Interest rates (or nominal yields) can be negative, this is an unavoidable fact which has already been visible during the Great Depression (1929–39). Nowadays we can find negative rates easily by e.g. auditing. Several theoretical and practical ideas how to model and eventually overcome empirical negative rates can be suggested, however, they are far beyond a simple practical realization. In this paper we discuss the dynamical reasons why negative interest rates can happen in the second order differential dynamics and how they can influence the variance and expectation of the interest rate process. Such issues are highly practical, involving e.g. the banking sector and pension securities.
We are indebted to Felix Fuders for his valuable comments.
First author was partially supported by grant VEGA MŠ SR 1/0344/14. Second author gratefully acknowledges support of ANR project Desire FWF I 833-N18. Corresponding author acknowledges Fondecyt Proyecto Regular No 1151441. We acknowledge also project LIT-2016-1-SEE-023.
Last but not the least, the authors are very grateful to the editor and the reviewers for their valuable comments.
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Appendix A. Appendix
We write for matrix Z.
([21: Theorem 5.2.1] LetT > 0 andb(⋅, ⋅): [0, T] × ℝn → ℝn, Σ(⋅,⋅): [0, T] × ℝn→ ℝn×mbe measurable functions satisfying
(x, t) ∈ ℝn × [0, T] for some constant C, and such that
x, y ∈ ℝn, t ∈ [0, T] for some constant D. Let Z be a random variable which is independent of theσ-algebragenerated byWs(⋅), s ≥ 0 and such that
Then the stochastic differential equation
has a unique t-continuous solutionXt(ω) with the property thatXt(ω) is adapted to the filtrationgenerated by Z andWs(⋅); s ≤ tand
(Itôs lemma). LetXtbe a process given by
Letψ(Xt, t) ∈ C(ℝn × [0, ∞)), then for a transformationZt = [ψ1 (Xt, t), …, ψn (Xt, t)], Ztis again an Itô process given by
where dXj, tdXi, tis calculated according to the standard rules (dt)2 = dt dWi, t = 0, dWj, t dWi, t = 0, j ≠ i, dWi, t dWi, t = dt.
(Higher moments of Wiener process, see e.g. ). For standard Brownian motion (Wiener process) the following formula for higher moments holds:
Let z be bivariate normal with zero mean ands1, s2be nonnegative integers, then
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