This thesis is devoted to the study of the applications of the error theory using Dirichlet forms. Our work is split into three parts. The first one deals with the models described by stochastic differential equations. After a short technical chapter, an innovative model for order books is proposed. We assume that the bid-ask spread is not an imperfection, but an intrinsic property of exchange ...

The solution of a nonlinear parabolic SPDE on the circle, with multiplicative Gaussian noise that is white-noise in time and a bonafide function in space, is approximated by a system of branching and interacting particles. Convergence of the system is established in the space of continuous-function-valued càdlàg processes via a mollification procedure.

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The socalled “method of the moving frame” allows us to reduce the SPDE problems to SDE problems.

For many practical problems it is useful to be able to sample conditioned diffusions on a computer (e.g. in filtering/smoothing to sample from the conditioned distribution of the unknown signal given the known observations). We present a recently developed, SPDE-based method to tackle this problem. The method is an infinite dimensional generalisation of the Langevin sampling technique.

Abstract We study the numerical approximation of invariant measure a viscous scalar conservation law, one-dimensional and periodic in space variable stochastically forced with white-in-time but spatially correlated noise. The flux function is assumed to be locally Lipschitz continuous have at most polynomial growth. scheme we employ discretizes stochastic partial differential equation (SPDE) ac...

Many real-world processes can naturally be modeled as systems of interacting agents. However, the long-term simulation such agent-based models is often intractable when system becomes too large. In this paper, starting from a stochastic spatio-temporal model (ABM), we present reduced in terms PDEs that describes evolution agent number densities for large populations. We discuss algorithmic deta...

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools control liquidity provision / consumption pool. In this paper, we study problem an exchange using incentives increase market liquidity. We model limit book as solution a stochastic partial differential equation (S...