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Tommi Sottinen

Publications and source records attributed to Tommi Sottinen.

At least 19 recordsLinked to original sources
arXiv2026

Walk-on-Cubes Monte Carlo Simulation for nonisotropic fractional Laplace, Helmholtz, and Yukawa equations

We study the nonisotropic fractional analogs of Laplace, Helmholtz and Yukawa equations. We provide a Duffin correspondence for the Yukawa equation and a Feynman--Kac reconstruction for the Helmholtz equation. The foundation of our analysis is the fact that the nonisotropic fractional Laplace equation is related to a symmetric $α$-stable Lévy process with independent identically distributed components. By using this relation we provide a Walk-on-Cubes algorithm that simulates the solutions of Helmholtz and Yukawa equations.

math.NA↗
arXiv2025

Data driven modeling of multiple interest rates with generalized Vasicek-type models

The Vasicek model is a commonly used interest rate model, and there exist many extensions and generalizations of it. However, most generalizations of the model are either univariate or assume the noise process to be Gaussian, or both. In this article, we study a generalized multivariate Vasicek model that allows simultaneous modeling of multiple interest rates while making minimal assumptions. In the model, we only assume that the noise process has stationary increments with a suitably decaying autocovariance structure. We provide estimators for the unknown parameters and prove their consistencies. We also derive limiting distributions for each estimator and provide theoretical examples. Furthermore, the model is tested empirically with both simulated data and real data.

econ.EM↗
arXiv2025

Tanaka formula for SDEs driven by fractional Brownian motion

We derive a Tanaka-type formula for the solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (fBm) with Hurst parameter $H > \frac{1}{2}$. While Tanaka formulas for the fractional Brownian motion itself have been established, a corresponding result for non-linear SDEs driven by fBm has so far been unavailable. Our formula reveals a structure not previously observed: it features both a Skorokhod integral and a Malliavin trace correction, where the analogue of the local time appears through a double integral involving the Dirac distribution and the Malliavin derivative of the solution. A second double integral captures the variation of the diffusion coefficient along the flow. A key step in our analysis is a novel method to establish $L^2$-convergence of the trace term, which avoids the use of white noise calculus and instead exploits Gaussian-type density estimates for the law of the solution. The result applies to a broad class of equations under suitable regularity assumptions and extends naturally to convex functionals. As special cases, we recover known identities for the fractional Brownian motion and the fractional Ornstein--Uhlenbeck process.

math.PR↗
arXiv2024

On optimal prediction of missing functional data with memory

This paper considers the problem of reconstructing missing parts of functions based on their observed segments. It provides, for Gaussian processes and arbitrary bijective transformations thereof, theoretical expressions for the $L^2$-optimal reconstruction of the missing parts. These functions are obtained as solutions of explicit integral equations. In the discrete case, approximations of the solutions provide consistent expressions of all missing values of the processes. Rates of convergence of these approximations, under extra assumptions on the transformation function, are provided. In the case of Gaussian processes with a parametric covariance structure, the estimation can be conducted separately for each function, and yields nonlinear solutions in presence of memory. Simulated examples show that the proposed reconstruction indeed fares better than the conventional interpolation methods in various situations.

math.ST↗
arXiv2024

Hedging in Jump Diffusion Model with Transaction Costs

We consider the jump-diffusion risky asset model and study its conditional prediction laws. Next, we explain the conditional least square hedging strategy and calculate its closed form for the jump-diffusion model, considering the Black-Scholes framework with interpretations related to investor priorities and transaction costs. We investigate the explicit form of this result for the particular case of the European call option under transaction costs and formulate recursive hedging strategies. Finally, we present a decision tree, table of values, and figures to support our results.

