arXiv subjects

Rafał Łochowski

Publications and source records attributed to Rafał Łochowski.

7 recordsLinked to original sources

arXiv2024

On the reducibility of affine models with dependent Lévy factor

The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t)) dt +\sum_{i=1}^{d}G(R(t-))dZ_i(t)$$ with deterministic functions $F,G_1,...,G_d$ and a multivariate Lévy process $Z=(Z_1,...,Z_d)$ with possibly dependent coordinates. The equation is supposed to have a nonnegative solution which generates an affine term structure model. The Lévy measure $ν$ of $Z$ is assumed to admit a spherical decomposition based on the representation $\mathbb{R}^d=S^{d-1}\times (0,+\infty)$, where $S^{d-1}$ stands for the unit sphere. Then $ν(dy)=λ(dξ)\times γ_ξ(dr)$, where $λ$ is a measure on $S^{d-1}$ and $γ_ξ$ on $(0,+\infty)$. Under some assumptions on spherical decomposition, a precise form of the generator of $R$ is determined and it is shown that the resulted term structure model is identical to that generated by the equation $$ d R(t)=(a R(t)+b) dt+C\cdot (R(t-))^{1/α} dZ^α(t), \quad R(0)=x, $$ with some constants $a,b,C$ and a one dimensional $α$-stable Lévy process $Z^α$, where $α\in(1,2)$. The case when $ν$ has a density is considered as a special case. The paper generalizes the classical results on the Cox-Ingersoll-Ross (CIR) model, \cite{CIR}, as well as on its extended version from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ is a one-dimensional Lévy process. It is the starting point for the classification in the spirit of \cite{DaiSingleton} and \cite{BarskiLochowski} for the affine models with dependent Lévy processes.

math.PR↗

arXiv2024

Local times of deterministic paths and self-similar processes with stationary increments as normalized numbers of interval crossings

We prove a general result on a relationship between a limit of normalized numbers of interval crossings by a càdlàg path and an occupation measure associated with this path. Using this result we define local times of fractional Brownian motions (classically defined as densities of relevant occupation measure) as weak limits of properly normalized numbers of interval crossings. We also discuss a similar result for càdlàg semimartingales, in particular for alpha-stable processes, and for Rosenblatt processes, and provide natural examples of deterministic paths which possess quadratic or higher order variation but no local times.

math.PR↗

arXiv2024

Affine term structure models driven by independent Lévy processes

We characterize affine term structure models of non-negative short rate $R$ which may be obtained as solutions of autonomous SDEs driven by independent, one-dimensional Lévy martingales, that is equations of the form $$ dR(r)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=r_0\geq 0, \quad t>0, \quad (1)$$ with deterministic real functions $F,G_1,...,G_d$ and independent one-dimensional Lévy martingales $Z_1,...,Z_d$. Using a general result on the form of the generators of affine term structure models due to Filipović, it is shown, under the assumption that the Laplace transforms of the driving noises are regularly varying, that all possible solutions $R$ of (1) may be obtained also as solutions of autonomous SDEs driven by independent stable processes with stability indices in the range $(1,2]$. The obtained models include in particular the $α$-CIR model, introduced by Jiao et al., which proved to be still simple yet more reliable than the classical CIR model. Results on heavy tails of $R$ and its limit distribution in terms of the stability indices are proven. Finally, results of numerical calibration of the obtained models to the market term structure of interest rates are presented and compared with the CIR and $α$-CIR models.

math.PR↗

arXiv2023

Level crossings of fractional Brownian motion

Since the classical work of Lévy, it is known that the local time of Brownian motion can be characterized through the limit of level crossings. While subsequent extensions of this characterization have primarily focused on Markovian or martingale settings, this work presents a highly anticipated extension to fractional Brownian motion -- a prominent non-Markovian and non-martingale process. Our result is viewed as a fractional analogue of Chacon et al. (1981). Consequently, it provides a global path-by-path construction of fractional Brownian local time. Due to the absence of conventional probabilistic tools in the fractional setting, our approach utilizes completely different argument with a flavor of the subadditive ergodic theorem, combined with the shifted stochastic sewing lemma recently obtained in Matsuda and Perkowski (22, arXiv:2206.01686). Furthermore, we prove an almost-sure convergence of the (1/H)-th variation of fractional Brownian motion with the Hurst parameter H, along random partitions defined by level crossings, called Lebesgue partitions. This result raises an interesting conjecture on the limit, which seems to capture non-Markovian nature of fractional Brownian motion.

math.PR↗

arXiv2023

Classification and calibration of affine models driven by independent Lévy processes

The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=x\geq 0, \quad t>0, $$ with deterministic functions $F,G_1,...,G_d$ and independent Lévy processes of infinite variation $Z_1,...,Z_d$ with regularly varying Laplace exponents. The equation is supposed to have a nonnegative solution which generates an affine term structure model. A precise form of the generator of $R$ is characterized and a related classification of equations which generate affine models introduced in the spirit of Dai and Singleton \cite{DaiSingleton}. Each class is shown to have its own canonical representation which is an equation with the same drift and the jump diffusion part based on a Lévy process taking values in $\mathbb{R}^{g}, 1\leq g\leq d$, with independent coordinates being stable processes with stability indices in the range $(1,2]$. Numerical calibration results of canonical representations to the market term structure of interest rates are presented and compared with the classical CIR model. The paper generalizes the classical results on the CIR model from \cite{CIR}, as well as on its extended version from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ was a one-dimensional Lévy process.

math.PR↗

arXiv2022

CIR equations with multivariate Lévy noise

The paper is devoted to the study of the short rate equation of the form $$ d R(t)=F(R(t)) dt+\sum_{i=1}^{d}G_i(R(t-)) dZ_i(t), \quad R(0)=x\geq 0,\quad t>0, $$ with deterministic functions $F,G_1,...,G_d$ and a multivariate Lévy process $Z=(Z_1,...,Z_d)$. The equation is supposed to have a nonnegative solution which generates an affine term structure model. Two classes of noise are considered. In the first one the coordinates of Z are independent processes with regularly varying Laplace exponents. In the second class Z is a spherical processes, which means that its Lévy measure has a similar structure as that of a stable process, but with radial part of a general form. For both classes a precise form of the short rate generator is characterized. Under mild assumptions it is shown that any equation of the considered type has the same solution as the equation driven by a Lévy process with independent stable coordinates. The paper generalizes the classical results on the Cox-Ingersoll-Ross (CIR) model as well as on its extended version where $Z$ is a one-dimensional Lévy process.

math.PR↗

arXiv2011

Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications

In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, $W_t = B_t + μt, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. Truncated variation differs from regular variation by neglecting jumps smaller than some fixed $c > 0$. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument. We also define two closely related quantities - upward truncated variation and downward truncated variation. The defined quantities may have some interpretation in financial mathematics. Exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process. We calculate the Laplace transform with respect to time parameter of the moment-generating functions of the upward and downward truncated variations. As an application of the obtained formula we give an exact formula for expected value of upward and downward truncated variations. We give also exact (up to universal constants) estimates of the expected values of the mentioned quantities.

math.PR↗