Effects of water currents on fish migration through a Feynman-type path integral approach under $$\sqrt{8/3}$$ Liouville-like quantum gravity surfaces

Abstract

A stochastic differential game theoretic model has been proposed to determine optimal behavior of a fish while migrating against water currents both in rivers and oceans. Then, a dynamic objective function is maximized subject to two stochastic dynamics, one represents its location and another its relative velocity against water currents. In relative velocity stochastic dynamics, a Cucker-Smale type stochastic differential equation is introduced under white noise. As the information regarding hydrodynamic environment is incomplete and imperfect, a Feynman type path integral under √ 8∕3 Liouville-like quantum gravity surface has been introduced to obtain a Wick-rotated Schrödinger type equation to determine an optimal strategy of a fish during its migration. The advantage of having Feynman type path integral is that, it can be used in more generalized nonlinear stochastic differential equations where constructing a Hamiltonian-Jacobi-Bellman (HJB) equation is impossible. The mathematical analytic results show exact expression of an optimal strategy of a fish under imperfect information and uncertainty.

Scoring a Goal optimally in a Soccer game under Liouville-like quantum gravity action

“…This surface is continuous but not differentiable. Furthermore, at 8/3 this behaves like a Brownian surface Sheffield, 2015, 2016;Hua, Polansky and Pramanik, 2019;Polansky, 2019, 2020b;Polansky and Pramanik, 2021;Pramanik and Polansky, 2021;Pramanik, 2021d). Furthermore, this approach can be used to obtain a solution for stability of an economy after pandemic crisis (Ahamed, 2021a), determine an optimal bank profitability (Islam, Alam and Chowdhury;Alam, Sultan and Afrin;Mohammad and Mohammad, 2010;Alam, Khondker and Molla, 2013;Hossain and Ahamed, 2015;Pramanik, 2016;ALAM and HOSSAIN, 2018;Ahamed, 2021b;Alam, 2021a,b).…”

Path integral control of a stochastic multi-risk SIR pandemic model

“…This path integral control solves a class a stochastic control problems with a Monte Carlo method for a HJB equation and this approach avoids the need of a global grid of the domain of HJB equation (Yang et al 2014a). If the objective function is quadratic and the differential equations are linear, then solution is given in terms of a number of Riccati equations which can be solved efficiently (Kappen 2007a;Pramanik and Polansky 2020a;Pramanik 2021a;Pramanik and Polansky 2021a). Although incorporate randomness with its HJB equation is straight forward but difficulties come due to dimensionality when a numerical solution is calculated for both of deterministic or stochastic HJB (Kappen 2007a).…”

Stochastic Control of a SIR Model with Non-linear Incidence Rate Through Euclidean Path Integral

“…Path integral control solves a class of stochastic control problems with a Monte Carlo method (Uddin et al, 2020;Islam, Alam and Afzal, 2021;Alam, 2021Alam, , 2022 for an HJB equation and this approach avoids the need for a global grid of the domain of the HJB equation (Yang et al, 2014). If the objective function is quadratic and the differential equations are linear, then the solution is given in terms of several Ricatti equations that can be solved efficiently (Kappen, 2007;Pramanik and Polansky, 2020b;Pramanik, 2021b;Pramanik and Polansky, 2021b).…”

“…2014/10/16 file: Covid_19.tex date: September 29, 2022 number of dimensions (Theodorou, Buchli and Schaal, 2010;Theodorou, 2011;Yang et al, 2014). Therefore, in order to determine the expected values, it is necessary to visit all states which leads to the inefficient summations of exponentially large sums (Kappen, 2007;Yang et al, 2014;Pramanik, 2021b). This is the main reason to implement a path integral control approach to deal with stochastic pandemic control.…”