In this work we are going to show weak convergence of a probability measure corresponding to the solution of the following nonlinear stochastic heat equation)η α with colored noise η α to the measure corresponding to the solution of the same equation but with white noise η as α ↑ 1 on the space of continuous functions with compact support. The noise η α is assumed to be colored in space and its covariance is given by E [η α (t, x)η α (s, y)] = δ(t − s)f α (x − y) where f α is the Riesz kernel f α (x) ∝ 1/ |x| α . We will also state a result about continuity of measure in α, for α ∈ (0, 1). We will work with the classical notion of weak convergence of measures.
The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation. The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. In this paper under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method and provide numerical solutions using the finite difference method.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.