Stratified Sampling Method

• Stratified sampling is a variance reduction technique that divides the sample space into non-overlapping strata to enhance estimator accuracy.
• Quantization-based stratification uses discrete approximations, like Voronoi cells and product quantizers, to minimize mean-square error in functional estimations.
• This method balances significant variance reduction with increased computational complexity, making it effective for high-dimensional, path-dependent, and Lipschitz continuous functional applications.

Stratified sampling is a variance reduction technique in stochastic simulation, numerical integration, survey sampling, and computational statistics that achieves more efficient estimation by partitioning the state, parameter, or spatial domain into disjoint "strata" and allocating simulation effort non-uniformly according to the variability or importance within each stratum. This approach encompasses a family of methods—including functional quantization for stochastic processes, product quantizers for functional data, adaptive stratification for non-smooth problems, and optimal or hybrid allocation rules—that target the minimization of estimator variance while maintaining unbiasedness across a range of applications.

1. Principles of Stratified Sampling and Variance Reduction

The fundamental idea in stratified sampling is to subdivide the sample space into non-overlapping subsets (strata), within each of which separate sampling occurs. Let $X$ denote a random variable on a measurable space $E$. Given a measurable partition $\{C_i\}_{i=1}^N$ of $E$, the goal is to estimate $\mathbb{E}[F(X)]$ as a weighted sum of conditional expectations:

$$\mathbb{E}[F(X)] = \sum_{i=1}^N p_i \mathbb{E}[F(X) \mid X \in C_i],$$

where $p_i = \mathbb{P}(X \in C_i)$. The variance of an estimator constructed via stratified sampling is reduced compared to naive Monte Carlo if the conditional variances $\mathrm{Var}(F(X) \mid X \in C_i)$ are not uniform. The variance-minimizing sample allocation—known as Neyman allocation—is proportional to $p_i \sigma_i$, where $\sigma_i^2$ is the within-stratum variance (Nguyen et al., 2018).

For functional or infinite-dimensional settings, the partition is typically constructed via quantization, yielding Voronoi cells (strata) built from a codebook $\Gamma$, with the projection $\mathrm{Proj}_\Gamma(X)$ defined as the nearest neighbor map. The estimator’s variance is then controlled by the quantization error, i.e., the squared $\mathrm{L}^2$ distance between $X$ and its projection.

2. Quantization-Based Stratification and Optimality

Stratification via quantization utilizes discrete approximations of a random variable or process to build "optimal" partitions for variance reduction. The quantization error in the mean-square sense measures the effectiveness of the stratification:

$$\mathcal{C}(N) = \min_{\mathrm{card}(\Gamma) \leq N} \mathbb{E}[|X - \mathrm{Proj}_\Gamma(X)|^2].$$

Optimal quadratic quantizers have a stationarity property:

$\gamma_i = \mathbb{E}[X \mid X \in C_i],$

ensuring that replacing $X$ by its cell-conditional mean preserves unbiasedness. The variance of any estimator of a Lipschitz functional $F$ under this scheme satisfies

$$\sup_{[F]_{\mathrm{Lip}} \leq 1}\sum_{i=1}^N p_i \sigma^2_{F,i} = \|X - \mathrm{Proj}(X)\|_2^2,$$

yielding a universal variance reduction over the class of Lipschitz continuous functionals (Corlay et al., 2010).

In finite-dimensional settings, the quantization error behaves as $O(N^{-1/d})$ (with $d$ the dimension); in infinite dimensions (e.g., Gaussian processes), the rates are logarithmic. A key aspect is that quantization-based partitions achieve uniform efficiency across all Lipschitz payoffs or functionals.

3. Product Functional Quantization for Stochastic Processes

Path-dependent and high-dimensional simulations, particularly of diffusions or Gaussian processes, benefit significantly from product functional quantization. Given a process $X$ with a Karhunen–Loève expansion,

$X(t) = \sum_{n} \sqrt{\lambda_n} \xi_n e_n(t),$

one constructs a product quantizer by optimally quantizing each independent coordinate $\xi_n$, thereby partitioning the space into hyperrectangular strata. For each stratum (indexed by quantized values of the first $d$ coordinates), the conditional expectation and conditional law of the process at sampling points or time grid nodes can be derived using the covariance structure. For example, the conditional law of $V = (X(t_0), ..., X(t_n))$ given the first $d$ K–L coordinates $Y$ is

$$\mathcal{L}(V \mid Y = y) = \mathcal{N}(A_{V|Y}(y), K),$$

with $A_{V|Y}(y)$ and $K$ given by explicit Gaussian regression formulas. This enables “guided” simulation of conditional paths within each stratum, which is crucial in accurately estimating prices for path-dependent options or other functionals (Corlay et al., 2010).

