Generalized Multivariate Vasicek Model

• The generalized multivariate Vasicek model is an extension of the classic Vasicek process that enables simultaneous dynamic modeling of multiple interest rates using flexible, non-Gaussian noise processes.
• Its formulation relies on minimal structural assumptions and robust estimation techniques that deliver consistent parameter estimates even under long-memory, heavy-tailed, or jump dynamics.
• Empirical studies and simulations validate its practical efficacy in calibrating interest rate systems, improving risk forecasting, and capturing inter-rate dependencies.

The generalized multivariate Vasicek model is an extension of the classic Vasicek interest rate model that allows for the simultaneous dynamic modeling of multiple interest rates driven by arbitrary stationary-increment noise processes with weak moment assumptions. Unlike traditional approaches that are typically univariate or restricted to Gaussian noise (e.g., Brownian motion or fractional Brownian motion), this framework—formally introduced in "Data driven modeling of multiple interest rates with generalized Vasicek-type models" (Ilmonen et al., 3 Sep 2025)—permits multidimensional dependence structures and application to real data with non-Gaussian, long-memory, or jump phenomena.

1. Formal Specification

The multivariate Vasicek model is defined via the stochastic differential equation: $dr_{t} = \Theta(b - r_{t})dt + \sigma\,dX_{t}$ where:

• $r_{t} \in \mathbb{R}^{d}$: vector of $d$ different interest rates at time $t$
• $b \in \mathbb{R}^{d}$: long-term mean vector
• $\Theta \in \mathbb{R}^{d \times d}$: positive definite mean-reversion matrix, encoding individual and cross-component reversion speeds
• $\sigma \in \mathbb{R}^{d \times d}$: positive definite diagonal volatility (or scale) matrix
• $X_{t}$: $d$-dimensional process with stationary increments, finite fourth moments, and suitably decaying autocovariance

This formulation generalizes the Vasicek process and accommodates wide classes of noise, including fractional Brownian motion, Hermite processes, Lévy processes, and other stationary-increment, non-Gaussian sources. By suitable transformations, the noise components may be assumed uncorrelated.

2. Structural and Distributional Assumptions

The model operates under only minimal conditions:

• The process $X_{t}$ must have stationary increments and finite fourth moments.
• The autocovariance $\gamma(t)$ of the stationary process

  $$U_{t} := e^{-\Theta t} \int_{-\infty}^{t} e^{\Theta s} dX_{s}$$

must decay as $\| \gamma(t) \| \rightarrow 0$ for $t \to \infty$, insuring that the time average is sharply concentrated around $b$.

• The known covariance structure $V(t)$ is required for invertibility and correct scaling of volatility estimates.

No Markov or Gaussian assumptions are imposed; this model is robust to persistent long-memory or heavy tail features, and can be specialized for ergodic or non-ergodic settings.

3. Estimation Methodology

Parameter estimation proceeds using continuous (or very finely sampled discrete) observations over $[0, T]$. The estimators are:

a) Mean Vector $b$

$\widehat{b}_T = \frac{1}{T} \int_0^T r_s\, ds$

Consistency is achieved via the negligible stationary error term $U_s$, which vanishes as $O_p(1/T)$.

b) Covariance Function $\gamma(s)$

$$\widehat{\gamma}_{r,T}(s) = \frac{1}{T} \int_0^T r_{s+v} r_v^\top\, dv - \left[ \frac{1}{T} \int_0^T r_v\,dv \right] \left[ \frac{1}{T} \int_0^T r_v\,dv \right]^\top$$

Uniform consistency over any bounded lag interval is obtained under the assumed autocovariance decay.

c) Noise Scale $\sigma$

Using high-frequency data in a fixed interval, the estimator is: $$\widehat{\sigma} \widehat{\sigma}^\top = V(1/N)^{-1}\, \frac{1}{N} \sum_{k=0}^{N-1}(r_{t_{k+1}} - r_{t_k})(r_{t_{k+1}} - r_{t_k})^\top$$ Appropriate behavior of $V(1/N)$ as $N \to \infty$ insures convergence.

d) Mean-Reversion Matrix $\Theta$

Estimated via solution of a continuous-time algebraic Riccati equation (CARE): $$B_t^\top\,\Theta + \Theta\,B_t - \Theta\,C_t\,\Theta + D_t = 0$$ where the matrices $B_t,\,C_t,\,D_t$ are defined by sample covariances $\gamma$, volatility estimates $\sigma$, and the sample estimate for $U_t$.

