arXiv subjects

David Criens

Publications and source records attributed to David Criens.

44 records · Page 3Linked to original sources

arXiv2018

Absolute Continuity of Semimartingales

We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov's theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Itô-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multi-dimensional Itô-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.

math.PR↗

arXiv2018

Cylindrical Martingale Problems Associated with Lévy Generators

We introduce and discuss Lévy-type cylindrical martingale problems on separable reflexive Banach spaces. Our main observations are the following: Cylindrical martingale problems have a one-to-one relation to weak solutions of stochastic partial differential equations. Moreover, well-posed problems possess the strong Markov property and a Cameron-Martin-Girsanov-type formula holds. As applications, we derive existence and uniqueness results.

math.PR↗

arXiv2017

Deterministic Criteria for the Absence and Existence of Arbitrage in Multi-Dimensional Diffusion Markets

We derive deterministic criteria for the existence and non-existence of equivalent (local) martingale measures for financial markets driven by multi-dimensional time-inhomogeneous diffusions. Our conditions can be used to construct financial markets in which the \emph{no unbounded profit with bounded risk} condition holds, while the classical \emph{no free lunch with vanishing risk} condition fails.

q-fin.MF↗

arXiv2017

Structure Preserving Equivalent Martingale Measures for $\mathscr{H}$-SII Models

In this article we relate the set of structure preserving equivalent martingale measures $(\mathcal{M})$ for financial models driven by semimartingales with conditionally independent increments to a set of measurable and integrable functions $(\mathscr{Y})$. More precisely, we prove that $(\mathcal{M}\not = \emptyset)$ if, and only if, $(\mathscr{Y}\not = \emptyset)$, and connect the sets $(\mathcal{M})$ and $(\mathscr{Y})$ to the semimartingale characteristics of the driving process. As examples we consider integrated Lévy models with independent stochastic factors and time-changed Lévy models and derive mild conditions for $(\mathcal{M} \not = \emptyset)$.

math.PR↗

arXiv2017

Monotone and Convex Stochastic Orders for Processes with Independent Increments

We study monotone and convex stochastic orders for processes with independent increments. Our contributions are twofold: First, we relate stochastic orders of the Poisson component to orders of their (generalized) Lévy measures. The relation is proven using an interpolation formula for infinitely divisible laws. Second, we derive explicit conditions on the characteristics of the processes. In this case, we prove the conditions via constructions of couplings.

math.PR↗

arXiv2017

A Note on Real-World and Risk-Neutral Dynamics for Heath-Jarrow-Morton Frameworks

As a consequence of the financial crises, risk management became more important and real-world dynamics of interest-rate models moved into the focus of interest. Since risk-neutral dynamics are classically important to compute prices of financial derivatives, it is interesting when real-world dynamics can be related to risk-neutral dynamics via an equivalent change of measures. In this article we give deterministic conditions in a general Heath-Jarrow-Morton framework driven by a Hilbert space valued Brownian motion and a Poisson random measure. Our conditions are of Lipschitz type and therefore easy to verify.

math.PR↗

arXiv2016

Martingale property of exponential semimartingales: a note on explicit conditions and applications to financial models

We give a collection of explicit sufficient conditions for the true martingale property of a wide class of exponentials of semimartingales. We express the conditions in terms of semimartingale characteristics. This turns out to be very convenient in financial modeling in general. Especially it allows us to carefully discuss the question of well-definedness of semimartingale Libor models, whose construction crucially relies on a sequence of measure changes.

q-fin.MF↗

arXiv2016

Martingale Property in Terms of Semimartingale Problems

Starting from the seventies mathematicians face the question whether a non-negative local martingale is a true or a strict local martingale. In this article we answer this question from a semimartingale perspective. We connect the martingale property to existence, uniqueness and topological properties of semimartingale problems. This not only leads to valuable characterizations of the martingale property, but also reveals new existence and uniqueness results for semimartingale problems. As a case study we derive explicit conditions for the martingale property of stochastic exponentials driven by infinite-dimensional Brownian motion.

math.PR↗