TY  - JOUR
A1  - Hellmuth, Kathrin
A1  - Klingenberg, Christian
T1  - Computing Black Scholes with uncertain volatility — a machine learning approach
JF  - Mathematics
N2  - In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.
KW  - numerical finance
KW  - Black Scholes equation
KW  - uncertainty quantification
KW  - uncertain volatility
KW  - polynomial chaos
KW  - Bi-Fidelity method
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-262280
SN  - 2227-7390
VL  - 10
IS  - 3
ER  - 
TY  - JOUR
A1  - Hellmuth, Kathrin
A1  - Klingenberg, Christian
A1  - Li, Qin
A1  - Tang, Min
T1  - Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting
JF  - Computation
N2  - Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism's movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.
KW  - inverse problems
KW  - Bayesian approach
KW  - kinetic chemotaxis equation
KW  - Keller–Segel model
KW  - multiscale modeling
KW  - asymptotic analysis
KW  - velocity jump process
KW  - mathematical biology
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-250216
SN  - 2079-3197
VL  - 9
IS  - 11
ER  -