Variance Explosion: Theory & Mitigation

• Variance explosion is a phenomenon where estimator variance increases exponentially with system parameters, posing challenges to simulation and inference.
• It occurs in settings such as Girsanov reweighting for Itô diffusions and stochastic volatility models, leading to sample collapse and biased estimators.
• Mitigation strategies like Marginal Girsanov Reweighting (MGR) decompose long-horizon weights to maintain variance growth at a manageable, linear rate.

Variance explosion refers to the rapid—often exponential or even super-exponential—increase in the variance of certain random variables or estimators as a system parameter, such as time or discretization step, increases. This phenomenon arises in a variety of applied contexts including importance sampling for stochastic processes and stochastic volatility modeling in quantitative finance. Variance explosion undermines the reliability of reweighting schemes, causes collapse of effective sample size, and is a central obstruction to unbiased inference and rare-event simulation methods over long timescales or with highly variable underlying state processes.

1. Variance Explosion in Girsanov Reweighting for Itô Diffusions

A canonical example of variance explosion appears in the Girsanov theorem, which gives the likelihood ratio between the law $\mu$ for a $d$-dimensional Itô diffusion with drift $a$ and reference law $\tilde\mu$ with drift $\tilde a$ but identical nondegenerate diffusion $b$ over a finite time horizon $[0,T]$. The Radon–Nikodym derivative is

$$L_T = \exp\Bigl(\int_0^T \theta_s^\top\,dW_s - \frac{1}{2}\int_0^T \|\theta_s\|^2 ds\Bigr),$$

with $\theta_s = b(X_s)^{-1} (a(X_s) - \tilde a(X_s))$ under reference law $\tilde\mu$.

Although $d$0 is an unbiased importance weight ($d$1), its variance conditional on $d$2,

$d$3

scales as $d$4 for ergodic systems ($d$5). Thus, $d$6 becomes rapidly unmanageable for long time horizons $d$7, rendering importance sampling estimators using $d$8 impractical (Wang et al., 30 Sep 2025).

2. Practical Impact: Sample Collapse, Biased Estimators, and Rare Event Inference

Variance explosion in practice leads to effective sample size (ESS) collapse, with nearly all weight concentrated on a vanishing fraction of trajectories and the remainder carrying negligible information. For long trajectories, as typically required for unbiased kinetic or Bayesian inference in stochastic dynamics or molecular simulation, the true variance of estimators based on classical Girsanov weights becomes so large that sampling noise dominates and estimators are unreliable. This effect is particularly acute in metastable systems, rare-event sampling, and Bayesian inference for SDEs with sparse observation schemes (Wang et al., 30 Sep 2025).

3. Mitigation Strategies and Marginal Girsanov Reweighting (MGR)

Several approaches have been developed to address variance explosion, among which Marginal Girsanov Reweighting (MGR) is prominent. MGR avoids direct computation of $d$9 by instead targeting the lag-$a$0 transition weight

$a$1

and decomposes long-horizon weights into compositions of short-lag, numerically stable Girsanov weights and learned binary classifiers that marginalize over intermediate states.

The key algorithmic step involves recursively accumulating stable short-lag weights using importance sampling and density ratio estimation, such that the variance of the weight estimator $a$2 grows at most linearly in $a$3 (or even sub-linearly in well-mixing cases), i.e.,

$a$4

in strong contrast to the exponential growth $a$5 in the classical pathwise scheme (Wang et al., 30 Sep 2025).

4. Variance Explosion in Stochastic Volatility Models: Randomised Heston

In mathematical finance, variance explosion arises in the context of short-maturity options pricing and implied volatility surfaces. The “randomised Heston model” introduced by Jacquier and Shi assumes a random initial variance $a$6 rather than a deterministic one: $a$7 with $a$8 independent Brownian motions (Jacquier et al., 2016).

The right-tail decay of the density of $a$9 sets the speed of explosion for the squared implied volatility $\tilde\mu$0 as $\tilde\mu$1: $\tilde\mu$2 with the “explosion exponent” $\tilde\mu$3 controlled by the moment generating function (mgf) or regular variation properties of $\tilde\mu$4. With sufficiently heavy right tails, one can recover empirically observed exponents $\tilde\mu$5, which are unattainable in classical (fixed-$\tilde\mu$6) models.

5. Quantitative Examples: Explosion Exponents and Probability Tail Behavior

In the randomised Heston model, three regimes are possible:

Regime	$\tilde\mu$7 Tail	Explosion Exponent $\tilde\mu$8	Asymptotics for $\tilde\mu$9
Bounded support	Fast cutoff	$\tilde a$0	No explosion
Fat-tail mgf	$\tilde a$1 for $\tilde a$2	$\tilde a$3	$\tilde a$4
Thin-tail	$\tilde a$5, $\tilde a$6	$\tilde a$7	$\tilde a$8

For instance, if $\tilde a$9 is exponential (mgf has finite cutoff), then $b$0; if $b$1 is Rayleigh (thin-tail: $b$2), then $b$3 (Jacquier et al., 2016).

6. Broader Implications and Methodological Significance

Variance explosion sets fundamental limits on the feasibility of unbiased rare-event simulation, inference for long-horizon dynamical systems, and pricing of short-maturity financial derivatives. It has driven the development of alternative reweighting schemes, variance reduction frameworks, and density ratio estimation tools (e.g., MGR using neural networks). In molecular dynamics, MGR demonstrated stable recovery of kinetic and thermodynamic properties at time horizons where classical Girsanov weights fail completely. In stochastic volatility, randomisation of initial variance provides a tractable mechanism to interpolate between insufficiently explosive classical volatility surfaces and observed steep option smiles.

A plausible implication is that controlling variance explosion—not merely reducing mean squared error—should be central in the design of sampling, inference, and asset pricing methodologies for high-dimensional and long-horizon stochastic systems. Challenges remain in optimizing step sizes, classifier architectures, and in extending these variance-controlling principles to more general state spaces and rare-event regimes (Wang et al., 30 Sep 2025, Jacquier et al., 2016).