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The question is inspired by a short passage on the LMM in Mark Joshi's book.

The LMM cannot be truly Markovian in the underlying Brownian motions due to the presence of state-dependent drifts. Nevertheless, the drifts can be approximated 'in a Markovian way' by using predictor-corrector schemes to make the rates functions of the underlying increments across a single step.

Ignoring the drifts, the LMM would be Markovian in the underlying Brownian motions if the volatility function is separable. The volatility function $\sigma_i(t)$ is called separable if it can be factored as follows $$\sigma_i(t)=v_i\times\nu(t),$$ where $v_i$ is a LIBOR specific scalar and $\nu(t)$ is a deterministic function which is the same across all LIBOR rates.

Questions.

  1. The separability condition above is sufficient for the LMM to be Markovian in the Brownian motion. How far is it from being a necessary one?
  2. What is the intuition behind the separability condition?
  3. Are there any weaker sufficient conditions?
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You can use a matrix type seperability condition as well. This is similar but the equation has more flexibiliity. The rates are then markovian in some combinations of the Brownian motion. See More Mathematical Finance for details.

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Re 2: It's a mathematical trick. Insisting on the separability of volatility function makes LMM useless. Its power lies in its powerful calibration abilities. If you constrain the vol function to separable form, you throw that ability out of the window. You might just as well use LGM then, and it will be more intuitive and faster.

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