Real-world models for multiple term structures: a unifying HJM semimartingale framework

We develop a unified framework for modeling multiple term structures arising in financial, insurance, and energy markets, adopting an extended Heath-Jarrow-Morton (HJM) approach under the real-world probability. We study market viability and characterize the set of local martingale deflators. We conduct an analysis of the associated stochastic partial differential equation (SPDE), addressing existence and uniqueness of solutions, invariance properties and existence of affine realizations.

💡 Research Summary

The paper develops a comprehensive real‑world probability framework for markets that simultaneously host several term structures, such as those found in foreign‑exchange, multi‑curve interest‑rate, credit, longevity‑bond, and energy‑forward markets. Starting from an abstract infinite‑dimensional market consisting of a risk‑free zero‑coupon bond curve (B_0(t,T)) and a family of risky curves ({B_i(t,T)}{i\in I}) together with spot processes ({S_i}{i\in I}), the authors model the discounted price processes (X^0_t{}^{-1}S_i(t)B_i(t,T)) as semimartingales.

Market viability is defined via the NUPBR (No Unbounded Profit with Bounded Risk) condition, appropriate for large markets with countably many assets. By extending the approach of large‑financial‑market theory, the authors prove a fundamental theorem of asset pricing: the market satisfies NUPBR if and only if there exists a strictly positive local martingale deflator (LMD). The LMD is explicitly characterized through a drift‑restriction that generalizes the classic HJM drift condition to the real‑world measure. In particular, the drift of each forward curve must be expressed as a linear functional of the market price of risk, which appears as the integrand of the LMD’s stochastic exponential.

The core of the work is the derivation and analysis of the real‑world HJMM stochastic partial differential equation (SPDE) governing the evolution of the forward curves (f_i(t,x)=B_i(t,t+x)). The authors prove a new existence‑and‑uniqueness theorem for semilinear SPDEs driven by a Brownian motion and a Poisson random measure, assuming only locally Lipschitz and locally bounded coefficients (randomly depending on the current state). This substantially weakens the usual global Lipschitz or linear‑growth assumptions found in earlier literature (e.g., Filipović, Teichmann, and others). Consequently, a wide class of multi‑dimensional HJM models—including those with jump components—are shown to be well‑posed.

Beyond well‑posedness, the paper investigates invariance properties of the SPDE. If the initial forward curve lies in a particular functional subspace (e.g., a Filipović‑type Sobolev space), the solution remains in that subspace for all future times. This is crucial for preserving structural properties such as monotonicity across term structures. The authors provide simple sufficient conditions on the volatility and jump kernels that guarantee that ordered term structures (e.g., increasing foreign‑currency yields or decreasing longevity‑bond prices with age) stay ordered throughout the evolution.

A major contribution is the analysis of affine realizations. The authors identify conditions under which the infinite‑dimensional SPDE admits a finite‑dimensional affine representation, i.e., the dynamics can be captured by a finite set of state variables evolving according to an affine diffusion‑jump process. This result opens the door to tractable calibration, simulation, and risk‑management applications, as it bridges the gap between the mathematically rigorous infinite‑dimensional HJM theory and the practical need for low‑dimensional models.

The paper also includes a concrete “minimal market model” example that illustrates how the real‑world framework recovers the classical risk‑neutral HJM model when the LMD is set to one, while simultaneously allowing for models where a risk‑neutral measure does not exist. Additional examples demonstrate the framework’s flexibility in handling multi‑curve interest‑rate markets, cross‑currency settings, and energy forward contracts with varying delivery lengths.

In summary, the authors deliver a unified HJM semimartingale framework that (1) works under the real‑world probability without assuming a risk‑neutral measure, (2) provides a full characterization of market viability via local martingale deflators, (3) establishes robust existence‑and‑uniqueness results for the associated SPDE under minimal regularity, (4) ensures structural monotonicity and invariance, and (5) identifies when finite‑dimensional affine realizations are possible. The results significantly broaden the applicability of HJM modelling to realistic multi‑term‑structure markets and lay a solid theoretical foundation for future work on estimation, control, and numerical methods in this setting.