Research 01
Berry–Esseen Bounds for Periodic Langevin Dynamics
Research poster
O. Smith, A. Valse, D. Wu (alphabetical). Manuscript coming soon. Mentor: Professor David P. Herzog.
This project establishes Berry–Esseen bounds for the Gaussian
approximation of unwrapped displacement in periodic Langevin
dynamics. My starting question for Proposition 11.1 was how to
control the Berry–Esseen constant in the main theorem as the
friction parameter \(\gamma\) tends to zero.
NSF-funded research ·
Award DMS-2246491
My contributions
- Formulated the Berry–Esseen framework for the unwrapped displacement and proved the overdamped results.
- Proved the main underdamped theorem and supporting propositions via a direct Markov-additive treatment of the displacement; verification of Assumption (A2) was carried out by O. Smith.
- Proved in Proposition 11.1 that, for one-dimensional periodic underdamped dynamics started from equilibrium and sufficiently small positive \(\gamma\), the Berry–Esseen constant C\(\gamma\) can be chosen with \(C_{\gamma} \le C\gamma^{-1/2}\), where C is independent of \(\gamma\).
- Simulated overdamped and underdamped transport, measured the friction-dependent constants, and compared them with the averaged energy dynamics.
The joint manuscript also includes additional numerical analysis by A. Valse.
Core contribution · Proposition 11.1
Controlling the Berry–Esseen constant in the low-friction regime
For the one-dimensional periodic underdamped system started
from equilibrium, I proved that for sufficiently small
positive \(\gamma\) the Berry–Esseen constant can be chosen so that
\[ C_{\gamma} \le C\gamma^{-1/2}. \]
Here C is independent of \(\gamma\) and observation time.
Equivalently, for \(T \ge 1\) the Kolmogorov-distance estimate reads
\[ d_K\!\left(\mathcal{L}(Y_T^{\gamma}),\,\mathcal{N}(0,\sigma_{\gamma}^{2})\right) \le \frac{C}{\sqrt{\gamma T}}. \]
The form \(C/\sqrt{\gamma T}\) highlights \(\gamma T\) as the combined parameter in
the error bound; the main contribution is the quantified
dependence of the Berry–Esseen constant on \(\gamma\).
Proof architecture
-
01
Sample at the relaxation scale.
Partition the process into blocks of length comparable to
\(\gamma^{-1}\) 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 \(\gamma\).
-
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
\(C/\sqrt{\gamma T}\) Kolmogorov-distance estimate.
Research 02
Random Prepayment Sensitivity in Nonlinear MBS Pricing
Working manuscript
I study how small random changes in prepayment intensity affect
mortgage-backed security prices. The current manuscript focuses
on the Papi–Briani model at \(p=2\), using a probabilistic
representation to identify the linear price response, control
its approximation error, and quantify the resulting mean and variance.
How do random prepayment environments change the pricing solution, its mean, and its variance?
Current manuscript · random coefficients at \(p=2\)
The model tracks a financial factor and cumulative prepayment on \(\mathbb{R}^2\). Prepayment intensity is perturbed as \(\eta^{\varepsilon}=\eta^0(1+\varepsilon\zeta)\), where \(\zeta\) has continuous paths and a deterministic uniform bound. Each coefficient scenario defines a degenerate nonlinear pricing PDE. The environmental noise selects the scenario; the Brownian motion in the BSDE is an auxiliary tool for representing its solution.
- Use Papi’s viscosity theory for well-posedness and price bounds, then remove the quadratic gradient term through a change of unknown to obtain a Lipschitz BSDE representation.
- Establish a bounded linear response operator and a uniform second-order remainder, giving Fréchet differentiability in the supremum norm.
- Derive mean-square sensitivity, mean-price bias, and Hilbert-space variance expansions; express the leading variance through the covariance of the input noise.
- Work with continuous bounded perturbations, without requiring their time derivatives or classical spatial derivatives of the price.
Manuscript results · Theorem 1.1 and Corollary 1.2
Linear price response with a quadratic remainder
\[ P^{\varepsilon}=P^0+\varepsilon Y+R^{\varepsilon},\qquad Y=L\zeta. \]
Here \(P^{\varepsilon}\) is the full security value and \(L\) is a deterministic bounded linear operator obtained from the forward–backward variational system. Under the manuscript’s assumptions, for sufficiently small \(\varepsilon\), almost surely,
\[ \lVert R^{\varepsilon}\rVert_{\infty}\le C\varepsilon^2\lVert\zeta\rVert_{\infty}^2. \]
The bound is uniform in the state and time, and also gives an \(O(\varepsilon^2)\) remainder in \(L^2(\Omega;C([0,T];H))\), where \(H=L^2(\mathbb{R}^2,\nu)\) for a probability measure \(\nu\). For centered perturbations, the mean-price bias is \(O(\varepsilon^2)\), and uniformly in \(t\),
\[ \operatorname{Var}_H(P^{\varepsilon}(t))=\varepsilon^2\mathbb{E}\lVert Y(t)\rVert_H^2+O(|\varepsilon|^3). \]
Proof approach · from the pricing PDE to its response
Viscosity well-posedness supplies a bounded price range. A power transform cancels the quadratic gradient term, and the transformed solution is identified with the value function of a Lipschitz BSDE. Linearizing the forward prepayment dynamics and the backward equation, then controlling the Taylor remainders and undoing the transform, yields the uniform sensitivity estimate. A deterministic response kernel links the leading price variance to the covariance of the prepayment perturbation.
Planned next stage · Spring 2027
SPDE extension with additive noise
In Spring 2027, I plan to introduce additive stochastic forcing directly into the MBS evolution equation. The proposed form is schematic:
\[ du_t^{\varepsilon}=\mathcal{F}(t,u_t^{\varepsilon})\,dt+\varepsilon G(t)\,dW_t. \]
Here \(\mathcal{F}\) denotes the chosen MBS PDE operator, \(W\) is a Wiener process, and the noise coefficient \(G(t)\) is independent of the solution. This adds a stochastic integral to the evolution itself. The analytical setting and noise regularity will be specified as part of this planned work.
01
Well-posedness and bounds
Establish existence, uniqueness, and suitable stochastic moment estimates for the additive-noise formulation.
02
Stability and the small-noise limit
Study sensitivity to the forcing and convergence to the deterministic baseline as the noise amplitude tends to zero.
03
Numerical exploration
Examine how direct stochastic forcing changes sample paths, mean prices, and variance, guided by the analytical estimates.