Showing papers by "David Criens published in 2022"
16 Apr 2022
TL;DR: In this paper , a family of nonlinear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics is studied, where the differential characteristics are prescribed by a setvalued function that depends on time and path in a non-Markovian way.
Abstract: 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.
6 citations
30 May 2022
TL;DR: In this article , a family of nonlinear (conditional) expectations with uncertain local characteristics is studied, where the differential characteristics are prescribed by a set-valued function that depends on time and path in a Markovian way.
Abstract: . 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 that depends on time and path in a Markovian way. We establish its Feller properties and examine how to linearize the associated sublinear Markovian semigroup. In particular, we observe a novel smoothing effect of nonlinear semigroups in frameworks which carry enough randomness. Furthermore, we link the value function corresponding to the semigroup to a nonlinear Kolmogorov equation.
5 citations
TL;DR: In this paper , a family of non-linear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics is studied, where the local characteristics are prescribed by a set-valued function that depends on time and path in a non-Markovian way.
Abstract: . In this paper we study a family of non-linear (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 non-linear expectation and we link the corresponding value function to a variational form of a non-linear path-dependent partial differential equation. Furthermore, we prove that the non-linear expectation solves a non-linear martingale problem, which confirms our interpretation as a non-linear semimartingale.
1 citations
11 Nov 2022
TL;DR: In this paper , the authors derived deterministic criteria for the no-arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below.
Abstract: The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before their separating time and singular afterwards. In this paper, we investigate the separating time for two laws of general one-dimensional regular continuous strong Markov processes, so-called general diffusions, which are parameterized via scale functions and speed measures. Our main result is a representation of the corresponding separating time as (loosely speaking) a hitting time of a deterministic set which is characterized via speed and scale. As hitting times are fairly easy to understand, our result gives access to explicit and easy-to-check sufficient and necessary conditions for two laws of general diffusions to be (locally) absolutely continuous and/or singular. Most of the related literature treats the case of stochastic differential equations. In our setting we encounter several novel features, which are due to general speed and scale on the one hand, and to the fact that we do not exclude (instantaneous or sticky) reflection on the other hand. These new features are discussed in a variety of examples. As an application of our main theorem, we investigate the no arbitrage concept no free lunch with vanishing risk (NFLVR) for a single asset financial market whose (discounted) asset is modeled as a general diffusion which is bounded from below (e.g., non-negative). More precisely, we derive deterministic criteria for NFLVR and we identify the (unique) equivalent local martingale measure as the law of a certain general diffusion on natural scale.
1 citations
31 May 2022
TL;DR: In this article , a family of nonlinear expectations that can be un-derstood as a multidimensional diffusion with uncertain drift and certain volatility is considered, where the drift is prescribed by a set-valued function that depends on time and path in a Markovian way.
Abstract: . In this note we consider a family of nonlinear (conditional) expectations that can be un-derstood as a multidimensional diffusion with uncertain drift and certain volatility. Here, the drift is prescribed by a set-valued function that depends on time and path in a Markovian way. We establish the Feller property for the associated sublinear Markovian semigroup and we observe a smoothing effect as our framework carries enough randomness. Furthermore, we link the corresponding value function to a semilinear Kolmogorov equation.
1 citations
21 Sep 2022
TL;DR: In this paper , a general framework for stochastic partial differential equations with parameter uncertainty is proposed. But the framework is restricted to nonlinear expectations and does not cover nonlinear L\'evy processes.
Abstract: 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. In particular, we show that our methodology includes the important class of nonlinear L\'evy processes as introduced by Neufeld and Nutz, and we introduce the new classes of stochastic partial differential equations and spin systems with 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) \(USC_b\)--Feller property and a strong Markov selection principle. Finally, we investigate ergodicity for spin systems with uncertain parameters such as the nonlinear contact process.
TL;DR: In this article , a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition is given. But the PHI is not optimal.
Abstract: 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.
21 Sep 2022
TL;DR: In this paper , a general framework for stochastic partial differential equations with uncertain parameters is proposed. But the framework is restricted to nonlinear expectations and does not consider nonlinear contact processes.
Abstract: . 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. In particular, we show that our methodology includes the important class of nonlinear L´evy processes as introduced by Neufeld and Nutz, and we introduce the new classes of stochastic partial differential equations and spin systems with 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) USC b –Feller property and a strong Markov selection principle. Finally, we investigate ergodicity for spin systems with uncertain parameters such as the nonlinear contact process.
28 Jun 2022
TL;DR: In this article , the robust utility maximization problem is studied in continuous time under model uncertainty, where the model uncertainty is governed by a continuous semimartingale with uncertain local characteristics.
Abstract: 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 study the existence of optimal portfolios for logarithmic, exponential and power utilities.