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.Comment: Forthcoming in "Stochastics

Limit Theorems for Cylindrical Martingale Problems associated with L\'evy Generators

We prove limit theorems for cylindrical martingale problems associated to L\'evy 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 L\'evy 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

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.Comment: The note has been changed in an applied directio

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

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) L\'evy 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

Structure Preserving Equivalent Martingale Measures for H\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 L\'evy models with independent stochastic factors and time-changed L\'evy models and derive mild conditions for $(\mathcal{M} \not = \emptyset)$.Comment: Forthcoming in the "Journal of Applied Probability

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

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.Comment: The article has been fully revised. To appear in "Mathematics and Financial Economics

Cylindrical Martingale Problems Associated with L\'evy Generators

We introduce and discuss L\'evy-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

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