Infinite-Series Weights in an OU Trade-Length Variance Formula

Summary

The document raises a mathematical question about calculating the variance of trade duration for a quantitative strategy based on an Ornstein–Uhlenbeck process. It refers to Bertram’s 2010 treatment and presents expressions for two weight terms, including an infinite series involving the gamma function and a second expression involving the digamma function. The author asks how to evaluate the series and interpret the weights in the cited equation.

No answer or derivation is included, so the document does not provide a method for summing the series, establishing convergence, or applying the result to a trading rule. It is useful as a pointer to a specific analytical issue in OU-based strategy research, but it offers no numerical evidence, implementation guidance, or discussion of the assumptions and limits behind the underlying model.

Key ideas

• The question concerns the variance of trade duration in an Ornstein–Uhlenbeck trading model.
• The cited formula defines weights using infinite sums and special functions.
• One expression includes the gamma function, while another invokes the digamma function.
• The document asks how to evaluate the weights but provides no derivation or solution.
• Any practical use requires consulting the original formula and verifying its assumptions and convergence.

Tags

# Variance of trade length for the OU-process


# Variance of trade length for the OU-process












Based on the article by Bertram (2010), I am trying to calculate the variance of the trade length stated in equation [10] of the paper. However, the weights used has a specification that I cannot quite get my head around. My question is, how do I evaluate the infinite sum for the weights as in this equation:

$w_1(z)=(\frac{1}{2}\sum_{k=1}^{\infty}\Gamma(k/2)(\sqrt{2}z)^k/k!) - (\frac{1}{2}\sum_{k=1}^{\infty}(-1)^k\Gamma(k/2)(\sqrt{2}z)^k/k!)^2$

$w_2(z)=\Gamma((2k-1)/2)\Psi((2k-1)/2)(\sqrt{2}x)$

Where $\Psi$ is the digamma function. I have considered the answers here and here, but it is still not clear to me how I should proceed.

Any help is appreciated!

I post here as the question is related to a quantitative trading strategy.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.