Mathematics

Notes that back the pipeline: targets, regimes, calibration, risk, and policy.

Targets

Forward returns r_t?t+h on a fixed horizon h. Classification uses sign(r) or quantiles; regression uses r directly with robust loss.

r_{t?t+h} = (P_{t+h} - P_t) / P_t

Regimes

Binary timeline with transition penalties. Regime labels can be market-implied (trend filter) or state-implied (hidden Markov). We validate stability across windows.

R_t ? {bull, bear},   cost = ?_t 1[R_t ? R_{t-1}]�?_trans

Calibration (ECE)

Reliability via Expected Calibration Error on binned probabilities.

ECE = ?_k (|B_k|/N) � | acc(B_k) - conf(B_k) |

• Monotone isotonic or temperature scaling as needed
• Drift monitor on rolling windows

Risk model (Greek proxies)

Delta/Gamma/Vega-like proxies from price curvature and variance surfaces; used for sanity bounds and portfolio mix.

? ? ?V/?S,  ? ? ?�V/?S�,  ? ? ?V/??
Proxies: finite diffs over smoothed returns / vol

Selection & aggregation

Cross-regime selection, regularized weighting, turnover-aware constraints.

argmax_w  E[r]  s.t.  risk(w) ? ?,  turnover(w) ? ?

Policy control

PID-style clamp around target exposure with leak and ceiling limits.

e_t = target - signal_t
u_t = Kp�e_t + Ki�?e + Kd�(e_t - e_{t-1})

Evaluation

• Windowed CAGR/Vol/Sharpe vs CB baseline
• Regime stratified stability
• ECE drift and OOS holdouts