For practical processes, such as Brownian bridge or the Ornstein–Uhlenbeck (OU) process, closed-form Karhunen–Loève expansions are available. For the OU process $dX_t = -\theta X_t dt + \sigma dW_t$, the eigenfunctions and eigenvalues are obtained by solving a characteristic equation of the form

$\theta \tan(\omega_n T) = -\omega_n$

with corresponding $$\lambda_n^{OU} = \sigma^2 / (\omega_n^2 + \theta^2)$$. This permits construction of explicit product quantizers for stratification of functional spaces.

4. Algorithmic Realization and Simulation Complexity

Practical realization of quantization-based stratified sampling requires careful consideration of computational complexity. The key steps include:

• Construction of product quantizers (including fast nearest-neighbor projections for Voronoi cells in product form).
• Calculation and storage of regression matrices for conditional simulation (e.g., $R_{V|Y}$ and its transpose).
• Efficient simulation within each stratum, leveraging the independence of quantized coordinates to maintain linear-in-$n$ cost where $n$ is the number of time discretization steps.

While the step of projecting a simulated path onto the codebook may represent a computational bottleneck when using generic nearest-neighbor searches, the use of product quantization restricts this to $O(d)$, where $d$ is the quantization dimension. For Gaussian models, regression and conditioning formulas can be precomputed analytically or via recurrence, keeping the overall cost per simulated conditional path at $O(n)$. Empirical results in derivative pricing (e.g., for path-dependent or barrier options) show variance reductions of approximately 50–90% compared to standard Monte Carlo at comparable simulation cost (Corlay et al., 2010).

5. Uniform Efficiency for Lipschitz Functionals

A salient property of quantization-based stratification is its universality for the class of Lipschitz continuous functionals. Proposition (“Universal Stratification”) states that the quantization-induced partition provides the smallest possible variance—quantified by the quantization error—uniformly over all functional payoffs with Lipschitz constants at most one. This guarantees that, for a broad spectrum of financial and statistical functionals, a single stratification design serves as a nearly optimal variance reduction mechanism (Corlay et al., 2010).

This is particularly significant in applications where the payoff functions can vary widely but are structurally Lipschitz (e.g., European and Asian options, functionals of stochastic integrals, or risk metrics).

6. Trade-Offs Between Variance Reduction and Computational Load

The substantial variance reductions of quantization-based stratified Monte Carlo methods must be juxtaposed with increased implementation complexity compared to basic Monte Carlo:

• Projection onto the quantization codebook, and computation of the corresponding stratum, may be nontrivial in high dimensions unless product structures are exploited.
• Regression matrices needed for conditional Gaussian simulation within each stratum must be computed, but can be done in closed form for standard processes.
• The method achieves linear scaling in number of time discretization steps for a fixed quantization dimension, but the total cost increases with the number and granularity of strata.

However, the trade-off is generally favorable: variance reduction of greater than 50% is achievable, and compared to naive variance reduction techniques or control variates, the method is robust and broadly applicable to path-dependent and infinite-dimensional settings. In scenarios such as high-dimensional derivative pricing, the total simulation cost remains of the same order as the cost per path of standard Monte Carlo while achieving much greater estimator efficiency.

7. Applications and Extensions

Quantization-based stratified sampling frameworks are particularly suited for:

• High-dimensional Monte Carlo simulation of path-dependent payoffs in finance and insurance, especially when the driving stochastic process is Gaussian (Brownian motion, Brownian bridge, OU process) or admits a tractable Karhunen–Loève expansion.
• Numerical evaluation of expectations of Lipschitz continuous functionals over function spaces.
• Accelerated uncertainty quantification for stochastic PDEs, where functional quantization leverages the smoothness or regularity of the underlying process.
• Broader classes of functionals through hybrid use with control variate or importance sampling techniques, leveraging the universality of variance reduction for Lipschitz payoffs.

Explicit derivations for classical processes, accompanied by regression and conditional simulation formulas, facilitate integration of this stratification technique into existing Monte Carlo frameworks. The approach is particularly robust for plug-in application to derivative pricing and risk management systems with path-dependent or high-dimensional requirements.

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This quantization-driven stratification paradigm thus provides an unbiased, robust, and computationally efficient mechanism for variance reduction in Monte Carlo simulation, achieving uniform optimality for a broad class of functional estimators relevant in applied probability, finance, and stochastic analysis (Corlay et al., 2010).