Compact embeddings for spaces of forward rate curves

TL;DR

The paper addresses approximating forward-rate evolutions by finite-dimensional dynamics within the Heath-Jarrow-Morton-Musiela (HJMM) framework. It proves a Rellich-type compact embedding for forward-curve spaces: for every $\gamma > \beta > 0$, the Sobolev-type space $H_{\gamma}$ embeds compactly into $L_{\beta}^2 \oplus \mathbb{R}$, using a reflection to $W^1(\mathbb{R})$ and Fourier-analytic techniques. This compactness yields a concrete approximation mechanism: finite-rank operators $T_n$ with $T_n \to \mathrm{Id}$ allow the HJMM dynamics in $H_{\gamma}$ to be approximated by finite-dimensional processes in $L_{\beta}^2 \oplus \mathbb{R}$, with error controlled by $\|T_n - \mathrm{Id}\|$ and, for SPDEs, by the availability of Itô-process approximations $r^{(n)}$. The results provide a rigorous path from infinite-dimensional forward-rate evolution to tractable finite-dimensional simulations, preserving the forward-curve structure in the larger state space.

Abstract

The goal of this note is to prove a compact embedding result for spaces of forward rate curves. As a consequence of this result, we show that any forward rate evolution can be approximated by a sequence of finite dimensional processes in the larger state space.

Theorems & Definitions (14)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 3.1