First-Passage Default Modeling for Prediction Market Collateral: Self-Exciting Dynamics and Adaptive Drift

We develop a first-passage framework for modeling loan default against prediction market collateral - a novel asset class characterized by bounded payoffs on [0, 1], event-driven price dynamics, and thin orderbook liquidity. The loan is modeled through its health factor, a stochastic process whose first passage below a critical threshold triggers liquidation. Price dynamics are specified in logit space as a jump-diffusion with adaptive drift estimated via Kalman filtering under a martingale prior, where the jump arrival rate evolves dynamically according to a multivariate Hawkes process with per-cluster decay rates calibrated from high-frequency orderbook data. The cross-excitation structure of the Hawkes process captures the empirically observed clustering of information events across related markets, producing correlated jumps across portfolio positions. We derive the first-passage default probability through a combination of analytical bounds and Monte Carlo simulation with Girsanov importance sampling, and introduce a real-time default risk surface that maps the current event intensity state to instantaneous default probability - providing a forward-looking risk indicator with no analogue in existing lending protocols. Empirical validation on live Polymarket orderbook data spanning 20 days across geopolitical event markets demonstrates that the first-passage default probability exceeds the terminal measure by approximately 63%, shared cluster jumps account for 52% of simulated defaults, and the Hawkes model with spectral radius 0.68 captures cross-market cascade dynamics that independent-jump models miss entirely.