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Interpreting the Undefined Function in Longstaff–Schwartz Convergence

Article Quant Q&A · Author: Caro
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Summary

The note examines an undefined function, F_X, in a theorem associated with the Longstaff–Schwartz method for valuing American options. It compares the theorem’s notation with the paper’s discussion of approximating the continuation value F using M basis functions, denoted F_M.

The answer argues that F_X is likely a typographical error for F_M. Its evidence is the surrounding explanation, which specifically describes uniform convergence of F_M to F under stated integrability conditions. The note is brief and relies on that contextual consistency rather than a separate proof or erratum, so the proposed correction is persuasive but not independently verified.

Key ideas

  • The continuation value F is approximated with a finite set of basis functions to form F_M.
  • The theorem’s undefined F_X appears inconsistent with the surrounding notation.
  • The paper’s convergence discussion supports reading F_X as F_M.
  • The suggested correction is based on context rather than independent verification.

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# What is this function in the Longstaff-Schwartz paper?


# What is this function in the Longstaff-Schwartz paper?












$F$ is the conditional expectation function (the "continuation value") and our approximate of this using $M$ basis functions is $F_M$... but in the paper, they have this theorem:

What is $F_X$? It has not been defined in the paper.

## Answer by Mats Lind (score 5)

https://quant.stackexchange.com/a/41347

From the arguments in the lines following the proposition in this paper where the proposition is made; it really looks as if $F_X$ is a typo which should actually read $F_M$, which stands for an approximation of $F$ using the first $M$ basis functions. It is argued that:

> The key to this result is that the convergence of $F_M(w, t)$ to $F(w;t)$ is uniform on (0,m) when the indicated integrability conditions are met.

which clearly points to that the integrabilitiy condition in the proposition made for $F$ and $F_X$ really was meant for $F$ and $F_M$

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.