A cops and robbers game (CR) on graphs was originated in 1983 by Quilliot and by Nowakowski and Winkler independently. This game has been applied in many problems in the area of theoretical computer science such as information seeking, robot motion planning or network security as evidenced by many conferences and publications. The CR game has also been extensively studied and varied to many versions. In this paper, we focus on CR game under the condition that the robber performs a random walk. Namely, he drunkenly, or randomly, chooses his next move to escape the cop, while the cop plays optimally in order to minimize the expected drunk capture times $${\textit{dct}}(G)$$dct ( G ) of a graph G. We apply the concepts of expected hitting time in Markov Chain and combinatorial technique to find $${\textit{dct}}(G)$$dct ( G ) for graphs G that have been used in many applications which are cycles, complete multipartite graphs, grid graphs and prism graphs.

A warrant is a financial derivative that grants the holder the right to purchase company shares at a predetermined price within a specified period. Generally, upon exercise, the total number of outstanding shares increases because of the issuance of new shares, reducing the stock price. In this study, an analytical formula for warrant valuation is developed without relying on the restrictive assumptions of log-normal asset return distributions or constant volatility. The model incorporates key financial variables, including the current asset price, the strike price, the risk-free interest rate, the time to maturity, and the dilution factor. To capture dynamic market conditions, asset return volatility is estimated using GARCH-type models. The performance of this analytical approach is evaluated by comparing its numerical results with those obtained using alternative methods, such as Monte Carlo simulations and the conventional warrant valuation framework. An empirical analysis based on data from the Stock Exchange of Thailand indicates that the proposed method yields improved pricing accuracy with lower estimation errors than existing benchmarks.