Probability · Stochastic dynamics · Financial mathematics
Research
I study quantitative questions in stochastic dynamics and
mathematical finance. My current work connects finite-time
probability bounds, nonlinear partial differential equations,
numerical experiments, and model validation against real data.
Research 01
Berry–Esseen Bounds for Periodic Langevin Dynamics
Working manuscript
How long must a periodic Langevin system be observed before its
unwrapped displacement is accurately approximated by a Gaussian
distribution? This project develops quantitative bounds for that
finite-time error, with particular attention to the singular
low-friction regime.
NSF-funded research ·
Award DMS-2246491
Analytical foundation. Measure-theoretic probability
is used throughout this work, together with Lp moment
estimates, conditional expectations, ergodic averages, and weak
convergence arguments. These tools connect my independent study in
measure theory directly to the Langevin research problem.
Core contribution · Proposition 11.1
The relaxation scale is γT
Equivalently, the friction-dependent Berry–Esseen constant may be chosen so that
In one dimension, I proved a Berry–Esseen estimate with a
constant that explicitly tracks the friction parameter. The
bound shows that the effective number of relaxation periods is
γT, rather than the observation time T alone.
Proof architecture
-
01
Sample at the relaxation scale.
Partition the process into blocks of length comparable to
γ-1 and form a discrete skeleton chain.
-
02
Recover uniform mixing.
Rescale the continuous-time semigroup estimate so the
sampled chain has a spectral gap uniform in small γ.
-
03
Control moments and variance.
Establish a uniform fourth-moment bound for block-averaged
velocity and use a one-dimensional diffusion lower bound to
prevent variance degeneration.
-
04
Apply a uniform Berry–Esseen theorem.
The Markov-additive formulation then yields the
C/√(γT) Kolmogorov-distance estimate.
Additional contributions
- Developed the Berry–Esseen framework for the displacement problem.
- Proved the projected overdamped result.
- Developed the direct Markov-additive treatment of underdamped displacement.
- Carried out the original numerical illustration and drafted Part I.
Research 02
Prediction, Model Risk, and Stochastic Sensitivity in Nonlinear MBS Pricing Models
Ongoing research
This project asks whether stochastic perturbations and nonlinear
p-Laplacian diffusion can improve the predictive performance of an
existing PDE model for mortgage-backed security prices.
How much predictive improvement do the proposed changes deliver when the model is evaluated against real market data?
Research trajectory
I initially proposed introducing stochastic noise into the pricing
equation to study how market uncertainty affects model predictions.
Through discussions with my advisor, the project expanded to include
a p-Laplacian operator as a nonlinear diffusion mechanism. The
resulting research program connects analytical foundations,
empirical backtesting, and stochastic model-risk analysis.
Completed · Analytical foundation
Literature reconstruction
I organized papers provided by my advisor into technical
sessions and presentations, reconstructing the main
derivations rather than treating the models as black boxes.
- Reduction of a nonlinear MBS pricing equation to a semilinear heat equation.
- Lipschitz control of the nonlinear mapping.
- Fixed-point proof of existence and uniqueness of weak solutions.
- Derivation of a p-diffusion extension of the Black–Scholes operator.
Completed · Empirical investigation
Market backtesting and numerical analysis
I built an initial market-backtesting workflow using real
transaction data, conducted numerical experiments, and
examined the financial interpretation of the model in
selected parameter regimes and special cases.
The central objective was to determine whether the stochastic
and nonlinear extensions produced a measurable improvement
over the original equation.
Initial empirical finding
An informative inconclusive result
The initial backtests did not yet yield a stable and economically
interpretable improvement over the baseline model. Instead, they
exposed important questions involving parameter identification,
alignment between model outputs and transaction observations,
and the separation of nonlinear-diffusion effects from other
market mechanisms. These limitations now help define the next
stage of the project.
Next stage: stochastic analysis
01
Mean-square sensitivity
Quantify how perturbations in model inputs and coefficients
propagate through the pricing solution, using estimates of
the form E||uε(t) − u0(t)||2.
02
Noise-driven simulation
Compare theoretical sensitivity estimates with numerical
behavior across noise levels, nonlinear-diffusion parameters,
and financially meaningful regimes.
03
Model risk and interpretation
Study whether nonlinear diffusion dampens or amplifies
uncertainty, and identify which mechanisms are responsible
for changes in predictive performance.
Financial perspective. The PDE analysis serves a
pricing and model-risk question: prediction under uncertainty,
sensitivity to assumptions, and economically interpretable behavior
under changing market conditions.
Research 03
Random Matrix Theory and Quantum Expanders
ISmART research
This project studies operator-theoretic questions arising from
Haar-distributed unitary matrices, with particular attention to
non-normal behavior, commutator lower bounds, and finite-dimensional
models related to quantum expanders.
Analytical focus
Haar unitaries and commutator bounds
I studied Haar-distributed unitary matrices, malnormal
matrices, commutator lower bounds, and their relationship to
quantum-expander constructions.
Computational contribution
Non-normal spectra and pseudospectra
I developed numerical experiments to investigate non-normal
spectral behavior and pseudospectra, and produced a final
technical note connecting the computations with the
operator-theoretic questions.