Distribution and Independence of a Deterministic-Integrand Brownian Integral

Summary

The document considers a stochastic integral over a time interval whose integrand is the exponential of an accumulated deterministic function involving a constant rate and a time-varying deterministic term. It states that, although the integral generally has no analytical expression, its distributional properties remain useful: it is normally distributed and independent of the information available at the interval’s starting time.

These properties can support calculations in interest rate models, including zero-coupon bond pricing in the Hull–White framework. The document gives no derivation, explicit mean or variance, or numerical example, so it is a brief statement of results rather than a complete solution. The conclusions rely on the rate and time-varying function being deterministic; stochastic inputs would require additional analysis.

Key ideas

• With a deterministic integrand, the Brownian integral is normally distributed.
• The integral is independent of information available at the start of its integration interval.
• These properties can be used in zero-coupon bond calculations under the Hull–White model.
• The document gives no closed-form evaluation or derivation of the integral.

Tags

# Answer by Gordon (score 1)


# How do one solve $ \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u $? Double integral with general deterministic function $\delta(t)$












How do one solve $ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $ ?

$\delta(t)$ is a general deterministic function. $r$ is constant.

## Answer by Gordon (score 1)

https://quant.stackexchange.com/a/22235

There is no analytical solutions to this integral. The conclusions we can draw about this integral are that, if $r$ and $\delta$ are deterministic, it is normal and is independent of the information set $\mathscr{F}_t$. These are probably the most needed properties, for example, in the computation of a zero-coupon bond price under the Hull-White interest rate model, as demonstrated in question. What else are you looking for?

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