Optimization of Market Stochastic Dynamics
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Optimization of Market Stochastic Dynamics Paramahansa Pramanik1 Received: 16 August 2020 / Accepted: 20 September 2020 / © Springer Nature Switzerland AG 2020
Abstract A Feynman-type path integral has been introduced to find an optimal strategy where a dynamic profit is maximized subject to a stochastic dynamics of a firm’s market share. This method is useful under a more generalized non-linear system such as the Merton-Garman-Hamiltonian process where constructing a Hamiltonian-JacobiBellman equation is very difficult. The path integral method also gives an optimal strategy without going through a value function and gives a different optimal strategy. The result obtained by a Feynman-type method is compared with that by the traditional Pontryagin’s maximum principle. Keywords Stochastic control · Feynman-type path integral
1 Introduction Non-linear stochastic optimal control theory has become popular in recent days as dynamical systems are non-linear and stochastic in nature. Following [8], we know that posterior inference of certain diffusion processes can be mapped onto a stochastic optimal control problem which is later termed as the path integral control problem [10, 18, 20]. Kappen (2005) shows the application of path integral control under reinforcement learning and foraging for food where animals are considered as automatons as they know how to breathe and digest by birth [10]. In [21], the path integral method has been extended to multi-agent dynamics [19]. In all of these papers, optimal controls have been obtained by Hamiltonian-Jacobi-Bellman (HJB) equations and solutions to the state variables are obtained by using the FeynmanKac method [9]. In this paper, Itˆo’s lemma has been used to find a Wick rotated Schr¨odinger equation and then Fourier transformation has been used to find its solution. Paramahansa Pramanik
[email protected] 1
Department of Mathematical Sciences, Northern Illinois University, 1425 Lincoln Highway, DeKalb, IL, USA
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SN Operations Research Forum
(2020) 1:31
There are three mathematical representations, partial differential equations (PDE), path integrals, and stochastic differential equations (SDE) for the same physical process [18]. PDEs give a macroscopic view of an underlying physical process, while path integrals and SDEs give a more microscopic view. The quantum path approach through Feynman path integral is connected to the Itˆo’s processes to stochastic calculus and yields a special set of HJB equations which are backward parabolic in nature [18]. As a general non-linear system such as Merton-Garman Hamiltonian is intractable to solve analytically, Feynman path integral approach attacks these problems and gives simplified solutions [1, 13]. Therefore, only a few problems in finance are directly tractable by the Pontryagin maximum principle as constructing a value function and solving for the HJB equation usually involve an unnecessary difficult task. Since a firm is very small compared to an economy, and is subject to many small stochastic pertur
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