Multiple G-It\^{o} integral in the G-expectation space

In this paper, motivated by mathematic finance we introduce the multiple G-It\^{o} integral in the G-expectation space, then investigate how to calculate. We get the the relationship between Hermite polynomials and multiple G-It\^{o} integrals which is a natural extension of the classical result obtained by It\^{o} in 1951.Comment: 9 page

An overview of Viscosity Solutions of Path-Dependent PDEs

This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12]

Analogy making and the structure of implied volatility skew

An analogy based option pricing model is put forward. If option prices are determined in accordance with the analogy model, and the Black Scholes model is used to back-out implied volatility, then the implied volatility skew arises, which flattens as time to expiry increases. The analogy based stochastic volatility and the analogy based jump diffusion models are also put forward. The analogy based stochastic volatility model generates the skew even when there is no correlation between the stock price and volatility processes, whereas, the analogy based jump diffusion model does not require asymmetric jumps for generating the skew

Further results on the common neighbourhood domination and some related graphs

In this paper we obtain some results and bounds on the common neighbourhood domination number (CN-domination number) of a graph G. Also we introduce the common neighbourhood domatic number (CN-domatic number). Common neighbourhood connectedness(CN-connectedness), Common neighbourhood regularity (CN-regularity), Common neighbourhood completeness (CN-completeness) are introduce by define the CN-connected graph, CN-regular graph, CN-complete graph and CN-complement. Some properties and interesting results of these graphs are established

The P3-domination in graphs

Let G be a graph and u,v be any vertices of G. Then u and v are said to be P3-adjacent vertices of G if there is a subgraph of G, isomorphic to P3, Containing u and v. A P3-dominating set of G is a set D of vertices such that every vertex of G belongs to D or is P 3-adjacent to a vertex of D. The P3-domination number of G denoted by γP3(G) is the minimum cardinality among the P 3-dominating sets of vertices of G. In this paper we introduce and study the P3-domination of a graph G and analogous to this concept we define the P3-independence number βP3(G), P 3-neighbourhood number ηP3(G) and P 3-domatic number dp3(G). Some bounds and interesting results are obtained. Also the P3-adjacency motivated us to define new graphs in particular P3-neighbourhood graph, P 3-complete graph, P3regular graph, P3- complement graph and P3-complementary graph, some basic properties of these graphs are introduce and new method to construct any r-regular graph is established, finally we generalize the domination of graphs

Strong Domination Critical and Stability in Graphs

ABSTRACT In general strong domination number s (G) can be made to decrease or increase by removal of vertices from G. In this paper our main objective is the study of this phenomenon. Further the stability of the strong domination number of a graph G is investigated. Mathematics subject classification : 05C7

Independent monopoly size in graphs

In a graph G = (V, E), a set D subset of V (G) is said to be a monopoly set of G if every vertex v is an element of V-D has at least d (v)/2 neighbors in D. The monopoly size of G, denoted mo(G), is the minimum cardinality of a monopoly set among all monopoly sets in G. The set D subset of V (G) is an independent monopoly set in G if it is both a monopoly set and an independent set in G. The number of vertices in a minimum independent monopoly set in a graph G is the independent monopoly size of G and is denoted by imo(G). In this paper, we study the existence of independent monopoly set in graphs, bounds for imo(G), and some exact values for some standard graphs are obtained. Finally we characterize all graphs of order n with imo(G) = 1; n-1 and n