Acknowledging many effects on humans, which are ignored in deterministic models for COVID-19, in this paper, we consider stochastic mathematical model for COVID-19. Firstly, the formulation of a stochastic susceptible-infected-recovered model is presented. Secondly, we devote with full strength our concentrated attention to sufficient conditions for extinction and persistence. Thirdly, we examine the threshold of the proposed stochastic COVID-19 model, when noise is small or large. Finally, we show the numerical simulations graphically using MATLAB.

We generalize the square integral estimate for the derivative of the convex function by Shashiashvili 2005 to the case of the family of the weight functions, satisfying certain conditions. This kind of generalization is especially valuable in the problems of mathematical finance for construction of the discrete time hedging strategies.

In a complete financial market we consider the discrete time hedging of the American option with a convex payoff. It is well known that for the perfect hedging the writer of the option must trade continuously in time, which is impossible in practice. In reality, the writer hedges only at some discrete time instants. The perfect hedging requires the knowledge of the partial derivative of the value function of the American option in the underlying asset, the explicit form of which is unknown in most cases of practical importance. Several approximation methods have been developed for the calculation of the value function of the American option. We claim in this paper that having at hand any uniform approximation of the American option value function at equidistant discrete rebalancing times it is possible to construct a discrete time hedging portfolio, the value process of which uniformly approximates the value process of the continuous time perfect delta-hedging portfolio. We are able to estimate the corresponding discrete time hedging error that leads to a complete justification of our hedging method for nonincreasing convex payoff functions including the important case of the American put. This method is essentially based on a new type square integral estimate for the derivative of an arbitrary convex function recently found by Shashiashvili.KEY WORDS: American option, discrete time hedging, hedging error, lower convex envelope. FORMULATION OF THE PROBLEM AND THE STATEMENT OF THE MAIN RESULTLet us consider a probability space ( , F, P R ) and a Brownian motion B = (B t ) 0≤t≤T on it. It is assumed that the time horizon T is finite and the notation P R corresponds to actual (i.e. real-world) probability measure. Denote by F B = (F B t ) 0≤t≤T the augmentation of the natural filtration of the Brownian motion by P R -null sets of F.On a filtered probability space ( , F, F B t , P R ) we will consider the financial market with two assets-a money market account (m t ) 0≤t≤T and a stock (S t ) 0≤t≤T , where m t denotes the price of the unit of the money market account at time t and S t is the price of a share of stock at time t. The dynamics of these prices obey respectively the following ordinary and the stochastic differential equationsThe support of Higher Education Commission of Islamabad is gratefully acknowledged. Manuscript

This work is devoted to the discrete time hedging of the American option on a dividendpaying stock with a convex payoff, the particular case of which is American call option. Perfect hedging requires continuous trading in time and knowledge of the partial derivative of the value function of the American option in the underlying asset. Neither one can trade continuously in time nor the closed-form expression of the value function of the American option is known. Several approximation methods have been developed for the calculation of this unknown value function. We justify in this work that having at hand any such nonnegative uniform approximation, it is possible to construct a discrete time hedging strategy the value process of which uniformly approximates the value process of the corresponding continuous time perfect hedging portfolio.

This work is devoted to a stochastic model on the spread and control of corona virus (COVID-19), in which the total population of a corona infected area is divided into susceptible, infected, and recovered classes. In reality, the number of individuals who get disease, the number of deaths due to corona virus, and the number of recovered are stochastic, because nobody can tell the exact value of these numbers in the future. The models containing these terms must be stochastic. Such numbers are estimated and counted by a random process called a Poisson process (or birth process). We construct an SIR-type model in which the above numbers are stochastic and counted by a Poisson process. To understand the spread and control of corona virus in a better way, we first study the stability of the corresponding deterministic model, investigate the unique nonnegative strong solution and an inequality managing of which leads to control of the virus. After this, we pass to the stochastic model and show the existence of a unique strong solution. Next, we use the supermartingale approach to investigate a bound managing of which also leads to decrease of the number of infected individuals. Finally, we use the data of the COVOD-19 in USA to calculate the intensity of Poisson processes and verify our results.