Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
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Asymptotics and Simulation · Chapter II-07

An Error Ledger for Monte Carlo and Forward Simulation

Simulation enters as numerical integration, forward value approximation, nested averaging, and optimization input. Centered simulation noise, bias, discretization, and common-random-number dependence scale differently. A useful computation reports each component against the estimator's statistical normalization.

Conceptual map

  1. II-07.01Numerical integration
  2. II-07.02Forward simulators
  3. II-07.03Nested averages
  4. II-07.04Dependence from random numbers
  5. II-07.05Computational budgets

1. Centered noise and bias occupy different rows

Simulation can approximate an expectation, a continuation value, or an optimizer input. Its centered variance, directional bias, discretization error, and induced dependence use different rates.

For unit or inequality \(i=1,\ldots,n_{I}\), let \(\mu _{i}=\mathbb{E}[g_{i}(U)]\) and estimate it with \(\hat{\mu }_{i}=n_{s}^{-1}\sum _{s=1}^{n_{s}}g_{i}(U_{is})\). The aggregate integration error is \(E=n_{I}^{-1}\sum(\hat{\mu}_{i}-\mu _{i})\).

Proposition 1 · Independent-draw variance

If all centered terms are independent, have variances bounded by \(\sigma ^{2}\), and have zero mean, then

\[\operatorname{Var}(E)\le \sigma ^{2}/(n_{I}n_{s}).\](1)

Proof. The variance of a sum of independent centered terms is the sum of their variances. Divide the bound \(n_{I}n_{s}\sigma ^{2}\) by the squared denominator \((n_{I}n_{s})^{2}\). ∎

2. Shared random numbers alter covariance

Suppose \(g_{i}(U_{s})-\mu _{i}=Z_{s}\) for every \(i\), with independent unit-variance \(Z_{s}\) across draws. Then averaging over \(i\) does nothing: \(E=n_{s}^{-1}\sum Z_{s}\) and \(\operatorname{Var}(E)=1/n_{s}\).

With independent draws across both indices, the variance is 1/(\(n_{I}n_{s}\)). At \(n_{I}=10\) and \(n_{s}=100\), the two variances are 0.01 and 0.001. Shared random numbers can reduce variance of paired differences when covariance cancels; their value depends on the estimand.

3. Averaging cannot remove a common directional bias

If \(\mathbb{E}[\hat{\mu}_{i}-\mu _{i}]=b_{i}\), averaging produces \(\bar{b}=n_{I}^{-1}\sum b_{i}\). A shared sign or common approximation method can keep \(\bar{b}\) at the same order as each component. Root-\(n\) inference requires \(\sqrt{n}\bar{b}\to0\) or an explicit bias correction.

Time discretization, value-function truncation, inner optimization tolerance, and interpolation error are deterministic conditional on their numerical inputs, even when those inputs were generated randomly. Each receives its own convergence study and scale comparison.

4. Nested and adaptive simulations need conditional accounting

Failure case · Draw count multiplied across dependent layers

Using the same outer path for many inner evaluations does not create the product number of independent paths. The variance must include covariance induced by shared outer states, common seeds, antithetic pairing, and adaptive stopping.

At zero simulation variance, Monte Carlo error disappears but discretization and model error can remain. At one draw, variance estimators based on within-unit replication are unavailable. Random stopping times require conditions for unbiasedness and finite expected cost. Seed sensitivity is a diagnostic rather than a probability theorem.

5. Implementation, exercises, and sources

Record seeds, stream construction, common-random-number groups, nesting, antithetic rules, draw counts, time grid, truncation horizon, and solver tolerance. Replicate the entire simulation under independent top-level seeds and compare empirical variance with the analytic covariance formula.

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Exercises

  1. Verify both variances in Section 2.
  2. Let pairwise cross-unit correlation equal \(\rho\) and derive \(\operatorname{Var}(E)\) for equal within-unit variances.
  3. Add bias 1/\(n_{s}\) and derive the two root-\(n\) requirements.
Partial solutions

1. Independent averaging uses 1,000 unit-draw terms; perfectly shared noise uses 100 distinct terms. 2. For unit averages with variance 1/\(n_{s}\), the aggregate variance is \([1+(n_{I}-1)\rho]/(n_{I}n_{s})\). 3. Require \(n/n_{s}^{2}\to0\) for bias and \(n/(n_{I}n_{s})\to0\) for independent centered noise.

  1. Christian P. Robert and George Casella, Monte Carlo Statistical Methods, chapters on variance reduction.Monte Carlo variance and common random numbers.
  2. Paul Glasserman, Monte Carlo Methods in Financial Engineering, Chapters 4–5.Variance reduction and nested simulation principles.
  3. Charles Gouriéroux and Alain Monfort, Simulation-Based Econometric Methods.Simulation error in estimators.

6. Audit checkpoint

State the simulated object, outer and inner sample sizes, independent stream count, covariance groups, bias, discretization, horizon, solver error, random stopping rule, seeds, statistical normalization, and empirical replication variance.

7. Scope boundary

The chapter covers forward Monte Carlo averages and their use as generated inputs. Markov-chain Monte Carlo and continuous-time discretization theory require separate analyses.

Prerequisites