Variance-Reduced Trajectory Sampling

• Variance-reduced trajectory sampling is a set of strategies that use optimal quantization-based stratification to reduce estimator variance in Monte Carlo simulations.
• The paper presents novel quantization techniques and optimal sample allocations that achieve significant reductions in sampling noise and computational cost.
• It demonstrates uniform efficiency for Lipschitz functionals and scalable algorithmic implementations applicable to high-dimensional diffusion and financial models.

Variance-reduced trajectory sampling comprises a set of strategies and theory-driven methodologies that aim to reduce sampling noise in trajectory-based Monte Carlo estimators. These approaches are essential for improving the efficiency, reliability, and scalability of simulations involving rare events, functional outputs of stochastic processes, or path-dependent functionals in mathematical finance, physics, and engineering. This article presents a comprehensive overview of fundamental concepts, theoretical underpinnings, core methodologies, algorithmic realizations, and practical considerations pertaining to variance-reduced trajectory sampling—focusing especially on quantization-based stratification as developed in (Corlay et al., 2010).

1. Foundations of Variance Reduction via Stratified Sampling

At its core, stratified sampling divides the state space of the process X into non-overlapping, measurable subsets ("strata"), with each stratum sampled independently. The fundamental estimator for a target expectation $E[F(X)]$ becomes

$E[F(X)] = \sum_{i} p_i \, E[F(X) \mid X \in C_i]$

where $C_i$ are the strata (typically defined via a partitioning scheme), and $p_i = P(X \in C_i)$.

The principal difficulty is to construct the partition $\{C_i\}$ so that the "local variance" $Var(F(X) \mid X \in C_i)$ is minimized, thus reducing the overall estimator variance. Optimal stratification has been a longstanding problem; it is resolved in (Corlay et al., 2010) through the introduction of optimal quadratic quantization, which provides an automated, data-driven stratification method applicable in both finite and infinite dimensions.

2. Functional Quantization as an Optimal Stratification Mechanism

Functional quantization replaces the continuous random variable $X$ (possibly taking values in a Hilbert space) by its nearest-neighbor projection onto a finite "codebook" $\Gamma=\{\gamma_1,\ldots,\gamma_N\}$: $$\text{Proj}_\Gamma(X) = \arg\min_{\gamma_i\in\Gamma} \| X - \gamma_i \| .$$ This induces a Voronoi partition $$C_i = \{ x : \|x-\gamma_i\| = \min_j \| x-\gamma_j\| \}$$. Optimal quantizers are computed to minimize mean squared error: $E[\| X - \text{Proj}_\Gamma(X) \|^2].$

A crucial theoretical result established in (Corlay et al., 2010) is that, for a functional $F$ which is Lipschitz with constant $[F]_{\text{lip}}$, the conditional variance inside each cell obeys: $$\sigma_{F,i}^2 = Var(F(X) \mid X \in C_i) \leq [F]_{\text{lip}}^2 \, Var(X \mid X \in C_i),$$ so that the total variance of the stratified estimator is directly controlled by the quantization error.

For a collection of strata generated by an optimal quadratic quantizer, this error is minimized globally, and the resulting stratified estimator achieves minimal variance among all stratified estimators with the same number of strata.

3. Uniform Efficiency and Consistency for Lipschitz Functionals

A unique property of quantization-based stratification is its uniform efficiency for the entire class of Lipschitz functionals. The "Universal stratification" proposition yields: $$\sup_{[F]_{\text{lip}} \leq 1} \sum_{i} p_i \sigma^2_{F,i} = \sum_{i} p_i \sigma_i^2 = \| X - \text{Proj}_\Gamma(X) \|^2_2,$$ with $\sigma_i^2$ the conditional variance of $X$ in $C_i$. Thus, the variance reduction benefits accrue to all Lipschitz functionals; the only pre-factor is the Lipschitz constant of $F$.

Furthermore, as the quantization level $N$ increases, the quantizer converges and the quantization error tends to zero, establishing the consistency of the method for partitioning in both finite and infinite-dimensional settings.

4. Algorithmic Realization for Gaussian and Diffusion Processes

For high-dimensional or infinite-dimensional stochastic processes (e.g., Brownian motion, bridges, Ornstein–Uhlenbeck processes), the state space is spanned using a Karhunen–Loève expansion: $X = \sum_{n=1}^\infty \sqrt{\lambda_n} \xi_n e_n$ with known orthonormal basis $\{e_n\}$ and independent standard normal coordinates $\xi_n$. Product functional quantization proceeds by quantizing each $\xi_n$ independently (often computed via iterative schemes or lookup tables), resulting in a stratification of the path space into hyperrectangular strata.

