We present a review and some new results on form methods for generating holomorphic semigroups on Hilbert spaces. In particular, we explain how the notion of closability can be avoided. As examples we include the Stokes operator, the Black–Scholes equation, degenerate differential equations and the Dirichlet-to-Neumann operator. Mathematics Subject Classification (2000). Primary 47A07; Secondar...

This paper will introduce the Ito’s lemma used in stochastic calculus to obtain Ito-Taylor expansion of a differential equations. The Euler-Maruyama and Milstein’s methods solving equations be discussed derived. We apply these two numerical Black-Scholes model values European call option stock at discretized time intervals. use computer simulation approximate while using formula exact solution....

We consider that the surplus of an insurer follows compound Poisson process and the insurer would invest its surplus in risky assets, whose prices satisfy the Black-Scholes model. In the risk process, we decompose the ruin probability into the sum of two ruin probabilities which are caused by the claim and the oscillation, respectively. We derive the integro-differential equations for these rui...

We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black–Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods.

This paper considers the pricing of a European option using a B, S -market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper 1999 . Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral...

This paper analyses the model of Black–Scholes option pricing from point view group theoretic approach. The study identified new independent variables that lead to transformation equation. Furthermore, corresponding determining equations were constructed and symmetries found. As a result, findings demonstrate integrability present an invariant solution for Ornstein–Uhlenbeck stochastic process.