Applying Tanaka’s Formula to the Absolute Value of a Diffusion

Summary

The question considers a variable following a geometric Brownian motion that can take positive or negative values, and asks how to derive the process of its absolute value. The obstacle is that the absolute value function has a kink at zero, so the ordinary version of Itô’s lemma does not apply there directly.

The answer points to the extended Itô formula based on Tanaka’s formula, which accounts for the nonsmooth point through local time. It also recommends a reference on local time. The document does not give the resulting stochastic differential or explain how to calculate its terms, so it serves as a pointer to the right mathematical tool rather than a worked derivation. Readers would need the cited reference or another treatment of Tanaka’s formula to complete the calculation.

Key ideas

• The absolute value function is not differentiable at zero, so ordinary Itô’s lemma cannot be applied there directly.
• Tanaka’s formula extends Itô’s method to functions with this type of kink.
• The extended formula includes a local time term at the point where the function is nonsmooth.
• The answer identifies the relevant method but does not work through the resulting process.

Tags

# Process for mod of a variable that follows some Stochastic Process


# Process for mod of a variable that follows some Stochastic Process












Assuming a variable $v$ follows some `Stochastic Process` as below -

$dv=\mu v dt + \sigma v dW_t, v \in \left( -\infty, \infty \right) $

I want to get the process of $|v|$

How can I use the `ito's lemma` in this context given that the function $|v|$ is not smooth?

## Answer by ir7 (score 3)

https://quant.stackexchange.com/a/55838

I think you are looking for extended Ito formula (based on Tanaka's formula).

Bjork's The Pedestrian’s Guide to Local Time should be useful.

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