arXiv subjects

Oscar Peralta

Publications and source records attributed to Oscar Peralta.

27 records · Page 2Linked to original sources

arXiv2022

Space-grid approximations of hybrid stochastic differential equations and first passage properties

Hybrid stochastic differential equations are a useful tool to model continuously varying stochastic systems which are modulated by a random environment that may depend on the system state itself. In this paper, we establish the pathwise convergence of the solutions to hybrid stochastic differential equations via space-grid discretizations. While time-grid discretizations are a classical approach for simulation purposes, our space-grid discretization provides a link with multi-regime Markov modulated Brownian motions, leading to computational tractability. We exploit our convergence result to obtain efficient approximations to first passage probabilities and expected occupation times of the solutions hybrid stochastic differential equations, results which are the first of their kind for such a robust framework. We finally illustrate the performance of the resulting approximations in numerical examples.

math.PR↗

arXiv2021

Multivariate matrix-exponential affine mixtures and their applications in risk theory

In this paper, a class of multivariate matrix-exponential affine mixtures with matrix-exponential marginals is proposed. The class is shown to possess various attractive properties such as closure under size-biased Esscher transform, order statistics, residual lifetime and higher order equilibrium distributions. This allows for explicit calculations of various actuarial quantities of interest. The results are applied in a wide range of actuarial problems including multivariate risk measures, aggregate loss, large claims reinsurance, weighted premium calculations and risk capital allocation. Furthermore, a multiplicative background risk model with dependent risks is considered and its capital allocation rules are provided as well. We finalize by discussing a calibration scheme based on complete data and potential avenues of research.

q-fin.RM↗

arXiv2021

Strong convergence to two-dimensional alternating Brownian motion processes

Flip-flop processes refer to a family of stochastic fluid processes which converge to either a standard Brownian motion (SBM) or to a Markov modulated Brownian motion (MMBM). In recent years, it has been shown that complex distributional aspects of the univariate SBM and MMBM can be studied through the limiting behaviour of flip-flop processes. Here, we construct two classes of bivariate flip-flop processes whose marginals converge strongly to SBMs and are dependent on each other, which we refer to as \emph{alternating} two-dimensional Brownian motion processes}. While the limiting bivariate processes are not Gaussian, they possess desirable qualities, such as being tractable and having a time-varying correlation coefficient function.

math.PR↗

arXiv2021

Wong--Zakai approximations with convergence rate for stochastic differential equations with regime switching

We construct Wong--Zakai approximations of time--inhomogeneous stochastic differential equations with regime switching (RSSDEs), and provide a convergence rate. %Given a family of finite-variation processes $\{\mathcal{F}^λ\}_{λ\ge 0}$ that converge strongly to a standard Brownian motion $\mathcal{B}$, we construct pathwise approximations for regime-switching, time-inhomogeneous stochastic differential equations in the Wong-Zakai sense. Moreover, we determine the rate of strong convergence to the solutions of such regime-switching SDEs, showing that this rate is almost as good as that of $\{\mathcal{F}^λ\}_{λ\ge 0}$ to $\mathcal{B}$. In the proposed approximations, the standard Brownian motion driving the time-inhomogeneous RSSDEs is replaced by a family of finite--variation processes $\{\mathcal{F}^λ\}_{λ> 0}$. We show that if $\mathcal{F}^λ$ strongly converges to $\mathcal{B}$ at rate $δ(λ)$, then the Wong--Zakai approximation strongly converges to the original solution of the time--inhomogeneous RSSDE at rate $δ(λ) λ^{\varepsilon}$, for any $\varepsilon > 0$. This is the first paper on Wong--Zakai approximations for time--inhomogeneous RSSDEs, and significantly extends the counterparts for time--homogeneous SDEs without regime switching in Römisch and Wakolbinger (1985).

math.PR↗

arXiv2021

A Markov jump process associated with the matrix-exponential distribution

Let $f$ be the density function associated to a matrix-exponential distribution of parameters $(α, T,s)$. By exponentially tilting $f$, we find a probabilistic interpretation which generalises the one associated to phase-type distributions. More specifically, we show that for any sufficiently large $λ\ge 0$, the function $x\mapsto \left(\int_0^\infty e^{-λr}f(r) dr\right)^{-1}e^{-λx}f(x)$ can be described in terms of a Markov jump process whose generator is tied to $T$. Finally, we show how to revert the exponential tilting in order to assign a probabilistic interpretation to $f$ itself.

math.PR↗

arXiv2021

RAP-modulated Fluid Processes: First Passages and the Stationary Distribution

We construct a stochastic fluid process with an underlying piecewise deterministic Markov process (PDMP) akin to the one used in the construction of the rational arrival process (RAP), which we call the RAP-modulated fluid process. As opposed to the classic stochastic fluid process driven by a Markov jump process, the underlying PDMP of a RAP-modulated fluid process has a continuous state space and is driven by matrix parameters which may not be related to an intensity matrix. Through novel techniques we show how well-known formulae associated to the classic stochastic fluid process, such as first passage probabilities and the stationary distribution of its queue, translate to its RAP-modulated counterpart.

math.PR↗

arXiv2020

An explicit solution to the Skorokhod embedding problem for double exponential increments

Strong approximations of uniform transport processes to the standard Brownian motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego et al. (1971)) and improves their rate of strong convergence (Gorostiza et al. (1980)). We finalise by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.

math.PR↗

arXiv2019

Rate of Strong Convergence to Markov-modulated Brownian motion

In Latouche and Nguyen (2015), the authors constructed a sequence of stochastic fluid processes and showed that it converges weakly to a Markov-modulated Brownian motion (MMBM). Here, we construct a different sequence of stochastic fluid processes and show that it converges strongly to an MMBM. To the best of our knowledge, this is the first result on strong convergence to a Markov-modulated Brownian motion. We also prove that the rate of this almost sure convergence is $o(n^{-1/2} \log n)$. When reduced to the special case of standard Brownian motion, our convergence rate is an improvement over that obtained by a different approximation in \cite{gorostiza1980rate}, which is $o(n^{-1/2}(\log n)^{5/2})$.

math.PR↗

arXiv2017

Approximation of Ruin Probabilities via Erlangized Scale Mixtures

In this paper, we extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér--Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a dense class of distributions that we denominate Erlangized scale mixtures (ESM) and correspond to nonnegative and absolutely continuous distributions which can be written as a Mellin--Stieltjes convolution $Π\star G$ of a nonnegative distribution $Π$ with an Erlang distribution $G$. A distinctive feature of such a class is that it contains heavy-tailed distributions. We suggest a simple methodology for constructing a sequence of distributions having the form $Π\star G$ to approximate the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.

math.PR↗