Structure in Motion

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dSt​=μ(St​,t)dt+σ(St​,t)dWt​

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On (Ω,F,(Ft)t≥0,P)(Ω,F,(Ft​)t≥0​,P), with WtWt​ a Brownian motion, the asset price StSt​ is an FtFt​-adapted semimartingale with drift μ(St,t)μ(St​,t) and diffusion σ(St,t)σ(St​,t). Under an equivalent martingale measure QQ, the discounted price is a martingale, yielding arbitrage-free valuation, pricing consistency, and tractability under both physical and risk-neutral measures. This forms the baseline from which broader and more refined modeling frameworks are developed.

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Notation

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• St​: asset price

• μ(St,t)μ(St​,t): drift (expected return)

• σ(St,t)σ(St​,t): volatility (diffusion coefficient)

• WtWt​: Wiener process (Brownian motion)

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Extensions

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Brownian models omit fat tails, clustering, and jumps; modeled through:

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• Stochastic volatility: Heston, CIR/OU factors

• Rough volatility: fractional Brownian components

• Jump–diffusion: Poisson, Merton, Bates

• Lévy processes: infinite-activity jumps

• Regime switching: Markov state models

• Multi-asset systems: correlated vector SDEs

• Control formulations: HJB-based optimal control

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Methodology

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• Measures: Girsanov, Esscher, numeraire change

• Filtering: Kalman, particle, Bayesian

• Simulation: Monte Carlo, quasi, variance reduction

• Estimation: MLE, GMM, Bayesian

• Numerics: PDE, FFT, BSDE

• Control: HJB, dynamic programming

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Calibration

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• Surfaces: skew, smile, term-structure

• High-freq: realized variance, bipower

• Estimation: MLE, Bayesian, spectral

• Robustness: stability, regime sensitivity

• Microstructure: order book, liquidity

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Applications

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• Option pricing: Heston, jumps, Lévy

• Portfolio: HJB, BSDEs

• Risk: tails, jumps, contagion

• Execution: trading, liquidity

• Digital assets: microstructure, fragmentation