86 research outputs found
On Kolmogorov equations for anisotropic multivariate Lévy processes
For d-dimensional exponential Lévy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate Lévy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Singularity-free representations of the Dirichlet forms are given which remain bounded for piecewise polynomial, continuous functions of finite element type. We prove that the variational problem can be localized to a bounded domain with explicit localization error bounds. Furthermore, we collect several analytical tools for further numerical analysi
The ANOVA decomposition and generalized sparse grid methods for the high-dimensional backward Kolmogorov equation
In this thesis, we discuss numerical methods for the solution of the high-dimensional backward Kolmogorov equation, which arises in the pricing of options on multi-dimensional jump-diffusion processes. First, we apply the ANOVA decomposition and approximate the high-dimensional problem by a sum of lower-dimensional ones, which we then discretize by a θ-scheme and generalized sparse grids in time and space, respectively. We solve the resultant systems of linear equations by iterative methods, which requires both preconditioning and fast matrix-vector multiplication algorithms. We make use of a Linear Program and an algebraic formula to compute optimal diagonal scaling parameters. Furthermore, we employ the OptiCom as non-linear preconditioner. We generalize the unidirectional principle to non-local operators and develop a new matrix-vector multiplication algorithm for the OptiCom. As application we focus on the Kou model. Using a new recurrence formula, the computational complexity of the operator application remains linear in the number of degrees of freedom. The combination of the above-mentioned methods allows us to efficiently approximate the solution of the backward Kolmogorov equation for a ten-dimensional Kou model.Die ANOVA-Zerlegung und verallgemeinerte dünne Gitter für die hochdimensionale Kolmogorov-Rückwärtsgleichung In der vorliegenden Arbeit betrachten wir numerische Verfahren zur Lösung der hochdimensionalen Kolmogorov-Rückwärtsgleichung, die beispielsweise bei der Bewertung von Optionen auf mehrdimensionalen Sprung-Diffusionsprozessen auftritt. Zuerst wenden wir eine ANOVA-Zerlegung an und approximieren das hochdimensionale Problem mit einer Summe von niederdimensionalen Problemen, die wir mit einem θ-Verfahren in der Zeit und mit verallgemeinerten dünnen Gittern im Ort diskretisieren. Wir lösen die entstehenden linearen Gleichungssysteme mit iterativen Verfahren, wofür eine Vorkonditionierung als auch schnelle Matrix-Vektor-Multiplikationsalgorithmen nötig sind. Wir entwickeln ein Lineares Programm und eine algebraische Formel, um optimale Diagonalskalierungen zu finden. Des Weiteren setzen wir die OptiCom als nicht-lineares Vorkonditionierungsverfahren ein. Wir verallgemeinern das unidirektionale Prinzip auf nicht-lokale Operatoren und entwickeln einen für die OptiCom optimierten Matrix-Vektor-Multiplikationsalgorithmus. Als Anwendungsbeispiel betrachten wir das Kou-Modell. Mit einer neuen Rekurrenzformel bleibt die Gesamtkomplexität der Operatoranwendung linear in der Anzahl der Freiheitsgrade. Unter Einbeziehung aller genannten Methoden ist es nun möglich, die Lösung der Kolmogorov-Rückwärtsgleichung für ein zehndimensionales Kou-Modell effizient zu approximieren
Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models
We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process X that ensure ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to O(ε−1), and with constants implied in O(⋅) which grow polynomially in d, thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.
In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide ε expression error in DNN size O(|log(ε)|a) with exponent a proportional to d, but with constants implied in O(⋅) growing exponentially with respect to d. Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.ISSN:0949-2984ISSN:1432-112
Numerical methods for Lévy processes
We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric Lévy model
Wavelet solution of variable order pseudodifferential equations
Sobolev spaces H m(x)(I) of variable order 0<m(x)<1 on an interval I⊂ℝ arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X t∈ℝ with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary boundary conditions, we establish multilevel norm equivalences in H m(x)(I) and prove preconditioning and wavelet matrix compression results for the variable order pseudodifferential generators A of X. Sufficient conditions on A to satisfy a Gårding inequality in H m(x)(I) and time-analyticity of the semigroup T t associated with the Feller process X t are established. As application, wavelet-based algorithms of log-linear complexity are obtained for the valuation of contingent claims on pure jump Feller-Lévy processes X t with state-dependent jump intensity by numerical solution of the corresponding Kolmogoroff equation
First-passage properties of asymmetric Lévy flights
Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the 'jump lengths'—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times
Field theoretic formulation and empirical tracking of spatial processes
Spatial processes are attacked on two fronts. On the one hand, tools from theoretical and
statistical physics can be used to understand behaviour in complex, spatially-extended
multi-body systems. On the other hand, computer vision and statistical analysis can be
used to study 4D microscopy data to observe and understand real spatial processes in
vivo.
On the rst of these fronts, analytical models are developed for abstract processes, which
can be simulated on graphs and lattices before considering real-world applications in elds
such as biology, epidemiology or ecology. In the eld theoretic formulation of spatial processes,
techniques originating in quantum eld theory such as canonical quantisation and
the renormalization group are applied to reaction-di usion processes by analogy. These
techniques are combined in the study of critical phenomena or critical dynamics. At this
level, one is often interested in the scaling behaviour; how the correlation functions scale
for di erent dimensions in geometric space. This can lead to a better understanding of how
macroscopic patterns relate to microscopic interactions. In this vein, the trace of a branching
random walk on various graphs is studied. In the thesis, a distinctly abstract approach
is emphasised in order to support an algorithmic approach to parts of the formalism.
A model of self-organised criticality, the Abelian sandpile model, is also considered. By
exploiting a bijection between recurrent con gurations and spanning trees, an e cient
Monte Carlo algorithm is developed to simulate sandpile processes on large lattices.
On the second front, two case studies are considered; migratory patterns of leukaemia cells
and mitotic events in Arabidopsis roots. In the rst case, tools from statistical physics
are used to study the spatial dynamics of di erent leukaemia cell lineages before and after
a treatment. One key result is that we can discriminate between migratory patterns in
response to treatment, classifying cell motility in terms of sup/super/di usive regimes.
For the second case study, a novel algorithm is developed to processes a 4D light-sheet
microscopy dataset. The combination of transient uorescent markers and a poorly localised
specimen in the eld of view leads to a challenging tracking problem. A fuzzy
registration-tracking algorithm is developed to track mitotic events so as to understand
their spatiotemporal dynamics under normal conditions and after tissue damage.Open Acces
Finite Element Model of the Innovation Diffusion: An Application to Photovoltaic Systems
This paper presents a Finite Element Model, which has been used for forecasting the diffusion of innovations in time and space. Unlike conventional models used in diffusion literature, the model considers the spatial heterogeneity. The implementation steps of the model are explained by applying it to the case of diffusion of photovoltaic systems in a local region in southern Germany. The applied model is based on a parabolic partial differential equation that describes the diffusion ratio of photovoltaic systems in a given region over time. The results of the application show that the Finite Element Model constitutes a powerful tool to better understand the diffusion of an innovation as a simultaneous space-time process. For future research, model limitations and possible extensions are also discussed
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