Efficient simulation is achieved as follows:

• These strata allow sampling $X$ by first sampling a representative quantized coordinate set, then simulating the conditional distribution of the remainder of the process (via fast “Bayesian simulation” using conditioning formulas for Gaussians).
• For a process observed at discrete times $(t_0, ..., t_n)$, the conditional mean and covariance for reconstructing the path are computed via

$E[V\mid Y]= E[V] + R_{V|Y}(Y - E[Y])$

where $V$ denotes the discretized path and $Y$ the K–L coordinates.

The estimator is unbiased: $$\overline{F(X)}_M = \sum_{i} p_i \frac{1}{M_i} \sum_{k=1}^{M_i} F(X^{(k)}|X^{(k)} \in A_i)$$ where $M_i$ samples are allocated to each stratum (natural allocation $q_i=p_i$ or Lipschitz-optimal allocation $q_i^*=p_i \sigma_i/\sum_j p_j \sigma_j$).

For Ornstein–Uhlenbeck processes, explicit formulas for eigenvalues and eigenfunctions (see equations in (Corlay et al., 2010); e.g., eigenvalues given by $$\lambda_n^{OU} = \frac{\sigma^2}{\omega_n^2+\theta^2}$$) enable fast and explicit construction of quantizers and the required conditional sampling formulas.

5. Trade-Offs: Computational Complexity versus Variance Reduction

The initial determination of optimal quantizers (and precomputation of regression matrices for conditional paths) constitutes an offline cost. However, once this quantization-based stratification is in place, each trajectory sample within a cell can be generated at $O(n)$ computational cost (for $n$ time steps), dramatically reducing the cost relative to approaches requiring Cholesky decompositions or unstratified control variates.

Variance reduction factors are significant—for typical applications in option pricing or functionals of diffusion processes, order-of-magnitude improvements in variance are observed for fixed sample size. The method's allocation rules (either natural or Lipschitz-optimal) and hyperrectangular strata efficiently distribute simulation effort in proportion to local variance.

6. Applications to Complex and Path-Dependent Functionals

The method is particularly suited to:

• Path-dependent functionals of multidimensional diffusions, where the quantization-based strata capture the key directions of variation in the process,
• Payoffs and functionals with only Lipschitz regularity,
• Gaussian processes where the K–L expansion is available (Brownian motion, bridge, Ornstein–Uhlenbeck),
• Problems in mathematical finance (derivative pricing), stochastic control, and high-dimensional simulation.

The approach is also robust to the curse of dimensionality to the extent quantization grids can be efficiently computed for the key process directions, and is complemented by practical conditional sampling schemes.

7. Algorithmic Summary and Theoretical Guarantees

Step	Description
1. Quantizer construction	Compute optimal quadratic quantizer (minimize $E[|X-\text{Proj}_\Gamma(X)|^2]$ via codebook $\Gamma$).
2. Strata definition	Induce Voronoi partition; compute cell probabilities $p_i$.
3. Sample allocation	Assign number of samples per stratum using $q_i=p_i$ or $q_i^*$ (Lipschitz-optimal).
4. Path simulation in each stratum	Simulate leading K–L coordinates as quantized, then sample conditional remainder with Bayesian simulation ($O(n)$).
5. Estimator construction	Combine stratum averages via $$\overline{F(X)}_M = \sum_i p_i (1/M_i) \sum_{k=1}^{M_i} F(X^{(k)}|X^{(k)} \in A_i)$$.
6. Variance assurance	Total variance controlled by quantization error; uniform and Lipschitz-optimal bounds proven in (Corlay et al., 2010).

The entire procedure achieves unbiasedness, consistency, and asymptotic variance control, and is supported by rigorous convergence rates as quantization levels increase.

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Variance-reduced trajectory sampling using functional quantization-based stratification delivers a principled, algorithmically efficient, and universally applicable route to variance reduction in high-dimensional and functional Monte Carlo simulations. Its theoretical guarantees apply uniformly to the class of Lipschitz continuous functionals, and its algorithmic structure enables deployment in classical as well as modern applications demanding scalable, accurate trajectory-based sampling.