scholarly journals PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black–Scholes Equation

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 105 ◽  
Author(s):  
Will Hicks
Keyword(s):  
Stochastic Processes ◽  
Diffusion Processes ◽  
Nonlocal Diffusion ◽  
Gaussian Kernel ◽  
Integral Formalism ◽  
Black Scholes ◽  
Symmetric Quantum ◽  
Quantum Stochastic Processes ◽  
Real Market ◽  
Non Gaussian

The Accardi–Boukas quantum Black–Scholes framework, provides a means by which one can apply the Hudson–Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers–Moyal expansion, and this provides useful tools to understand their behaviour. In this paper we develop further links between quantum stochastic processes, and nonlocal diffusions, by inverting the question, and showing how certain nonlocal diffusions can be written as quantum stochastic processes. We then go on to show how one can use path integral formalism, and PT symmetric quantum mechanics, to build a non-Gaussian kernel function for the Accardi–Boukas quantum Black–Scholes. Behaviours observed in the real market are a natural model output, rather than something that must be deliberately included.

scholarly journals All adapted topologies are equal

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1125-1172
Author(s):  
Julio Backhoff-Veraguas ◽  
Daniel Bartl ◽  
Mathias Beiglböck ◽  
Manu Eder
Keyword(s):  
Stochastic Processes ◽  
Weak Topology ◽  
Equilibrium Problems ◽  
Diffusion Processes ◽  
Temporal Structure ◽  
Wasserstein Distance ◽  
Optimal Stopping Problems ◽  
Reward Functions ◽  
Nested Distance ◽  
The Stability

Abstract A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

Quantification of Model-Form Uncertainty in Drift-Diffusion Simulation Using Fractional Derivatives

2015 ◽  
Author(s):  
Yan Wang
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Optimization Procedure ◽  
Physical World ◽  
Drift Diffusion ◽  
Long Range Correlations ◽  
Modeling Process ◽  
Model Form ◽  
Generic Description ◽  
Non Gaussian

In modeling and simulation, model-form uncertainty arises from the lack of knowledge and simplification during modeling process and numerical treatment for ease of computation. Traditional uncertainty quantification approaches are based on assumptions of stochasticity in real, reciprocal, or functional spaces to make them computationally tractable. This makes the prediction of important quantities of interest such as rare events difficult. In this paper, a new approach to capture model-form uncertainty is proposed. It is based on fractional calculus, and its flexibility allows us to model a family of non-Gaussian processes, which provides a more generic description of the physical world. A generalized fractional Fokker-Planck equation (fFPE) is proposed to describe the drift-diffusion processes under long-range correlations and memory effects. A new model calibration approach based on the maximum accumulative mutual information is also proposed to reduce model-form uncertainty, where an optimization procedure is taken.

Tolerant control for non-Gaussian stochastic processes with unknown faults via two-step fuzzy modeling

2018 ◽  
Vol 95 (4) ◽  
pp. 2703-2716 ◽  
Author(s):  
Yang Yi ◽  
Liren Shao ◽  
Xiangxiang Fan ◽  
Tianping Zhang
Keyword(s):  
Stochastic Processes ◽  
Fuzzy Modeling ◽  
Non Gaussian ◽  
Gaussian Stochastic Processes
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