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20 pages, 351 KiB

Open AccessFeature PaperArticle

Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

by Luca Di Persio and Matteo Garbelli

Symmetry 2021, 13(1), 14; https://doi.org/10.3390/sym13010014 - 23 Dec 2020

Cited by 6 | Viewed by 3441

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the [...] Read more.

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton–Jacobi–Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

13 pages, 370 KiB

Open AccessArticle

Pricing Various Types of Power Options under Stochastic Volatility

by Youngrok Lee, Yehun Kim and Jaesung Lee

Symmetry 2020, 12(11), 1911; https://doi.org/10.3390/sym12111911 - 20 Nov 2020

Cited by 7 | Viewed by 1617

Abstract

The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in [...] Read more.

The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in markets in order to provide high leverage strategy. In pricing power options, the classical Black–Scholes model which assumes a constant volatility is simple and easy to handle, but it has a limit in reflecting movements of real financial markets. As the alternatives of constant volatility, we focus on the stochastic volatility, finding more exact prices for power options. In this paper, we use the stochastic volatility model introduced by Schöbel and Zhu to drive the closed-form expressions for the prices of various power options including soft strike options. We also show the sensitivity of power option prices under changes in the values of each parameter by calculating the resulting values obtained from the formulas. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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18 pages, 961 KiB

Open AccessArticle

A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

by Jorge E. Macías-Díaz

Symmetry 2020, 12(11), 1907; https://doi.org/10.3390/sym12111907 - 20 Nov 2020

Cited by 1 | Viewed by 1741

Abstract

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for [...] Read more.

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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24 pages, 1439 KiB

Open AccessArticle

Approximation Formula for Option Prices under Rough Heston Model and Short-Time Implied Volatility Behavior

by Siow Woon Jeng and Adem Kilicman

Symmetry 2020, 12(11), 1878; https://doi.org/10.3390/sym12111878 - 14 Nov 2020

Cited by 3 | Viewed by 2240

Abstract

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative [...] Read more.

Rough Heston model possesses some stylized facts that can be used to describe the stock market, i.e., markets are highly endogenous, no statistical arbitrage mechanism, liquidity asymmetry for buy and sell order, and the presence of metaorders. This paper presents an efficient alternative to compute option prices under the rough Heston model. Through the decomposition formula of the option price under the rough Heston model, we manage to obtain an approximation formula for option prices that is simpler to compute and requires less computational effort than the Fourier inversion method. In addition, we establish finite error bounds of approximation formula of option prices under the rough Heston model for

0.1 \leq H < 0.5

under a simple assumption. Then, the second part of the work focuses on the short-time implied volatility behavior where we use a second-order approximation on the implied volatility to match the terms of Taylor expansion of call option prices. One of the key results that we manage to obtain is that the second-order approximation for implied volatility (derived by matching coefficients of the Taylor expansion) possesses explosive behavior for the short-time term structure of at-the-money implied volatility skew, which is also present in the short-time option prices under rough Heston dynamics. Numerical experiments were conducted to verify the effectiveness of the approximation formula of option prices and the formulas for the short-time term structure of at-the-money implied volatility skew. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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11 pages, 1987 KiB

Open AccessArticle

Effective Boundary of Innovation Subsidy: Searching by Stochastic Evolutionary Game Model

by Junqiang Li, Jingyi Yi and Yingmei Zhao

Symmetry 2020, 12(9), 1531; https://doi.org/10.3390/sym12091531 - 16 Sep 2020

Cited by 4 | Viewed by 1626

Abstract

Relationship between innovation subsidies and corporate strategic choices has been extensively studied. Public innovation subsidies are by no means a certain value, existing in the form of an effective range instead. This means that the public innovation subsidies existing within the reasonable range [...] Read more.

Relationship between innovation subsidies and corporate strategic choices has been extensively studied. Public innovation subsidies are by no means a certain value, existing in the form of an effective range instead. This means that the public innovation subsidies existing within the reasonable range can achieve the same incentive effect. So, what is the reasonable range or the effective boundaries of public innovation subsidies to promote enterprises that adopt cooperation strategies? There is no definite answer. Based on classical game theory, a stochastic evolutionary game model is proposed in this paper, which takes into account the influence of random disturbance on the strategy evolution process. An effective boundary of public innovation subsidy is provided as the main contribution based on a mature game scenario. A set of experimental data is subsequently selected as the sample for numerical simulation and result verification. The results showed that the probability of noncooperation within the effective value range will successfully converge to zero, which also means that the agents will adopt a collaborative cooperation strategy. The regulation effect of the combination of multiple variables is also discussed. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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12 pages, 791 KiB

Open AccessArticle

Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

by He Yang

Symmetry 2020, 12(6), 1031; https://doi.org/10.3390/sym12061031 - 19 Jun 2020

Cited by 4 | Viewed by 1614

Abstract

In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we first investigate the existence as well as the uniqueness of mild solutions for a class of

α \in (1, 2)

-order [...] Read more.

In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we first investigate the existence as well as the uniqueness of mild solutions for a class of

α \in (1, 2)

-order Riemann–Liouville fractional stochastic evolution equations of Sobolev type in abstract spaces. Then the symmetrical technique is used to deal with the

α \in (1, 2)

-order Caputo fractional stochastic evolution equations of Sobolev type in abstract spaces. Two examples are given as applications to the obtained results. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

17 pages, 918 KiB

Open AccessArticle

Space-Time Inversion of Stochastic Dynamics

by Massimiliano Giona, Antonio Brasiello and Alessandra Adrover

Symmetry 2020, 12(5), 839; https://doi.org/10.3390/sym12050839 - 20 May 2020

Viewed by 1912

Abstract

This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is [...] Read more.

