Why Orthogonalizing Brownian Path Samples Can Distort SDE Inputs

Summary

This question examines whether QR or Gram–Schmidt orthogonalization can make Sobol Brownian bridge increments more stable across random seeds when simulating stochastic differential equations. The author reports improved sample stability after orthogonalizing the increments and asks whether this is standard practice or whether it changes their distribution.

The key concern is sound: a nonlinear transformation of Gaussian samples generally does not preserve their joint Gaussian distribution, and forcing sample vectors to be orthogonal also changes their dependence structure. That can undermine the assumptions behind Brownian increments. The document poses the issue but provides no answer, derivation, comparison, or simulation evidence, so it does not establish that orthogonalization is appropriate. Its value is as a question highlighting the tradeoff between variance reduction or sample stability and faithfully preserving the intended stochastic process.

Key ideas

• Orthogonalizing simulated Brownian increments may improve apparent stability across seeds.
• Gram–Schmidt and QR transformations can alter the joint distribution of Gaussian samples.
• The question does not provide an answer or evidence that the transformation preserves Brownian properties.

Tags

# Orthogonalizing brownian path


# Orthogonalizing brownian path












I want to improve the stability of my SDE sample (statistical properties do not change much when using a different seed). I am using a sobol brownian bridge to generate the brownian path increments dw. I have seen a noticeable improvement when I use qr/gram-schmidt orthogonalization of my dw samples as it makes the samples completely independent. Is this a general practise or am I doing something wrong (as gram schmidt is a nonlinear process making the samples non gaussian).

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