arXiv subjects

Frank Proske

Publications and source records attributed to Frank Proske.

At least 37 records · Page 2Linked to original sources

arXiv2022

Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs in arbitrary moments

We deduce stability and pathwise uniqueness for a McKean-Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz drift coefficient and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients we provide unique strong solutions, even if the drift fails to be of affine growth. The theory that we develop rests on Itô's formula and leads to $p$-th moment and pathwise $α$-exponential stability for $p\geq 2$ and $α> 0$ with explicit Lyapunov exponents, regardless of whether a Lyapunov function exists.

math.PR↗

arXiv2022

On the Analysis of a Generalised Rough Ait-Sahalia Interest Rate Model

Fractional Brownian motion with the Hurst parameter $H<\frac{1}{2}$ is used widely, for instance, to describe a 'rough' stochastic volatility process in finance. In this paper, we examine an Ait-Sahalia-type interest rate model driven by a fractional Brownian motion with $H<\frac{1}{2}$ and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.

math.PR↗

arXiv2021

Sensitivity Analysis with respect to a Stock Price Model with Rough Volatility via a Bismut-Elworthy-Li Formula for Singular SDEs

In this paper, we show the existence of unique Malliavin differentiable solutions to SDE`s driven by a fractional Brownian motion with Hurst parameter H<1/2 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we also derive a Bismut-Elworthy-Li formula with respect to our stock price model for certain classes of vector fields.

math.PR↗

arXiv2021

Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths

In this article we prove path-by-path uniqueness in the sense of Davie \cite{Davie07} and Shaposhnikov \cite{Shaposhnikov16} for SDE's driven by a fractional Brownian motion with a Hurst parameter $H\in(0,\frac{1}{2})$, uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.\par Using this result, we construct weak unique regular solutions in $W_{loc}^{k,p}\left([0,1]\times\mathbb{R}^d\right)$, $p>d$ of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.\par The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons \cite{DiPernaLions89}, Ambrosio \cite{Ambrosio04} or Crippa-De Lellis \cite{CrippaDeLellis08}.\par Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from \cite{DMN92} as well as supremum concentration inequalities. \emph{keywords}: Transport equation, Compactness criterion, Singular vector fields, Regularization by noise.

math.AP↗

arXiv2021

Optimal stopping, randomized stopping and singular control with partial information flow

The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available information flow is a filtration with no a priori assumed relation to the filtration of the process. We call these problems optimal stopping and randomized stopping with general information. Following an idea of Krylov [K] we introduce a special singular stochastic control problem with general information and show that this is also equivalent to the partial information optimal stopping and randomized stopping problems. Then we show that the solution of this singular control problem can be expressed in terms of partial information variational inequalities.

math.OC↗

arXiv2021

Thiele's Differential Equation Based on Markov Jump Processes with Non-countable State Space

In modern life insurance, Markov processes in continuous time on a finite or at least countable state space have been over the years an important tool for the modelling of the states of an insured. Motivated by applications in disability insurance, we propose in this paper a model for insurance states based on Markov jump processes with more general state spaces. We use this model to derive a new type of Thiele's differential equation which e.g. allows for a consistent calculation of reserves in disability insurance based on two-parameter continuous time rehabilitation rates.

q-fin.RM↗

arXiv2019

Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin and A. Y. Veretennikov on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

math.PR↗

arXiv2018

Singular control and optimal stopping of memory mean-field processes

The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory, (ii) reflected advanced mean-field backward stochastic differential equations, and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: - We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations (AMBSDEs), - we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information, and - we deduce a relation between the optimal singular control of a MMSDE, and the optimal stopping of such processes.

math.OC↗

arXiv2018

Girsanov Theorem for Multifractional Brownian Processes

In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a regularity function. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the last 20 years. We develop a multifractional derivative operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional derivative operator, we are able to analyze a variety of new problems, both in the field of stochastic analysis and in fractional and functional analysis, ranging from regularization properties of noise to solutions to multifractional differential equations. In this paper, we will focus on application of the derivative operator to the construction of strong solutions to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of (0,1/2). The application of multifractional calculus to SDE's is based on a generalization of the works of D. Nualart and Y. Ouknine from 2002.

math.PR↗

arXiv2018

Strong Solutions of Mean-Field Stochastic Differential Equations with irregular drift

We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.

math.PR↗

arXiv2018

C-infinity-regularization by Noise of Singular ODE's

In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds to the Nash-Moser principle combined with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

math.FA↗

arXiv2018

Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift

In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. Furthermore, we prove the rather surprising result of the higher order Frechet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

math.PR↗

arXiv2018

A Bismut-Elworthy-Li Formula for Singular SDE's Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling

In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H<1/2$. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed "local time variational calculus".

math.PR↗

arXiv2018

Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

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

Sensitivity analysis in a market with memory

A general market model with memory is considered in terms of stochastic functional differential equations. We aim at representation formulae for the sensitivity analysis of the dependence of option prices on the memory. This implies a generalization of the concept of delta.

math.PR↗

arXiv2017

Stochastic functional differential equations and sensitivity to their initial path

We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of {\it randomisation of the initial condition}.

math.PR↗

arXiv2015

Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada-Watanabe principle

In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart for square integrable Brownian functionals to construct unique strong solutions of SDE's under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut-Elworthy-Li formula for solutions of the Kolmogorov equation.

math.PR↗