q-fin.MF↗
arXiv2024

On the existence and regularity of local times

We study the existence and regularity of local times for general $d$-dimensional stochastic processes. We give a general condition for their existence and regularity properties. To emphasize the contribution of our results, we show that they include various prominent examples, among others solutions to stochastic differential equations driven by fractional Brownian motion, where the behavior of the local time was not fully understood up to now and remained as an open problem in the stochastic analysis literature. In particular this completes the picture regarding the local time behavior of such equations, above all includes high dimensions and both large and small Hurst parameters. As other main examples, we also show that by using our general approach, one can quite easily cover and extend some recently obtained results on the local times of the Rosenblatt process and Gaussian quasi-helices.

math.PR↗
arXiv2024

Efficient simulation of mixed boundary value problems and conformal mappings

In this paper, we present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on-Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an L-shaped insulated polyhedron.

math.NA↗
arXiv2023

Transfer principle for fractional Ornstein-Uhlenbeck processes

We prove the transfer principle for fractional Ornstein-Uhlenbeck processes, i.e., we construct a Brownian motion that has the same filtration as the fractional Ornstein-Uhlenbeck process and then represent the fractional Ornstein-Uhlenbeck process by using the constructed Brownian motion. As applications of the transfer principle we consider prediction.

math.PR↗
arXiv2022

Multi-Mixed Fractional Brownian Motions and Orstein-Uhlenbeck Processes

We study the so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein--Ulhenbeck (mmfOU) processes. These processes are constructed by mixing by superimposing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein--Uhlenbeck processes (fOU), respectively. We prove their existence as $L^2$ processes and study their path properties, viz. long-range and short-range dependence, Hölder continuity, $p$-variation, and conditional full support.

math.PR↗
arXiv2022

On sharp rate of convergence for discretisation of integrals driven by fractional Brownian motions and related processes with discontinuous integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $H>\frac12$ with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $L^1$-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $n^{1-2H}$ that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

math.PR↗
arXiv2021

Long-range dependent completely correlated mixed fractional Brownian motion

In this paper we introduce the long-range dependent completely correlated mixed fractional Brownian motion (ccmfBm). This is a process that is driven by a mixture of Brownian motion (Bm) and a long-range dependent completely correlated fractional Brownian motion (fBm, ccfBm) that is constructed from the Brownian motion via the Molchan--Golosov representation. Thus, there is a single Bm driving the mixed process. In the short time-scales the ccmfBm behaves like the Bm (it has Brownian Hölder index and quadratic variation). However, in the long time-scales it behaves like the fBm (it has long-range dependence governed by the fBm's Hurst index). We provide a transfer principle for the ccmfBm and use it to construct the Cameron--Martin--Girsanov--Hitsuda theorem and prediction formulas. Finally, we illustrate the ccmfBm by simulations.

math.PR↗
arXiv2019

Decomposition formula for rough Volterra stochastic volatility models

The research presented in this article provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, even simple non-linear financial derivatives as vanilla European options are typically priced by means of Monte-Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility -- a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models with rough fractional volatility and we derive an explicit semi-closed approximation formula. Numerical properties of the approximation for a popular model -- the rBergomi model -- are studied and we propose a hybrid calibration scheme which combines the approximation formula alongside MC simulations. This scheme can significantly speed up the calibration to financial markets as illustrated on a set of AAPL options.

q-fin.PR↗
arXiv2019

Prediction Law of Mixed Gaussian Volterra Processes

We study the regular conditional law of mixed Gaussian Volterra processes under the influence of model disturbances. More precisely, we study prediction of Gaussian Volterra processes driven by a Brownian motion in a case where the Brownian motion is not observable, but only a noisy version is observed. As an application, we discuss how our result can be applied to variance reduction in the presence of measurement errors.

math.PR↗
arXiv2019

Integration-by-Parts Characterizations of Gaussian Processes

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.

math.PR↗
arXiv2018

Transfer Principle for nth Order Fractional Brownian Motion with Applications to Prediction and Equivalence in Law

The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the $n$the order fractional Brownian motion and also a representation formula for all the Gaussian processes that are equivalent in law to the $n$th order fractional Brownian motion.

math.PR↗