikov) | \(\gamma \to 0\) | risk-neutral (linear-utility) market making | risk-neutral reduction |
| finite-horizon market making | \(T \to \infty\) | stationary market making (Perron eigenvector) | stationary solution |
| market making with drift | \(\mu \to 0\) | base market making | drift extension |
| market making with permanent impact | \(\xi \to 0\) | base market making | impact extension |
| market making with stochastic volatility | \(\xi_\nu \to 0\) | base market making (constant volatility) | Heston extension |
| market making with Hawkes order flow | \(\alpha \to 0\) | base market making (constant intensity) | Hawkes extension |
| American put | \(r \to 0\) | Black-Scholes European put | American put |
| Merton with deterministic jumps | \(\lambda \to 0\) or \(y \to 1\) | no-jump Merton | deterministic-jump Merton |
| Merton with log-normal jumps | \(\delta \to 0\) | deterministic-jump Merton | log-normal-jump Merton |
| correlated linear-quadratic regulator | \(C\) block-diagonal | uncorrelated regulator | correlated LQ |
| finite-horizon linear-quadratic regulator | \(T \to \infty\), \(\rho > 0\) | stationary regulator (algebraic Riccati) | stationary reduction |
| regulator with Poisson jumps | \(\lambda \to 0\) or symmetric jump law | no-jump regulator | jump LQ |
Scope
Only limits that produce a model with a stated exact solution on the exact-solutions page are listed. Asymptotic approximations (the near-maturity \(\tau \to 0\) spread) and research limits without an implemented closed form (mean-field \(m \to \infty\), the no-loss default limit) are catalogued in Rigor Reference and Symmetries in Stochastic Optimal Control respectively.