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Leibniz Rule for Integrated Instantaneous Forward Rates

Article Quant Q&A · Author: Younes S
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Summary

The document asks how to differentiate the accumulated instantaneous forward rate from the current time to a fixed maturity in the Heath–Jarrow–Morton framework. The expression combines a moving lower integration bound with a stochastic forward-rate process, so its differential contains a boundary contribution as well as the integral of the process differential.

The responses identify this as the Leibniz integral rule in differential form. Since maturity is fixed while current time advances, differentiating the lower bound contributes the negative forward rate evaluated at the current time; the fixed upper bound contributes no boundary term. The remaining term comes from integrating the forward-rate differential across maturities. The explanation is conceptual and points to the classical rule rather than developing technical conditions for exchanging stochastic integration and differentiation, which may matter in rigorous treatments.

Key ideas

  • The integrated forward rate has a moving lower bound and a fixed maturity bound.
  • The Leibniz rule produces a negative boundary term at the current time.
  • A fixed upper limit contributes no boundary term as current time changes.
  • The forward-rate differential is integrated over the maturity interval to obtain the remaining term.
  • The discussion gives an intuition for the identity but does not state conditions for stochastic interchange operations.

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Fixed incomeDerivatives pricingStatistics
Full text
# Instantaneous forward rate within the HJM framework


# Instantaneous forward rate within the HJM framework












within the HJM framework, the dynamics of the instantaneous forward rate are defined by:

$$f_t(T)=f_0(T) + \int_0^t\alpha_s(T)ds+\int_0^t\sigma_s(T)dW_s$$

or in differential form: $$df_t(T)=\alpha_t(T)dt+\sigma_t(T)dW_t$$

In the litterature (like Tankov, you can find the url below), it is written that: $$d\left(\int_t^Tf_t(u)du\right)= -f_t(t)dt+\int_t^Tdf_t(u)du $$ I could not find a proof and Tankov mentions it like it is trivial.

page 96 in :https://masterfinance.math.univ-paris-diderot.fr/attachments/article/47/processus_en_finance_6_7.pdf

Thank you for your help.

## Answer by Daneel Olivaw (score 5)

https://quant.stackexchange.com/a/49142

This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is: $$\text{d}\left(\int_{g(x)}^{h(x)}f(x,s)\text{d}s\right)=h'(x)f(x,h(x))\text{d}x-g'(x)f(x,g(x))\text{d}x+\int_{g(x)}^{h(x)}\text{d}f(x,s)\text{d}s$$

## Answer by Magic is in the chain (score 2)

https://quant.stackexchange.com/a/49143

It is just an application of the Leibniz integral rule, written in differential form. Please see here: https://en.m.wikipedia.org/wiki/Leibniz_integral_rule

Capital T is constant, t is changing, so the second term on the right hand side is the exchange of integral and differential, the first term on the right hand side is the function value at the lower integration limit times derivative of t wrt t (which is 1), the function value at upper integration limit term that you see in the Leibniz rule is zero here because T is considered constant.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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