Search results for "stochastic differential equations"
showing 10 items of 24 documents
Stochastic models for phytoplankton dynamics in marine ecosystems
2014
In this thesis, the stochastic advection-reaction-diffusion models are analyzed to obtain the vertical stationary spatial distributions of the main groups of picophytoplankton, which account about for 80% of total chlorophyll on average in Mediterranean Sea. In Chapter 1 we give a short presentation of the experimental and phytoplanktonic data collected during different oceanographic surveys in Mediterranean Sea. In Chapter 2 we introduce the deterministic and stochastic approaches (one-population model) adopted to describe the picoeukaryotes dynamics in Sicily Channel. Moreover, numerical results for the biomass concentration are compared with experimental data by using chi-squared goodnes…
Decoupling on the Wiener Space, Related Besov Spaces, and Applications to BSDEs
2021
We introduce a decoupling method on the Wiener space to define a wide class of anisotropic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivat…
$L_2$-variation of L\'{e}vy driven BSDEs with non-smooth terminal conditions
2016
We consider the $L_2$-regularity of solutions to backward stochastic differential equations (BSDEs) with Lipschitz generators driven by a Brownian motion and a Poisson random measure associated with a L\'{e}vy process $(X_t)_{t\in[0,T]}$. The terminal condition may be a Borel function of finitely many increments of the L\'{e}vy process which is not necessarily Lipschitz but only satisfies a fractional smoothness condition. The results are obtained by investigating how the special structure appearing in the chaos expansion of the terminal condition is inherited by the solution to the BSDE.
Mean square rate of convergence for random walk approximation of forward-backward SDEs
2020
AbstractLet (Y,Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk$B^n$from the underlying Brownian motionBby Skorokhod embedding, one can show$L_2$-convergence of the corresponding solutions$(Y^n,Z^n)$to$(Y, Z).$We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in$C^{2,\alpha}$. The proof relies on an approximative representation of$Z^n$and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to t…
Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior
2012
Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479-497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.
Simulation of BSDEs with jumps by Wiener Chaos Expansion
2016
International audience; We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.
Donsker-Type Theorem for BSDEs: Rate of Convergence
2019
In this paper, we study in the Markovian case the rate of convergence in Wasserstein distance when the solution to a BSDE is approximated by a solution to a BSDE driven by a scaled random walk as introduced in Briand, Delyon and Mémin (Electron. Commun. Probab. 6 (2001) Art. ID 1). This is related to the approximation of solutions to semilinear second order parabolic PDEs by solutions to their associated finite difference schemes and the speed of convergence. peerReviewed
Random walk approximation of BSDEs with H{\"o}lder continuous terminal condition
2018
In this paper, we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space. peerReviewed
Spatio-temporal behaviour of the deep chlorophyll maximum in Mediterranean Sea: Development of a stochastic model for picophytoplankton dynamics
2013
In this paper, by using a stochastic reaction-diffusion-taxis model, we analyze the picophytoplankton dynamics in the basin of the Mediterranean Sea, characterized by poorly mixed waters. The model includes intraspecific competition of picophytoplankton for light and nutrients. The multiplicative noise sources present in the model account for random fluctuations of environmental variables. Phytoplankton distributions obtained from the model show a good agreement with experimental data sampled in two different sites of the Sicily Channel. The results could be extended to analyze data collected in different sites of the Mediterranean Sea and to devise predictive models for phytoplankton dynam…
A stochastic reaction-diffusion-taxis model for picophytoplankton dynamics
2011
The dynamics of picophytoplankton communities in marine environment is studied by astochastic reaction-dìffusìon-taxis model, analyzing the time evolution of the biomass concentration along a water column. The model is based on two stochastic differentìal equations, where the random fluctuations of the environmental variables are considered by inserting two multiplicative noise terms. Specifically, the model describes the dynamics of diffusion of picophytoplankton biomass and nutrient concentrations. In the proposed model the marine environment is characterized by poorly mixed waters and picophytoplankton is subject to intraspecific competition for light and nutrients. By numerically solvin…