I recently started reading Filipovic's Consistency problems for HJM interest rate models and came across the Musiela reparametrization

$$r_t(x)=f(t,x+t)$$ so the forward curve can be thought of as a map $x\to r_t(x)$. The book goes on to tell that this maybe thought of as an infinite dimensional state variable.

Does anyone have a good explanation for this? Is it because we pick $r_t$ from a space of functions?


Keeping it simple, you know HJM SDE gives the dynamics of an instantaneous forward referencing a fixed maturity T, $f\left(t, T\right)$, but there is a continuum of such maturities - the whole forward curve as a function of T.

You can have each forward driven by a different brownian for example, so in general the HJM approach will be infinite dimensional. to visualise infinite dimensions, it may be helpful to recall one dimensional SDE, two dimensional SDE, and so on. But there are special cases for which it an be viewed as finite dimensional.

Please also see the discussion here: Musiela parameterization

  • $\begingroup$ Say we want to price a set of bonds with maturities $T$ varying on a certain interval so we have a continuum of bonds. In this case I see why we end up with a continuum of forwards. But what if we just take a finite set of bonds? $\endgroup$ – Heisenberg Aug 25 '19 at 16:43
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    $\begingroup$ But then even a single maturity zero coupon depends on the entire range of instantaneous forwards, remember the relationship: $B=e^{-\int_{0}^{T}{f(0,u)du}}$ $\endgroup$ – Magic is in the chain Aug 25 '19 at 16:58
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    $\begingroup$ Oh yes!! Thank you so much it makes sense now! $\endgroup$ – Heisenberg Aug 25 '19 at 17:03

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