All estimators are proved consistent, i.e., $\widehat{b}_T \rightarrow b$, $\widehat{\sigma} \rightarrow \sigma$, $\widehat{\Theta} \rightarrow \Theta$ in probability as $T$, $N$ increase.

4. Limiting Distributions and Asymptotics

Central limit theorems (CLTs) are established for each estimator under general conditions:

• For the mean estimator,

$l_1(T)(\widehat{b}_T - b) \Longrightarrow Y_1$

• For the covariance estimator,

$$l_2(T)(\widehat{\gamma}_{U,T}(s) - \gamma(s)) \Longrightarrow Y_2(s)$$

• For the volatility estimator,

$$l_3(N(T))(\widehat{\sigma} \widehat{\sigma}^\top - \sigma \sigma^\top) \Longrightarrow Y_3$$

Rate functions $l_i(\cdot)$ depend on the memory structure of the noise. For Gaussian processes with autocovariance $\|\gamma(t)\| \sim t^{2H-2}$, the rates are:

• Mean: $l_1(T) = \sqrt T$ for $H < 0.5$, $\sqrt{T/\log T}$ for $H=0.5$, $T^{1-H}$ for $H>0.5$.

These intermediate limiting distributions are then linearly combined and further mapped (using the delta method) to yield the asymptotic behavior of the CARE solution for $\Theta$, resulting in

$$l_\Theta(T)\, \mathrm{vec}(\widehat{\Theta} - \Theta) \Longrightarrow L_2(L_1(\mathrm{vec}(Y_1 Y_1^\top, Y_2, Y_3)))$$

5. Model Flexibility and Empirical Performance

Simulations

• Diagonal and non-diagonal mean-reversion cases, with fractional Brownian noise ($H = 0.35, 0.5, 0.6, 0.8$), demonstrate proper convergence rates and broader error distributions for larger $H$.
• For non-diagonal $\Theta$ and standard Brownian motion $X_t$, empirical histograms of estimation errors are nearly Gaussian, and the overall error's Frobenius norm is well-described by a weighted chi-squared distribution.

Applications to Real Data

• Using bivariate data (1-month Euribor and US Federal Funds Rate), the model yields interpretable parameter estimates:
  • Diagonal elements of $\Theta$ reflect the force of mean reversion.
  • Off-diagonal elements capture inter-rate dependence, with negative off-diagonal values reflecting joint movement.
• Predictive accuracy for one-step ahead forecasts is strong, illustrating practical usability.

6. Theoretical Examples and Extensions

• For Gaussian noise (e.g., Brownian motion or fBm), explicit convergence rates for all estimators are calculated based on the Hurst index.
• For non-Gaussian noise (e.g., Hermite, Lévy), weak moment and decay conditions still yield consistency; rates may differ and require individualized calculation.
• In a generalized setting with Lévy noise, increment independence yields standard $\sqrt T$ rates for volatility estimation.

7. Mathematical Structure and Core Formulas

Key formulas:

Concept	Definition	Key Formula
Model SDE	Multivariate Vasicek	$dr_{t} = \Theta(b - r_{t})dt + \sigma dX_{t}$
Stationary Process	$U_{t}$	$$U_{t} = e^{-\Theta t} \int_{-\infty}^t e^{\Theta s} dX_{s}$$
Covariance	$\gamma(t)$	Autocovariance of $U_t$
CARE Estimation	Riccati Solution	$$B_t^\top \Theta + \Theta B_t - \Theta C_t \Theta + D_t = 0$$
Mean Estimator	$\widehat{b}_T$	$(1/T) \int_0^T r_s ds$

8. Significance and Outlook

The generalized multivariate Vasicek framework (Ilmonen et al., 3 Sep 2025) provides a robust, data-driven approach to modeling the joint evolution of multiple interest rates under minimal structural assumptions on the driving noise. Its flexible parameter estimation and characterization of limiting distributions enable both theoretical rigor and empirical applicability in contexts with persistent memory, heavy tails, or jumps—extending well beyond the limitations of classical Gaussian, Markovian, or univariate models. The approach is broadly compatible with practical calibration, risk management, and forecasting in multi-rate environments across finance and economics.