Symmetries in Stochastic Optimal Control
This appendix catalogs the symmetry structure of the stochastic optimal control (SOC) problems in this repository: which symmetries hold, which are gained or restored in a limit, and which model effect breaks each. It is a companion to Exact Solutions in Stochastic Optimal Control, which derives the solutions; this page only lists the symmetries and points to the derivations. The general framework follows [@pham2009continuous, @fleming2006controlled].
The symmetries labelled "research" below belong to the multi-asset market-making and mean-field problems, which are not yet realized by an implemented model; they are catalogued here so the symmetry structure is not lost, but they do not yet have a closed form on the exact-solutions page.
A symmetry is a transformation of the state (possibly paired with a relabeling of controls) under which the value transforms covariantly, \(V(t, \phi(x)) = \Phi(V(t, x))\) for a fixed \(\Phi\). The three covariance types that occur are multiplicative (CARA), additive (log), and power (CRRA). A symmetry is useful when it reduces the independent coordinates of \(V\), or forces a parity constraint a solver or network must satisfy. The Lie-symmetry treatment follows [@olver1993applications]; the conservation content follows [@noether1918invariante].
Symmetries of the base problems
| Symmetry | Transformation / group | Value covariance | Economic meaning | Derived in |
|---|---|---|---|---|
| Numeraire (translation) gauge | \((S, X) \mapsto (S + c,\; X - q c)\), with \(Y = X + q S\) invariant | CARA: \(V = -\exp(-\gamma (X + q S + \theta(t, q)))\); risk-neutral: \(V = X + q S + \theta(t, q)\) | a mid-priced fill is a fair cash-for-inventory exchange, so the quote is a spread, not a price | soc_exact.md |
| Wealth scale (no money illusion) | \(x \mapsto c x\) | log: \(V \mapsto V + \ln c\); CRRA: homogeneous of degree \(1 - \eta\) | only relative quantities matter; the control is a scale-free fraction (Kelly \(w^* = \mu / \sigma^2\)) | soc_exact.md |
| Bid-ask reflection | \(q \mapsto -q\) with bid/ask exchange, group \(\mathbb{Z}_2\) | \(\theta(t, q) = \theta(t, -q)\), \(\delta^{a}(t, q) = \delta^{b}(t, -q)\) | no preferred direction; long and short are interchangeable | soc_exact.md |
| Asset permutation and sign flip | \(B_m = (\mathbb{Z}_2)^m \rtimes S_m\) (hyperoctahedral) | value is a symmetric function of \(\lbrace | q_1 | , \dots, |
| Isotropic rotation (partial) | \(O(m)\) | penalty invariant in \(\sum_i q_i^2\) and \((\sum_i q_i)^2\) | rotation reduces only the penalty/diffusion, not the fill (jump) operator, which is only \(B_m\)-invariant | research |
Symmetries obtained in limits
A limit acts on a symmetry in one of three ways: it restores a symmetry removed by a finite-size or finite-horizon effect, deforms the realization of a symmetry, or exposes the deterministic skeleton by removing dissipation.
| Limit | Symmetry gained or restored | Note | Derived in |
|---|---|---|---|
| Infinite horizon \(T \to \infty\) | time-translation (autonomy); gauge scale \(\mathbf{v} \mapsto c \mathbf{v}\); Perron-Frobenius forces the symmetric sector | the terminal anchor recedes; the principal eigenvector is even in \(q\) (and \(B_m\)-symmetric) without any symmetric terminal data | soc_exact.md |
| Mean-field \(m \to \infty\) | permutation \(S_m\) becomes measure symmetry | de Finetti / Hewitt-Savage: symmetric functions of many coordinates become functionals of the empirical measure \(\tfrac1m \sum_i \delta_{q_i}\) | research |
| Risk-neutral \(\gamma \to 0\) | exponential gauge deforms to affine | the inventory penalty vanishes and \(\delta_0 \to 1/k\) | soc_exact.md |
| Zero volatility \(\sigma \to 0\) | dissipation removed | the HJB becomes a first-order Hamilton-Jacobi equation; characteristics are exact and \(V\) is non-smooth | stated here |
| Unbounded inventory \(q_{\max} \to \infty\) | jump operator becomes a lattice convolution | the truncation boundary disappears | stated here |
| Jump intensity \(\lambda \to 0\) | diffusion symmetry recovered | jumps vanish | soc_exact.md |
| Fluid / CLT \(\lambda \to \infty\) | continuity restored | discrete jumps converge to a diffusion | soc_exact.md |
| Perfect correlation \(\rho \to 1\) | penalty collapses to \(\sum_i q_i\) | only the penalty reduces; per-asset fills still distinguish assets | stated here |
| Zero correlation \(\rho \to 0\) | decoupling | the value separates into independent single-asset problems | stated here |
Symmetry breaking
Each model effect breaks a specific symmetry; the break is the economic correction a solver must reproduce.
| Effect | Symmetry broken | Economic correction |
|---|---|---|
| Constant drift \(\mu \neq 0\) | \(\mathbb{Z}_2\) (bid-ask reflection) | reservation-price lean; target \(q^* = \mu / (\gamma \sigma^2)\) |
| Mean-reverting drift \(\theta(\bar S - S)\) | numeraire gauge | value depends on \(S - \bar S\); no pure CARA separation; the process is instead reflection-symmetric about \(\bar S\) |
| Permanent impact \(\xi \neq 0\) | \(\mathbb{Z}_2\) (per-asset sign) | bid-ask skew; asymmetric up/down couplings |
| Unequal fill decay \(k_i\) | \(S_m\) (permutation) | idiosyncratic allocation; no single log-linearization |
| Non-exchangeable \(\Sigma\) | \(S_m\) | factor structure; risk along eigenvectors |
| Finite horizon / terminal cost | time translation | time-dependent spread schedule |
| Non-stationary intensity | time translation | time-of-day quoting |
What is and is not conserved
The HJB is a parabolic, dissipative equation; there is no physical energy conserved along optimal trajectories. The objects that play the role of a conserved structure are:
- Comparison principle. If \(V_1 \le V_2\) at the terminal time then \(V_1 \le V_2\) everywhere [@crandall1983viscosity]; monotone schemes preserve it [@barles1991convergence]. This is the correct law to preserve, not an energy.
- Martingale value process. Along the optimal path, the value plus the accumulated running reward is a martingale, conserved in expectation.
- Hamiltonian along characteristics. Pontryagin's Hamiltonian \(H(x, p) = \sup_u \lbrace f(x, u) + b(x, u)^\top p \rbrace\) generates a symplectic flow on \((x, p)\); a continuous symmetry of \(H\) yields a Noether-conserved quantity along characteristics [@noether1918invariante].
- Discrete parity. A reflection symmetry is a \(\mathbb{Z}_2\) charge, preserved if the terminal data and the generator preserve evenness.