Martingale Conditions for the CEV Diffusion Model

Summary

The document asks whether a constant-elasticity-of-variance diffusion of the form dS = σS^βdW is a martingale when β is below one. The answer recalls that Ito integrals are martingales under suitable integrability conditions and links those conditions to square integrability of the integrand.

It argues informally that for exponents above one, the relevant squared process may have an infinite integral, preventing a straightforward square-integrability argument. It associates this concern with the variance of the Ito integral and quadratic variation. However, the response does not give a precise theorem, proof, or complete characterization across values of β, and its explanation of quadratic variation is not rigorous. Readers should treat it as a prompt to examine the needed integrability conditions rather than a definitive result about the CEV model.

Key ideas

• The question concerns whether the CEV diffusion is a martingale for exponents below one.
• Ito-integral martingale results depend on appropriate integrability conditions.
• Square integrability of the integrand is connected to the variance of the stochastic integral.
• The response raises integrability concerns for higher exponents but does not establish a complete criterion.
• A rigorous answer requires specifying assumptions and checking the relevant stochastic integral conditions.

Tags

# Martingale property of the CEV model


# Martingale property of the CEV model












I am a bit confused about the martingale property of the CEV model. Given $dS(t)=σS(t)^βdW(t)$, is $S$ a martingale for values of $β<1$?

## Answer by Arshdeep (score 0)

https://quant.stackexchange.com/a/78848

Ito integrals are martingales. I guess the problem for beta>1 is the square of process has an infinite integral so the limits you use to prove this don't follow. The square integral is relevant because it is the variance of the ito integral.

It is intuitive because all of Ito's work relies on quadratic variation converging almost surely to $t$. If process is not square integrable, it's vol is infinite and thus it's quadratic var. does not converge to $t$, making it difficult to call it an "Ito integral" as it cannot be computed by ito's lemma.

Ultimately CLT determines everything.

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