RESEARCH_SPECIFICATION.md

📐 Advanced Quantitative Stochastic Calculus & Non-Markovian Microstructure Suite

Author: Ramprasad Atluri
Affiliation: Quantitative Research & Advanced Econometric Architectures
License: MIT License

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1. Merton Compound Poisson Jump-Diffusion Model

The underlying asset price $S_t$ satisfies the jump-diffusion stochastic differential equation (SDE):

$$\frac{d{S}_t}{S_{t-}}=(\mu -\lambda \kappa )dt+\sigma d{W}_t+(Y-1)d{N}_t\backslash frac\{dS\_t\}\{S\_\{t-\}\}\; =\; (\backslash mu\; -\; \backslash lambda\; \backslash kappa)\; dt\; +\; \backslash sigma\; dW\_t\; +\; (Y\; -\; 1)\; dN\_t$$

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.

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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:

$$\mathbb{E}[B_H(t)B_H(s)]=\frac{1}{2}(\mathrm{\mid }t{\mathrm{\mid }}^{2H}+\mathrm{\mid }s{\mathrm{\mid }}^{2H}-\mathrm{\mid }t-s{\mathrm{\mid }}^{2H})\backslash mathbb\{E\}[B\_H(t)\; B\_H(s)]\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\; |t|^\{2H\}\; +\; |s|^\{2H\}\; -\; |t\; -\; s|^\{2H\}\; \backslash right)$$

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 $s$:

$$F(s)=\sqrt{\frac{1}{N}\sum _{k=1}^N{(X(k)-p_v(k))}^2}\sim s^{H}F(s)\; =\; \backslash sqrt\{\backslash frac\{1\}\{N\}\; \backslash sum\_\{k=1\}^N\; \backslash left(\; X(k)\; -\; p\_v(k)\; \backslash right)^2\}\; \backslash sim\; s^H$$

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3. Hawkes Self-Exciting Point Process

The conditional intensity function $\lambda(t)$ of high-frequency orderbook arrival events satisfies:

Subcriticality Condition:

The spectral radius of the transition kernel satisfies the subcritical stability criterion:

The total log-likelihood over interval $[0, T]$ is evaluated via exact recursion:

$$\ln \mathcal{L}=\sum _{i=1}^N\ln \lambda (t_i)-\mu T-\frac{\alpha }{\beta }\sum _{i=1}^N(1-e^{-\beta (T-t_i)})\backslash ln\; \backslash mathcal\{L\}\; =\; \backslash sum\_\{i=1\}^N\; \backslash ln\; \backslash lambda(t\_i)\; -\; \backslash mu\; T\; -\; \backslash frac\{\backslash alpha\}\{\backslash beta\}\; \backslash sum\_\{i=1\}^N\; \backslash left(\; 1\; -\; e^\{-\backslash beta(T\; -\; t\_i)\}\; \backslash right)$$

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4. Optimal Stopping via Longstaff-Schwartz LSM Dynamic Programming

Resolves the backward Snell envelope for early-exercise American contingent claims:

$$V(S_t,t)=\max \{h(S_t),e^{-r\mathrm{\Delta }t}{\mathbb{E}}^{\mathbb{Q}}[V(S_{t+1},t+1)\mid {\mathcal{F}}_t]\}V(S\_t,\; t)\; =\; \backslash max\backslash left\backslash \{\; h(S\_t),\; \backslash ,\; e^\{-r\; \backslash Delta\; t\}\; \backslash mathbb\{E\}^\backslash mathbb\{Q\}\; \backslash left[\; V(S\_\{t+1\},\; t+1)\; \backslash mid\; \backslash mathcal\{F\}\_t\; \backslash right]\; \backslash right\backslash \}$$

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 ${L_k(x)}_{k=0}^M$:

$$C(S_t)\approx \sum _{k=0}^M{\beta }_k{L}_k(S_t),\mathit{\beta }=({\mathbf{X}}^{\mathrm{\top }}\mathbf{X}{)}^{-1}{\mathbf{X}}^{\mathrm{\top }}\mathbf{Y}C(S\_t)\; \backslash approx\; \backslash sum\_\{k=0\}^M\; \backslash beta\_k\; L\_k(S\_t),\; \backslash quad\; \backslash boldsymbol\{\backslash beta\}\; =\; (\backslash mathbf\{X\}^\backslash top\; \backslash mathbf\{X\})^\{-1\}\; \backslash mathbf\{X\}^\backslash top\; \backslash mathbf\{Y\}$$