Markovian Lifting

Discrete-time Recipe for Markovian Lifting

Intuition

Markovian lifting approximates fractional and rough-volatility dynamics by replacing a singular, non-Markov kernel with a Laplace mixture of exponentials. Informally, one writes a memory kernel \(G(\tau)\) as a Laplace transform of a measure on mean-reversion rates \(\lambda\): $$ G(\tau) \approx \int_0^\infty e^{-\lambda \tau}\,\nu(d\lambda), \qquad \tau \ge 0. $$ Each exponential corresponds to an Ornstein–Uhlenbeck (OU) factor. Stacking \(M\) OU states driven by a single Brownian motion \(W\) gives a Gaussian Markov process \(Z_t = (X^{(1)}_t,\ldots,X^{(M)}_t)^\top\), and the linear aggregate $$ Y_t = \sum_{k=1}^M w_k\, X^{(k)}_t $$ mimics rough fractional covariance when the pairs \((\lambda_k, w_k)\) are chosen by quadrature in the Laplace domain—analogous in spirit to the Volterra recipe, where a causal kernel weights Gaussian noise, except here the kernel is represented by a finite exponential mixture and the state vector makes the dynamics Markovian for filtering and simulation.

In continuous time, with rates \(\lambda_k > 0\) and weights \(w_k\), $$ dX^{(k)}_t = -\lambda_k X^{(k)}_t\,dt + w_k\,dW_t, \qquad X^{(k)}_0 = 0. $$ The observable \(Y_t\) is the rough-vol-style factor used in lifted stochastic-volatility models when coupled with price.

Discretization

Fix a time grid \( \{t_i\}_{i=0}^n \) with uniform step \(\Delta t\). Over each interval, the exact OU transition driven by the same \(W\) implies correlated Gaussian innovations \(\eta^{(k)}_{i}\) for each factor. Conditionally on the past, one updates $$ X^{(k)}_{i+1} = e^{-\lambda_k \Delta t}\, X^{(k)}_i + \eta^{(k)}_{i+1}, $$ where the innovation vector \(\eta\) is Gaussian with mean zero and $$ \mathrm{Cov}\big(\eta^{(j)},\eta^{(k)}\big) = w_j w_k\,\frac{1 - e^{-(\lambda_j+\lambda_k)\Delta t}}{\lambda_j+\lambda_k}. $$ In practice one forms the \(M \times M\) covariance matrix of \(\eta\), computes its Cholesky factor \(L\), draws \(z \sim \mathcal{N}(0,I_M)\), sets \(\eta = L z\), then updates each \(X^{(k)}\) and sets \(Y_{i} = \sum_k w_k X^{(k)}_{i}\). For \(H \in (0,\tfrac{1}{2})\), nodes \(\lambda_k\) are typically log-spaced and weights scale like \(w_k \propto \lambda_k^{-(H+1/2)}\) (normalized) as a heuristic Laplace-domain quadrature; larger \(M\) tightens the match to a target fractional covariance, and in applications \(w_k\) may be calibrated to market smiles.

General Recipe

1. Choose Hurst parameter \(H \in (0,1)\) (rough regimes use \(H < \tfrac{1}{2}\)) and the number of OU factors \(M\).
2. Define the time grid \( \{t_i\}_{i=0}^n \) with uniform spacing \(\Delta t\).
3. Select mean-reversion rates \(\lambda_1,\ldots,\lambda_M\) (e.g. log-spaced on \([2/T, 400/T]\)) and weights \(w_k \propto \lambda_k^{-(H+1/2)}\) with \(\sum_k w_k = 1\).
4. Build the innovation covariance \(\mathrm{Cov}(\eta^{(j)},\eta^{(k)})\) for step \(\Delta t\) and decompose it with Cholesky: \(\Sigma = L L^\top\).
5. For each path and each time step: draw \(z \sim \mathcal{N}(0,I_M)\), \(\eta = L z\), update \(X^{(k)} \leftarrow e^{-\lambda_k \Delta t} X^{(k)} + \eta^{(k)}\), then \(Y \leftarrow \sum_k w_k X^{(k)}\).

Python Implementation

import numpy as np

def laplace_ou_weights(M, H, T=1.0):
    """Log-spaced rates lambda_k and weights w_k propto lambda_k^{-(H+1/2)}."""
    lam = np.logspace(np.log10(2.0 / T), np.log10(400.0 / T), M)
    nu = H + 0.5
    w = lam ** (-nu)
    w /= w.sum()
    return lam, w

def ou_innovation_cov(lam, w, dt):
    M = len(lam)
    S = np.zeros((M, M))
    for j in range(M):
        for k in range(M):
            s = lam[j] + lam[k]
            S[j, k] = w[j] * w[k] * (1.0 - np.exp(-s * dt)) / s if s > 1e-14 else w[j] * w[k] * dt
    return S

def markov_lift_ou_paths(steps, paths, H, M, T=1.0, rng=None):
    rng = rng or np.random.default_rng()
    dt = T / (steps - 1)
    lam, w = laplace_ou_weights(M, H, T)
    L = np.linalg.cholesky(ou_innovation_cov(lam, w, dt) + 1e-12 * np.eye(M))
    a = np.exp(-lam * dt)
    X = np.zeros(M)
    Y = np.zeros((paths, steps))
    for p in range(paths):
        X[:] = 0.0
        for n in range(steps - 1):
            eta = L @ rng.standard_normal(M)
            X = a * X + eta
            Y[p, n + 1] = np.dot(w, X)
    return Y, lam, w

# When comparing to fBM, scale each time column so Var(Y(t_i)) matches t_i^{2H}
# (raw lift uses \\sum w_k=1 and is far smaller than fractional Brownian motion).

Summary: This approach yields a Gaussian Markov chain in the lifted state \(Z_n\) whose linear functional \(Y_n\) approximates fractional / rough covariance when \((\lambda_k,w_k)\) are chosen in the Laplace domain, enabling efficient simulation and filtering for rough-volatility-style models without full-path fBM Cholesky at every step.