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| # 📐 Advanced Quantitative Stochastic Calculus & Non-Markovian Microstructure Suite | |
| **Author**: [Ramprasad Atluri](https://huggingface.co/At1uri) | |
| **Affiliation**: Quantitative Research & Advanced Econometric Architectures | |
| **License**: MIT License | |
| --- | |
| ## 1. Merton Compound Poisson Jump-Diffusion Model | |
| The underlying asset price $S_t$ satisfies the jump-diffusion stochastic differential equation (SDE): | |
| $$\frac{dS_t}{S_{t-}} = (\mu - \lambda \kappa) dt + \sigma dW_t + (Y - 1) dN_t$$ | |
| Where: | |
| - $W_t$ is a standard standard Brownian motion on $(\Omega, \mathcal{F}, (\mathcal{F}_t), \mathbb{P})$. | |
| - $N_t$ is a homogeneous Poisson counting process with constant intensity $\lambda > 0$, independent of $W_t$. | |
| - $Y$ is a log-normal jump size distribution: $\ln Y \sim \mathcal{N}(\gamma, \delta^2)$. | |
| - $\kappa = \mathbb{E}[Y - 1] = \exp\left(\gamma + \frac{1}{2} \delta^2\right) - 1$ is the expected relative jump percentage, ensuring the compensated process is a $\mathbb{Q}$-martingale. | |
| --- | |
| ## 2. Fractional Brownian Motion & Detrended Fluctuation Analysis (DFA) | |
| The fractional Brownian motion (fBM) $B_H(t)$ with Hurst index $H \in (0, 1)$ possesses covariance structure: | |
| $$\mathbb{E}[B_H(t) B_H(s)] = \frac{1}{2} \left( |t|^{2H} + |s|^{2H} - |t - s|^{2H} \right)$$ | |
| ### Fractional Integration Properties: | |
| - $H > 1/2$: Non-Markovian long-range dependence, $\sum_{k=1}^\infty \gamma(k) = \infty$, exhibiting directional persistence. | |
| - $H = 1/2$: Classical Brownian motion with independent increments. | |
| - $H < 1/2$: Anti-persistent, negative autocorrelation, anomalous sub-diffusion. | |
| The Detrended Fluctuation Analysis computes root-mean-square fluctuation $F(s)$ over window length $s$: | |
| $$F(s) = \sqrt{\frac{1}{N} \sum_{k=1}^N \left( X(k) - p_v(k) \right)^2} \sim s^H$$ | |
| --- | |
| ## 3. Hawkes Self-Exciting Point Process | |
| The conditional intensity function $\lambda(t)$ of high-frequency orderbook arrival events satisfies: | |
| $$\lambda(t) = \mu + \int_0^t \alpha e^{-\beta (t - s)} dN(s) = \mu + \sum_{t_i < t} \alpha e^{-\beta(t - t_i)}$$ | |
| ### Subcriticality Condition: | |
| The spectral radius of the transition kernel satisfies the subcritical stability criterion: | |
| $$\eta = \frac{\alpha}{\beta} < 1.0$$ | |
| The total log-likelihood over interval $[0, T]$ is evaluated via exact recursion: | |
| $$\ln \mathcal{L} = \sum_{i=1}^N \ln \lambda(t_i) - \mu T - \frac{\alpha}{\beta} \sum_{i=1}^N \left( 1 - e^{-\beta(T - t_i)} \right)$$ | |
| --- | |
| ## 4. Optimal Stopping via Longstaff-Schwartz LSM Dynamic Programming | |
| Resolves the backward Snell envelope for early-exercise American contingent claims: | |
| $$V(S_t, t) = \max\left\{ h(S_t), \, e^{-r \Delta t} \mathbb{E}^\mathbb{Q} \left[ V(S_{t+1}, t+1) \mid \mathcal{F}_t \right] \right\}$$ | |
| The conditional continuation value $C(S_t) = \mathbb{E}^\mathbb{Q}[e^{-r \Delta t} V_{t+1} \mid S_t]$ is projected onto an orthogonal basis of weighted Laguerre polynomials $\{L_k(x)\}_{k=0}^M$: | |
| $$C(S_t) \approx \sum_{k=0}^M \beta_k L_k(S_t), \quad \boldsymbol{\beta} = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{Y}$$ | |
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