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Ludger Overbeck

Publications and source records attributed to Ludger Overbeck.

7 recordsLinked to original sources
arXiv2026

Path-dependent Affine Processes

We extend the classical theory of affine processes to a path-dependent setting by introducing path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform in terms of generalized Riccati-type equations. In the proposed framework, we define path-dependent affine processes through their exponential-affine Fourier--Laplace transform on the path space and establish a characterization theorem. Conversely, for path-dependent stochastic differential equations with affine path-dependent coefficients, we also provide explicit exponential-affine representations of the Fourier--Laplace functional in terms of those Riccati equations. Moreover, we derive a condition ensuring non-negativity of the path-dependent diffusion coefficient, guaranteeing well-posedness of the model. Finally, we apply these results to a path-dependent volatility model and a path-dependent extension of the Heston model, including a delayed Heston model as a special case.

math.PR↗
arXiv2022

Scalar systemic risk measures and Aumann-Shapley allocations

We study two different contributions to the theory of (scalar) systemic risk measures. Namely the first aggregate or axiomatic approach and the first inject capital approach. For this purpose we establish a general framework, which is rich enough to embed both approaches. It turns out that in most relevant situations systemic risk measures of the first inject capital approach have a representation in the more general axiomatic approach. Moreover, we study capital allocation rules (CARs). In both situations there exist canonical ways to answer the capital allocation problem. Additionally, a capital allocation rule (CAR) in the spirit of Aumann-Shapley is introduced, which gives us the opportunity to compute systemic capital allocations regardless of the risk measurement approach. This CAR also serves as an instrument to compare both approaches and to identify commonalities.

q-fin.MF↗
arXiv2020

Inhomogeneous affine Volterra processes

We extend recent results on affine Volterra processes to the inhomogeneous case. This includes moment bounds of solutions of Volterra equations driven by a Brownian motion with an inhomogeneous kernel $K(t,s)$ and inhomogeneous drift and diffusion coefficients $b(s,X_s)$ and $σ(s,X_s)$. In the case of affine $b$ and $σσ^T$ we show how the conditional Fourier-Laplace functional can be represented by a solution of an inhomogeneous Riccati-Volterra integral equation. For a kernel of convolution type $K(t,s)=\overline{K}(t-s)$ we establish existence of a solution to the stochastic inhomogeneous Volterra equation. If in addition $b$ and $σσ^T$ are affine, we prove that the conditional Fourier-Laplace functional is exponential-affine in the past path. Finally, we apply these results to an inhomogeneous extension of the rough Heston model used in mathematical finance.

math.PR↗
arXiv2020

A Functional Ito-Formula for Dawson-Watanabe Superprocesses

We derive an Ito-formula for the Dawson-Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Ito-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process $(X(t))_{t\in [0,T]}$ to an infinite-dimensional one - the space of finite measure. Secondly, we extend the formula to functions $F(t,X_t)$ depending on the entire paths $X_t=(X(s\wedge t))_{s \in [0,T]}$ up to times $t$. This later extension is usually called functional Ito-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses.

math.PR↗
arXiv2018

Option Pricing in a Regime Switching Stochastic Volatility Model

In the classical model of stock prices which is assumed to be Geometric Brownian motion, the drift and the volatility of the prices are held constant. However, in reality, the volatility does vary. In quantitative finance, the Heston model has been successfully used where the volatility is expressed as a stochastic differential equation. In addition, we consider a regime switching model where the stock volatility dynamics depends on an underlying process which is possibly a non-Markov pure jump process. Under this model assumption, we find the locally risk minimizing pricing of European type vanilla options. The price function is shown to satisfy a Heston type PDE.

q-fin.PR↗
arXiv2014

Suitability of Capital Allocations for Performance Measurement

Capital allocation principles are used in various contexts in which a risk capital or a cost of an aggregate position has to be allocated among its constituent parts. We study capital allocation principles in a performance measurement framework. We introduce the notation of suitability of allocations for performance measurement and show under different assumptions on the involved reward and risk measures that there exist suitable allocation methods. The existence of certain suitable allocation principles generally is given under rather strict assumptions on the underlying risk measure. Therefore we show, with a reformulated definition of suitability and in a slightly modified setting, that there is a known suitable allocation principle that does not require any properties of the underlying risk measure. Additionally we extend a previous characterization result from the literature from a mean-risk to a reward-risk setting. Formulations of this theory are also possible in a game theoretic setting.

q-fin.RM↗