arXiv subjects

Salvador Ortiz-Latorre

Publications and source records attributed to Salvador Ortiz-Latorre.

At least 19 recordsLinked to original sources

arXiv2026

Signature McKean-Vlasov stochastic differential equations

McKean-Vlasov-type stochastic differential equations (SDEs) are characterized by coefficients depending on both the state and the law of the solution. In this work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric $p$-rough path lift of its solution, with $p\in(2,3)$. After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean-Vlasov SDEs. Finally, we consider the associated particle system and propagation of chaos is established.

math.PR↗

arXiv2025

Convergence in law for quasi-linear SPDEs

We consider the quasi-linear stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $μ_n$. We allow the Fourier transform of $μ_n$ to be a genuine distribution. Let $u^n$ be the mild solution to these equations. We provide sufficient conditions on the measures $μ_n$ and the initial data to ensure that $u^n$ converges in law, in the space of continuous functions, to the solution of our equations driven by a noise with spectral measure $μ$, where $μ_n\toμ$ in some sense. We apply our main result to various types of noises, such as the anisotropic fractional noise. We also show that we cover existing results in the literature, such as the case of Riesz kernels and the fractional noise with $d=1$.

math.PR↗

arXiv2024

A functional variational approach to pricing path dependent insurance policies

The main purpose of this work is the derivation of a functional partial differential equation (FPDE) for the calculations of equity-linked insurance policies, where the payment stream may depend on the whole past history of the financial asset. To this end, we employ variational techniques from the theory of functional Itô calculus.

q-fin.PR↗

arXiv2024

Thiele's PIDE for unit-linked policies in the Heston-Hawkes stochastic volatility model

The main purpose of the paper is to derive Thiele's differential equation for unit-linked policies in the Heston-Hawkes stochastic volatility model introduced in arXiv:2210.15343. This model is an extension of the well-known Heston model that incorporates the volatility clustering feature by adding a compound Hawkes process in the volatility. Since the model is arbitrage-free, pricing unit-linked policies via the equivalence principle under a risk neutral probability measure is possible. Studying the moments of the variance and certain stochastic exponentials, a suitable family of risk neutral probability measures is found. The established and practical method to compute reserves in life insurance is by solving Thiele's equation, which is crucial to guarantee the solvency of the insurance company.

q-fin.PR↗

arXiv2023

SPDE bridges with observation noise and their spatial approximation

This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.

math.NA↗

arXiv2022

Change of measure in a Heston-Hawkes stochastic volatility model

We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed.

math.PR↗

arXiv2022

An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations

The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors. This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter.

math.PR↗

arXiv2021

Particle representation for the solution of the filtering problem. Application to the error expansion of filtering discretizations

We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical consequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution.

math.PR↗

arXiv2021

Pathwise approximations for the solution of the non-linear filtering problem

We consider high order approximations of the solution of the stochastic filtering problem, derive their pathwise representation in the spirit of the earlier work of Clark and Davis and prove their robustness property. In particular, we show that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space. This property is important from the practical point of view as it is in fact the pathwise version of the filtering functional that is sought in numerical applications. Moreover, the pathwise viewpoint will be a stepping stone into the rigorous development of machine learning methods for the filtering problem. This work is a continuation of a recent work by two of the authors where a discretisation of the solution of the filtering problem of arbitrary order has been established. We expand the previous work by showing that robust approximations can be derived from the discretisations therein.

math.NA↗

arXiv2020

A decomposition formula for fractional Heston jump diffusion models

We present an option pricing formula for European options in a stochastic volatility model. In particular, the volatility process is defined using a fractional integral of a diffusion process and both the stock price and the volatility processes have jumps in order to capture the market effect known as leverage effect. We show how to compute a martingale representation for the volatility process. Finally, using Itô calculus for processes with discontinuous trajectories, we develop a first order approximation formula for option prices. There are two main advantages in the usage of such approximating formulas to traditional pricing methods. First, to improve computational effciency, and second, to have a deeper understanding of the option price changes in terms of changes in the model parameters.

q-fin.PR↗

arXiv2020

Variance and interest rate risk in unit-linked insurance policies

One of the risks derived from selling long term policies that any insurance company has, arises from interest rates. In this paper we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to obtain the risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedging strategy by completing the market. We conclude with a simulation experiment, where we price unit-linked policies using Norwegian mortality rates. In addition we compare prices for the classical Black-Scholes model against the Heston stochastic volatility model with a Vasicek interest rate model.

q-fin.PR↗

arXiv2019

Self-Exciting Multifractional Processes

We propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this through a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as give bounds on the p-order moments, for all p>=1. We show convergence of an Euler-Maruyama scheme for the process, and also give the rate of convergence, which is depending on the self-exciting dynamics of the process. Moreover, we discuss different applications of this process, and give examples of different functions to model self-exciting behavior.

math.PR↗

arXiv2018

Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise

In this paper we prove the existence of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<1/2. Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a "local time variational calculus" argument.

math.PR↗

arXiv2017

A high order time discretization of the solution of the non-linear filtering problem

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the $m$-power of the mesh of the partition for arbitrary $m\in\mathbb{N}$. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem.

math.PR↗

arXiv2014

A second order time discretization of the solution of the non-linear filtering problem

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a new time discretisation of these functionals corresponding to a chosen partition of the time interval and show that the convergence rate of discretisation is proportional with the square of the mesh of the partition.

math.PR↗

arXiv2014

A change of measure preserving the affine structure in the BNS model for commodity markets

For a commodity spot price dynamics given by an Ornstein-Uhlenbeck process with Barndorff-Nielsen and Shephard stochastic volatility, we price forwards using a class of pricing measures that simultaneously allow for change of level and speed in the mean reversion of both the price and the volatility. The risk premium is derived in the case of arithmetic and geometric spot price processes, and it is demonstrated that we can provide flexible shapes that is typically observed in energy markets. In particular, our pricing measure preserves the affine model structure and decomposes into a price and volatility risk premium, and in the geometric spot price model we need to resort to a detailed analysis of a system of Riccati equations, for which we show existence and uniqueness of solution and asymptotic properties that explains the possible risk premium profiles. Among the typical shapes, the risk premium allows for a stochastic change of sign, and can attain positive values in the short end of the forward market and negative in the long end.

q-fin.PR↗

arXiv2013

A pricing measure to explain the risk premium in power markets

In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump Lévy process and models the characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricing measure is given by the Esscher transform that preserves the probabilistic structure of the driving Lévy processes, while changing the levels of mean reversion. Using this choice one can generate stochastic risk premiums (in geometric spot models) but with (deterministically) changing sign. In this paper we introduce a pricing change of measure, which is an extension of the Esscher transform. With this new change of measure we also can slow down the speed of mean reversion and generate stochastic risk premiums with stochastic non constant sign, even in arithmetic spot models. In particular, we can generate risk profiles with positive values in the short end of the forward curve and negative values in the long end. Finally, our pricing measure allows us to have a stationary spot dynamics while still having randomly fluctuating forward prices for contracts far from maturity.

q-fin.PR↗

arXiv2012

Optimal simulation schemes for Lévy driven stochastic differential equations

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied between the jump times. The overall approximation is analyzed using a stochastic splitting argument. The resulting error bound involves separate contributions of the compound Poisson approximation and of the discretization scheme for the Brownian part, and allows, on one hand, to balance the two contributions in order to minimize the computational time, and on the other hand, to study the optimal design of the approximating compound Poisson process. For driving processes whose Lévy measure explodes near zero in a regularly varying way, this procedure allows to construct discretization schemes with arbitrary order of convergence.

math.PR↗