Convergence in probability of numerical solutions of a highly non-linear delayed stochastic interest rate model

TL;DR

This work analyzes a highly non-linear delayed stochastic interest-rate model with super-linear drift and diffusion, formulating it as $dx(t)=\alpha(\mu-x(t)^{\gamma})dt+\sigma x(t-\tau)^{r}x(t)^{\theta}dB(t)$ and proving well-posedness and positivity. It introduces a truncated EM scheme to cope with super-linear growth, proving that the true SDDE solution converges in probability to the truncated EM solution as the time step $\Delta$ vanishes, and providing finite-time error bounds. Theoretical results are complemented by numerical simulations, including comparisons with a backward EM method and Monte Carlo-based pricing of bonds and lookback options, demonstrating practical viability for financial applications. Overall, the paper delivers a rigorous numerical-analytic framework for SDDEs with delays and super-linear growth in a financial modeling context, with concrete convergence guarantees and pricing implications.

Abstract

We examine a delayed stochastic interest rate model with super-linearly growing coefficients and develop several new mathematical tools to establish the properties of its true and truncated EM solutions. Moreover, we show that the true solution converges to the truncated EM solutions in probability as the step size tends to zero. Further, we support the convergence result with some illustrative numerical examples and justify the convergence result for the Monte Carlo evaluation of some financial quantities.

Figures (3)

• Figure 1: Plot of the empirical distribution of the TEM and BEM solutions
• Figure 2: Convergence of the TEM and BEM solutions
• Figure 3: Errors between the TEM and BEM solutions

Theorems & Definitions (20)

• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Lemma 4.1
• proof