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David Criens

Publications and source records attributed to David Criens.

At least 37 records · Page 2Linked to original sources
arXiv2023

Nonlinear Semimartingales and Markov Processes with Jumps

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a time and path-dependent set-valued function. We show that the associated control problem coincides with both its weak and relaxed counterparts. Furthermore, we establish regularity properties of the value function and discuss their relation to Feller properties of nonlinear semigroups. In the Markovian case we provide conditions that allow us to identify the corresponding semigroup as the unique viscosity solution to a nonlinear Hamilton-Jacobi-Bellman equation. To illustrate our results we discuss a random $G$-double exponential Lévy setting.

math.PR↗
arXiv2023

Nonlinear Continuous Semimartingales

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued function that depends on time and path in a non-Markovian way. We provide a dynamic programming principle for the nonlinear expectation and we link the corresponding value function to a variational form of a nonlinear path-dependent partial differential equation. In particular, we establish conditions that allow us to identify the value function as the unique viscosity solution. Furthermore, we prove that the nonlinear expectation solves a nonlinear martingale problem, which confirms our interpretation as a nonlinear semimartingale.

math.PR↗
arXiv2023

Markov selections and Feller properties of nonlinear diffusions

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a diffusion with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued function. We establish its Feller properties and examine how to linearize the associated sublinear Markovian semigroup. In particular, we observe a novel smoothing effect of sublinear semigroups in frameworks which carry enough randomness. Furthermore, we link the value function corresponding to the semigroup to a nonlinear Kolmogorov equation. This provides a connection to the so-called Nisio semigroup.

math.PR↗
arXiv2023

Robust utility maximization with nonlinear continuous semimartingales

In this paper we study a robust utility maximization problem in continuous time under model uncertainty. The model uncertainty is governed by a continuous semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued function that depends on time and path. We show that the robust utility maximization problem is in duality with a conjugate problem, and we study the existence of optimal portfolios for logarithmic, exponential and power utilities.

q-fin.MF↗
arXiv2023

Stochastic Processes under Parameter Uncertainty

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a stochastic process with uncertain parameters. We develop a general framework which can be seen as a version of the martingale problem method of Stroock and Varadhan with parameter uncertainty. To illustrate our methodology, we explain how it can be used to model nonlinear Lévy processes in the sense of Neufeld and Nutz, and we introduce the new class of stochastic partial differential equations under parameter uncertainty. Moreover, we study properties of the nonlinear expectations. We prove the dynamic programming principle, i.e., the tower property, and we establish conditions for the (strong) $\textit{USC}_b$-Feller property and a strong Markov selection principle.

math.PR↗
arXiv2022

Propagation of Chaos for Weakly Interacting Mild Solutions to Stochastic Partial Differential Equations

This article investigates the propagation of chaos property for weakly interacting mild solutions to semilinear stochastic partial differential equations whose coefficients might not satisfy Lipschitz conditions. Furthermore, we derive continuity and linear growth conditions for the existence and uniqueness of mild solutions to SPDEs with distribution dependent coefficients, so-called McKean-Vlasov SPDEs.

math.PR↗
arXiv2021

On the relation of one-dimensional diffusions on natural scale and their speed measures

It is well-known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. C. Stone proved a continuous dependence of diffusions on their speed measures. In this paper we establish the converse direction, i.e. we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures.

math.PR↗
arXiv2021

On the Feller-Dynkin and the Martingale Property of One-Dimensional Diffusions

We show that a one-dimensional regular continuous Markov process \(\X\) with scale function \(s\) is a Feller--Dynkin process precisely if the space transformed process \(s (X)\) is a martingale when stopped at the boundaries of its state space. As a consequence, the Feller--Dynkin and the martingale property are equivalent for regular diffusions on natural scale with open state space. By means of a counterexample, we also show that this equivalence fails for multi-dimensional diffusions. Moreover, for Itô diffusions we discuss relations to Cauchy problems.

math.PR↗
arXiv2021

The Martingale Problem Method Revisited

We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales), nor to continuous or cadlag paths. We illustrate our findings both, by finding generalizations of known results, and proving new results. For the latter, we work on processes with fixed times of discontinuity.

math.PR↗
arXiv2021

A parabolic Harnack principle for balanced difference equations in random environments

We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.

math.PR↗
arXiv2021

A Dual Yamada-Watanabe Theorem for Levy driven stochastic differential equations

We prove a dual Yamada-Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Levy processes. More precisely, we prove that weak uniqueness, i.e. uniqueness in law, implies weak joint uniqueness, i.e. joint uniqueness in law for the solution process and its driver.

math.PR↗
arXiv2021

On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

We consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by cylindrical Brownian motion. The solutions are allowed to take values in general separable Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny's and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada-Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

math.PR↗
arXiv2020

On Absolute Continuity and Singularity of Multidimensional Diffusions

Consider two laws \(P\) and \(Q\) of multidimensional possibly explosive diffusions with common diffusion coefficient \(\mathfrak{a}\) and drift coefficients \(\mathfrak{b}\) and \(\mathfrak{b} + \mathfrak{a} \mathfrak{c}\), respectively, and the law \(P^\circ\) of an auxiliary diffusion with diffusion coefficient \(\langle \mathfrak{c},\mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{a}\) and drift coefficient \(\langle \mathfrak{c}, \mathfrak{a}\mathfrak{c}\rangle^{-1}\mathfrak{b}\). We show that \(P \ll Q\) if and only if the auxiliary diffusion \(P^\circ\) explodes almost surely and that \(P\perp Q\) if and only if the auxiliary diffusion \(P^\circ\) almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.

math.PR↗
arXiv2020

No Arbitrage in Continuous Financial Markets

We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Ito process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We also show that for Markov switching models the minimal martingale measure preserves the independence of the noise and we study how the minimal martingale measure can be modified to change the structure of the switching mechanism. Our main mathematical tools are new criteria for the martingale and strict local martingale property of certain stochastic exponentials.

q-fin.MF↗
arXiv2019

Limit Theorems for Cylindrical Martingale Problems associated with Lévy Generators

We prove limit theorems for cylindrical martingale problems associated to Lévy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients. We discuss two applications. First, we derive continuity and linear growth conditions for the existence of weak solutions to infinite-dimensional stochastic differential equations driven by Lévy noise. Second, we derive continuity, local boundedness and linear growth conditions for limit theorems and the Feller property of weak solutions to stochastic partial differential equations driven by Wiener noise.

math.PR↗
arXiv2019

On the Existence of Semimartingales with Continuous Characteristics

We prove the existence of quasi-left continuous semimartingales with continuous local semimartingale characteristics which satisfy a Lyapunov-type or a linear growth condition, where latter takes the whole history of the paths into consideration. The proof is based on an approximation and a tightness argument and the martingale problem method.

math.PR↗
arXiv2019

Lyapunov Criteria for the Feller-Dynkin Property of Martingale Problems

We give necessary and sufficient criteria for the Feller-Dynkin property of solutions to martingale problems in terms of Lyapunov functions. Moreover, we derive a Khasminskii-type integral test for the Feller-Dynkin property of multidimensional diffusions with random switching. For one dimensional switching diffusions with state-independent switching, we provide an integral-test for the Feller-Dynkin property.

math.PR↗