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Using Integrability to Justify CIR Process Expectations

Article Quant Q&A · Author: user39119
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Summary

The document examines a mathematical issue in deriving the expected value of the Cox–Ingersoll–Ross short-rate process. Applying Itô’s lemma to an exponentially scaled rate gives a drift term that leads to the familiar expected-rate expression, provided the stochastic integral has expectation zero. The question focuses on the condition needed to treat that integral as a true martingale rather than only a local martingale.

The answer points to proofs of the expectation result and suggests establishing finiteness of the relevant Lebesgue integral, using the nonnegativity of the integrand to justify moving expectation inside the integral with Fubini’s theorem. The discussion highlights a useful distinction: an Itô integral does not automatically have zero expectation without suitable integrability conditions. It remains a brief pointer rather than a full proof, and it does not spell out the bounds needed to verify those conditions for the CIR process. The topic is relevant to interest-rate modeling and stochastic calculus, though the document gives no empirical trading analysis.

Key ideas

  • The CIR model specifies mean-reverting short-rate dynamics with volatility proportional to the square root of the rate.
  • The expectation derivation requires the stochastic integral to have zero expectation.
  • A local martingale need not be a true martingale without appropriate integrability.
  • Fubini’s theorem may help establish finiteness when the integrand is nonnegative.

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Fixed incomeStatistics
Full text
# Expectation of the CIR process


# Expectation of the CIR process












The CIR process follows

$$dr_t = (\alpha - \beta r_t)dt + \sigma \sqrt{r_t}dW_t.$$

It can be proved that there exists a unique solution for this SDE but it's not possible to get an expression for that solution. However, according to the Shreve (page 152) we can obtain the expected value of the solution. Applying Ito's lemma with the function $f(t,x)=e^{\beta t}x$ we obtain $$e^{\beta t}r_t = r_0 + \frac{\alpha}{\beta}(e^{\beta t} -1) + \sigma \int_0^t (e^{\beta t}-1)+ \sigma \int_0^t e^{\beta u} \sqrt{r_u}dW_u.$$ So he obtains $$e^{\beta t}E [r_t] = r_0 + \frac{\alpha}{\beta}(e^{\beta t} -1)$$ because he says that $$E \left[\sigma \int_0^t e^{\beta u} \sqrt{r_u}dW_u \right]=0.$$ He says that the expectation of an Ito integral is zero, but this is not always true. If $$E\left[ \int_0^T |\sigma e^{\beta u} \sqrt{r_u}|^2du \right]< \infty \tag*{($\star$)}$$ then the process $\{I_t\}:=\left\{\sigma \int_0^t e^{\beta u} \sqrt{r_u}dW_u ; 0 \leq t \leq T \right\}$ is a martingale and the expectation is zero. But if ($\star$) is not true, then the process $\{I_t\}$ is just a local martingale, not necessarily a (true) martingale and the expectation above may not be zero. Since we don't have the expression for the solution, I don't think the condition ($\star$) can be verified. Is there any other argument that allow us to conclude that the expectation is zero? Do you know of any papers/books where they treat this issue?

## Answer by Valometrics.com (score 1, accepted)

https://quant.stackexchange.com/a/50966

Please have a look at this question: https://math.stackexchange.com/questions/944181/martingality-theorem-solving-expectation-of-a-stochastic-integral/953779#953779 You have there two different proofs for your question. But in general case, you can try to prove that your lebesgues integral is finite by using the fubini theorem that allows you to move the expectation inside the integral as the stuff inside the integral is positive.

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