Stochastic Fubini Conditions for Brownian Integrals

Summary

The document asks when a time integral of a Brownian stochastic integral can be rewritten by reversing the order of integration. Its example uses an exponential kernel over nested time intervals and proposes the resulting single stochastic integral with an analytically integrated kernel. The question arises in an application to zero-coupon bond pricing under the Vasicek interest-rate model.

It seeks a rigorous stochastic Fubini theorem and a general criterion for interchanging an ordinary integral with a Wiener integral. The document itself supplies no theorem, integrability assumptions, or proof, so the displayed transformation should be read as a question rather than a verified result. In practice, such an interchange requires conditions that make both integrals well defined and justify exchanging their order; those conditions are not developed here. Its value is identifying the mathematical step that needs justification in a fixed-income modeling derivation.

Key ideas

• The example combines an ordinary time integral with a Brownian stochastic integral.
• Stochastic Fubini can permit switching the integration order under suitable assumptions.
• The exponential kernel example is connected to Vasicek zero-coupon bond pricing.
• The document asks for the needed theorem but does not state or verify its integrability conditions.

Tags

# When can I switch the order of a Riemann integral and a stochastic integral?


# When can I switch the order of a Riemann integral and a stochastic integral?












I'm working with an expression that combines a Lebesgue integral and a stochastic integral, and I'm trying to understand under what conditions it's valid to switch the order of integration.

Consider the following process: $$X := \int_t^T \left( \int_t^s e^{-a(s - u)} \, dW_u \right) ds $$

I am wondering if I can switch both integrals to become: $$X = \int_t^T \left( \int_u^T e^{-a(s - u)} \, ds \right) dW_u = \int_t^T \frac{1 - e^{-a(T - u)}}{a} \, dW_u $$

Is there a clean and rigorous statement of the stochastic Fubini theorem that justifies this operation?

More generally, if I have an expression like: $$\int_a^b \left( \int_c^d f(s,u) \, dW_u \right) ds $$

how can I determine when it's valid to write: $$\int_c^d \left( \int_a^b f(s,u) \, ds \right) dW_u \, ? $$

I came across this problem while trying to price zero-coupon bonds under the Vasicek model.

Thanks in advance!

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