Stochastic Leibniz Rule in an HJM Bond-Price Derivation

Summary

The question concerns how to differentiate a bond-price expression in a single-factor Heath–Jarrow–Morton interest-rate model. The expression includes a time-indexed stochastic integral whose upper integration boundary also depends on the Brownian integration time. The author compares it with a stochastic Leibniz rule for an integral whose bounds evolve differently and asks how that rule applies to the bond-price term.

The document sets out the forward-rate dynamics, the bond-price representation, and an Itô differential already obtained for the bond. It does not include an answer or derivation, so it provides no resolution of the boundary-term or interchange-of-integration issue. Its value is as a mathematical problem statement about differentiating parameter-dependent stochastic integrals in fixed-income modeling, rather than as a self-contained explanation or validated derivation.

Key ideas

• The problem arises in an HJM model with stochastic forward rates and a bond price expressed through integrated forward rates.
• The stochastic integral has a maturity-dependent integration boundary.
• A standard stochastic Leibniz rule is cited, but its limits differ from those in the bond-price expression.
• The document poses the derivation question without providing a solution.

Tags

# Stochastic Leibniz rule


# Stochastic Leibniz rule












We have the following single-factor HJM model $$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$ $$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$ The discounted T bond is then \begin{align} Z(t,T)&=\exp-\bigg(\int_0^Tf(0,u)du+\int_0^t\int_s^T\sigma(s,u)dudW_s+\int_0^t\int_s^T\alpha(s,u)duds\bigg)\\ &=\exp\bigg(-\int_0^Tf(0,u)du+\int_0^t\underbrace{-\int_s^T\sigma(s,u)du}_{\Sigma(s,T)}dW_s-\int_0^t\int_s^T\alpha(s,u)duds\bigg) \end{align} By Ito's Lemma $$d_tZ(t,T)=Z(t,T)\Bigg(\bigg(\frac{1}{2}\Sigma^2(t,T)-\int_t^T\alpha(t,u)du\bigg)dt+\Sigma(t,T)dW_t\Bigg)$$ My question is, how was Ito's Lemma applied to the term $\int_0^t\int_s^T\sigma(s,u)dudW_s$, which contains an integral of Brownian motion?

Claus Munk's Fixed Income Modelling proves the following stochastic Leibniz rule on page 57-58 $$Y_t=\int_t^Tf(0,u)du+\int_t^T\int_0^t\alpha(s,u)dsdu+\int_t^T\int_0^t\sigma(s,u)dW_sdu$$ $$dY_t=\bigg(\int_t^T\alpha(t,u)du-f(t,t)\bigg)dt+\bigg(\int_t^T\sigma(t,u)du\bigg)dW_t$$ however due to the different limits in the integrals, I am unable to extrapolate from Munk's example to solve my question.

Any help is appreciated.

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