Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one-sided estimate type that is formulated in terms of so-called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1-complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so-called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration

Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2011We consider Forward Backward Stochastic Differential Equations (FBSDEs for short) with different assumptions on its coefficients. In a first part we present results of existence, uniqueness and dependence upon initial conditions and on the coefficients. There are two main methodologies employed in this study. The first one presented is the Four Step Scheme, which makes very clear the connection of FBSDEs with quasilinear parabolic systems of Partial Differential Equations (PDEs for short). The weakness of this methodology is the smoothness and regularity assumptions recquired on the coefficients of the system, which motivate the employment of Banach`s Fixed Point Theorem in the study of existence and uniqueness results. This classic analytical tool requires less regularity on the coefficients, but gives only local existence of solution in a small time duration. In a second stage, with the help of the previous work with a running-down induction on time, we can assure the existence and uniqueness of solution for the FBSDE problem in global time. The second goal of this work is the study of the assymptotic behaviour of the FBSDEs solutions when the diffusion coefficient of the forward equation is multiplicatively perturbed with a small parameter that goes to zero. This question adresses the problem of the convergence of the classical/viscosity solutions of the quasilinear parabolic system of PDEs associated to the system. When this quasilinear parabolic system of PDEs takes the form of the backward Burgers Equation, the problem is the convergence of the solution when the viscosity parameter goes to zero. To study conveniently this problem with a probabilistic approach , we present a concise survey of the classical Large Deviations Principles and the basics of the so-called "Freidlin-Wentzell Theory". This theory is mainly concerned with the study of the Itô Diffusions with the diffusion term perturbed by a small parameter that converges to zero and the richness of properties of the FBSDEs shows us that (even in a coupled FBSDE system) this approach is a good one, since we can extract for the solutions of the perturbed systems a Large Deviations Principle and state convergence of the perturbed solutions to a solution of a deterministic system of ordinary differential equations

Backward stochastic dynamics on a filtered probability space

We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation formulate are needed. This approach provides new tools for the study of BSDE, and is particularly useful for the study of BSDE with partial information. The approach allows us to study the following type of backward stochastic differential equations: $$dY_t^j=-f_0^j(t,Y_t,L(M)_t) dt-\sum_{i=1}^df_i^j(t,Y_t), dB_t^i+dM_t^j$$ with $Y_T=\xi$, on a general filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_t,P)$, where $B$ is a $d$-dimensional Brownian motion, $L$ is a prescribed (nonlinear) mapping which sends a square-integrable $M$ to an adapted process $L(M)$ and $M$, a correction term, is a square-integrable martingale to be determined. Under certain technical conditions, we prove that the system admits a unique solution $(Y,M)$. In general, the associated partial differential equations are not only nonlinear, but also may be nonlocal and involve integral operators.Comment: Published in at http://dx.doi.org/10.1214/10-AOP588 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A dynamical approximation for stochastic partial differential equations

Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states (invariant measures) is considered for one example of stochastic partial differential equations.Comment: 28 pages, no figure

Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian

We study the long-time behavior of the unique viscosity solution $u$ of the viscous Hamilton-Jacobi Equation $u_t-\Delta u + |Du|^m = f\hbox{in }\Omega\times (0,+\infty)$ with inhomogeneous Dirichlet boundary conditions, where $\Omega$ is a bounded domain of $\mathbb{R}^N$. We mainly focus on the superquadratic case ($m>2$) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on $f,$ the initial and boundary data, we connect the problem studied to its associated stationary generalized Dirichlet problem on one hand and to a stationary problem with a state constraint boundary condition on the other hand

Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations

We establish Schauder a priori estimates and regularity for solutions to a class of boundary-degenerate elliptic linear second-order partial differential equations. Furthermore, given a smooth source function, we prove regularity of solutions up to the portion of the boundary where the operator is degenerate. Degenerate-elliptic operators of the kind described in our article appear in a diverse range of applications, including as generators of affine diffusion processes employed in stochastic volatility models in mathematical finance, generators of diffusion processes arising in mathematical biology, and the study of porous media.Comment: 58 pages, 1 figure. To appear in the Journal of Differential Equations. Incorporates final galley proof corrections corresponding to published versio