Stochastic calculus, statistical asymptotics, Taylor strings and phyla
Stochastic differential equations with jumps.
Stochastic fuzzy differential equations with an application
In this paper we present the existence and uniqueness of solutions to the stochastic fuzzy differential equations driven by Brownian motion. The continuous dependence on initial condition and stability properties are also established. As an example of application we use some stochastic fuzzy differential equation in a model of population dynamics.
Stochastic Inverse Problem with Noisy Simulator. Application to aeronautical model
Inverse problem is a current practice in engineering where the goal is to identify parameters from observed data through numerical models. These numerical models, also called Simulators, are built to represent the phenomenon making possible the inference. However, such representation can include some part of variability or commonly called uncertainty (see [4]), arising from some variables of the model. The phenomenon we study is the fuel mass needed to link two given countries with a commercial...
Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions
In this paper we explicit the derivative of the flows of one-dimensional reflected diffusion processes. We then get stochastic representations for derivatives of viscosity solutions of one-dimensional semilinear parabolic partial differential equations and parabolic variational inequalities with Neumann boundary conditions.
Stochastic Taylor expansions and heat kernel asymptotics
These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.
Strict concavity of the intersection exponent for Brownian motion in two and three dimensions.
Surface stretching for Ornstein Uhlenbeck velocity fields.
Système de processus auto-stabilisants [Book]
Systemic risk through contagion in a core-periphery structured banking network
We contribute to the understanding of how systemic risk arises in a network of credit-interlinked agents. Motivated by empirical studies we formulate a network model which, despite its simplicity, depicts the nature of interbank markets better than a symmetric model. The components of a vector Ornstein-Uhlenbeck process living on the nodes of the network describe the financial robustnesses of the agents. For this system, we prove a LLN for growing network size leading to a propagation of chaos result....
The Bellman equation related to the minimal entropy martingale measure.
The Feynman-Kac formula for a system of parabolic equations
The infinite Brownian loop on a symmetric space.
The multi-dimensional super-replication problem under gamma constraints
The norm of the product of a large matrix and a random vector.
The pricing of credit risky securities under stochastic interest rate model with default correlation
In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R. A. Jarrow and F. Yu (2001), S. Y. Leung and Y. K. Kwok (2005), A. Wang and Z. Ye (2011)). By using the method of change of measure and...
Time dependent subordination and Markov processes with jumps
Ultrafast Subordinators and Their Hitting Times
Uncertainty orders on the sublinear expectation space
In this paper, we introduce some definitions of uncertainty orders for random vectors in a sublinear expectation space. We all know that, under some continuity conditions, each sublinear expectation 𝔼 has a robust representation as the supremum of a family of probability measures. We describe uncertainty orders from two different viewpoints. One is from sublinear operator viewpoint. After giving definitions such as monotonic orders, convex orders and increasing convex orders, we use these uncertainty...
Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem
The reliable and effective assimilation of measurements and numerical simulations in engineering applications involving computational fluid dynamics is an emerging problem as soon as new devices provide more data. In this paper we are mainly driven by hemodynamics applications, a field where the progressive increment of measures and numerical tools makes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusion of noisy data in the incompressible steady Navier-Stokes equations...