Research & Publications_

• Generic chaining theory, geometric functional analysis, maximal inequalities, and majorizing measures for controlling suprema of stochastic processes.
• Empirical process theory, weak convergence, Glivenko–Cantelli theorems, and Donsker–Skorokhod type results.
• Stochastic integration, martingale theory, stochastic differential equations, and stochastic optimal control.
• Quantitative finance, including optimal market making, statistical arbitrage, and algorithmic trading.

My current research concerns decomposition theorems, generic chaining, and majorizing measures as tools for controlling suprema of stochastic processes, applied to weak convergence and Donsker–Skorokhod-type results for processes satisfying absolute regularity. My supervisor is dr. A.M. Dürre.

I organize and lead a weekly graduate seminar on weak convergence and empirical process theory.

In short: I study how to bound the supremum of a stochastic process from its geometry alone, and how much of that machinery survives once you drop independence. My favorite result in the toolkit is Talagrand's majorizing measure theorem — the full proof is written up here.

• Developed likelihood-based estimation procedures for functional scale models, with vectorized and parallel implementations to improve computational efficiency.
• Reduced the runtime of large-scale Monte Carlo simulations by 92.3% on average through numpy vectorization and parallel computation.
• Implemented the main Python components: bernstein_basis, functional_operator, and qmle_filter, including a QMLE filter for the functional GARCH(1,1) model.
• Used non-negative Bernstein coefficients to ensure a positive volatility surface by construction, avoiding constrained optimization over the entire operator.
• Estimated a functional GARCH(1,1) model using a Bernstein-basis QMLE approach with three basis functions on simulated intraday data. The fitted model captures the main structure of the true volatility surface, while introducing additional day-to-day variation. Several operator coefficients reach their imposed bounds, indicating that the model is operating near the edge of the parameter space. The results are illustrated through comparisons of the true and fitted GARCH surfaces, as well as the GARCH and GAS-GARCH volatility surfaces.

def bernstein_basis(u, M, k):
    return comb(M - 1, k - 1) * u ** (k - 1) * (1 - u) ** (M - k)

def functional_operator(grid, coefs, M):
    phi = np.stack([bernstein_basis(grid, M, k) for k in range(1, M + 1)])
    return sum(c * np.outer(phi[i], phi[j]) for c, (i, j) in zip(coefs, product(range(M), repeat=2)))

def qmle_filter(returns, theta, M):
    delta, alpha, beta = theta[:M], theta[M:M + M**2], theta[M + M**2:]
    grid = np.linspace(0, 1, returns.shape[0])
    A, B = functional_operator(grid, alpha, M), functional_operator(grid, beta, M)
    d = sum(c * bernstein_basis(grid, M, k) for k, c in enumerate(delta, 1))

    sigma2 = np.ones(returns.shape[0])
    surface = np.zeros_like(returns)
    for t in range(1, returns.shape[1]):
        sigma2 = d + (A @ returns[:, t - 1] ** 2 + B @ sigma2) / returns.shape[0]
        surface[:, t] = sigma2
    return surface

theta_hat = minimize(lambda theta: qmle_loss(returns, theta, M), theta0, method='SLSQP').x

Non-negative Bernstein coefficients guarantee a positive volatility surface directly, without constrained optimization over the full operator.

The first comparison figure shows the estimated GAS-GARCH volatility surface against the true volatility surface to assess the model’s ability to recover the underlying dynamics.

The second comparison figure compares the functional GARCH and GAS-GARCH volatility surfaces, demonstrating the increased flexibility of the GAS-GARCH specification relative to the traditional functional GARCH model.

Developed a functional stationarity test for multidimensional diffusion processes, implementing and packaging the underlying mathematical framework as an open-source library.

Open-source implementation of a functional stationarity test for multidimensional diffusion processes, developed for my MSc thesis.

Estimation library for functional GARCH/GAS models of time-varying intraday volatility surfaces, using B-splines and numba JIT compilation.