This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is the analysis of solute dispersion in long straight tubes (Taylor-Aris dispersion), where the time-parametrization of the dynamics is recast in that of the axial coordinate. This allows the connection between the analysis of the forward (in time) evolution of the process and that of its exit-time statistics. The derivation of the Fokker-Planck equation for the inverted dynamics requires attention: it can be deduced using a mollified approach of the Wiener perturbations “a-la Wong-Zakai” by considering a sequence of almost everywhere smooth stochastic processes (in the present case, Poisson-Kac processes), converging to the Wiener processes in some limit (the Kac limit). The mathematical interpretation of the resulting Fokker-Planck equation can be obtained by introducing a new way of considering the stochastic integrals over the increments of a Wiener process, referred to as stochastic Stjelties integrals of mixed order. Several examples ranging from stochastic thermodynamics and fractal-time models are also analyzed. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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24 pages, 397 KiB

Open AccessArticle

Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition

by Marek T. Malinowski

Symmetry 2020, 12(5), 819; https://doi.org/10.3390/sym12050819 - 16 May 2020

Cited by 4 | Viewed by 1727

Abstract

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case [...] Read more.

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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18 pages, 793 KiB

Open AccessArticle

Dynamical Behavior of a Stochastic SIRC Model for Influenza A

by Tongqian Zhang, Tingting Ding, Ning Gao and Yi Song

Symmetry 2020, 12(5), 745; https://doi.org/10.3390/sym12050745 - 05 May 2020

Cited by 7 | Viewed by 2304

Abstract

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the [...] Read more.

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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33 pages, 467 KiB

Open AccessArticle

Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

by Haesung Lee and Gerald Trutnau

Symmetry 2020, 12(4), 570; https://doi.org/10.3390/sym12040570 - 05 Apr 2020

Cited by 3 | Viewed by 2103

Abstract

We show uniqueness in law for a general class of stochastic differential equations in

R^{d}

,

d \geq 2

, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy [...] Read more.

We show uniqueness in law for a general class of stochastic differential equations in

R^{d}

,

d \geq 2

, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Points of degeneracy have a d-dimensional Lebesgue–Borel measure zero. Weak existence is obtained for a more general, but not necessarily locally bounded drift coefficient. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

19 pages, 1195 KiB

Open AccessFeature PaperArticle

Threshold Analysis and Stationary Distribution of a Stochastic Model with Relapse and Temporary Immunity

by Peng Liu, Xinzhu Meng and Haokun Qi

Symmetry 2020, 12(3), 331; https://doi.org/10.3390/sym12030331 - 26 Feb 2020

Cited by 4 | Viewed by 2057 | Correction

Abstract

In this paper, a stochastic model with relapse and temporary immunity is formulated. The main purpose of this model is to investigate the stochastic properties. For two incidence rate terms, we apply the ideas of a symmetric method to obtain the results. First, [...] Read more.

In this paper, a stochastic model with relapse and temporary immunity is formulated. The main purpose of this model is to investigate the stochastic properties. For two incidence rate terms, we apply the ideas of a symmetric method to obtain the results. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction and persistence of this system. Then, we investigate the existence of a stationary distribution for this model by employing the theory of an integral Markov semigroup. Finally, the numerical examples are presented to illustrate the analytical findings. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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13 pages, 1356 KiB

Open AccessArticle

Estimation for the Discretely Observed Cox–Ingersoll–Ross Model Driven by Small Symmetrical Stable Noises

by Chao Wei

Symmetry 2020, 12(3), 327; https://doi.org/10.3390/sym12030327 - 25 Feb 2020

Cited by 10 | Viewed by 2128

Abstract

This paper is concerned with the least squares estimation of drift parameters for the Cox–Ingersoll–Ross (CIR) model driven by small symmetrical α-stable noises from discrete observations. The contrast function is introduced to obtain the explicit formula of the estimators and the error [...] Read more.

This paper is concerned with the least squares estimation of drift parameters for the Cox–Ingersoll–Ross (CIR) model driven by small symmetrical α-stable noises from discrete observations. The contrast function is introduced to obtain the explicit formula of the estimators and the error of estimation is given. The consistency and the rate of convergence of the estimators are proved. The asymptotic distribution of the estimators is studied as well. Finally, some numerical calculus examples and simulations are given. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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23 pages, 1425 KiB

Open AccessFeature PaperReview

Symmetry Analysis of the Stochastic Logistic Equation

by Giuseppe Gaeta

Symmetry 2020, 12(6), 973; https://doi.org/10.3390/sym12060973 - 08 Jun 2020

Cited by 1 | Viewed by 1538

Abstract

We apply the recently developed theory of symmetry of stochastic differential equations to stochastic versions of the logistic equation; these may have environmental or demographical noise, or both—in which case we speak of the complete model. We study all these cases, both [...] Read more.

We apply the recently developed theory of symmetry of stochastic differential equations to stochastic versions of the logistic equation; these may have environmental or demographical noise, or both—in which case we speak of the complete model. We study all these cases, both with constant and with non-constant noise amplitude, and show that the only one in which there are nontrivial symmetries is that of the stochastic logistic equation with (constant amplitude) environmental noise. In this case, the general theory of symmetry of stochastic differential equations is used to obtain an explicit integration, i.e., an explicit formula for the process in terms of any single realization of the driving Wiener process. Full article

(This article belongs to the Special Issue Advances in Stochastic Differential Equations)

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