Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions

Using the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type $$dX_t = dW_t + \beta (X_t )dt$$ for possibly very singular drifts β. Here (X t ) t ≧0 takes values in some topological vector spaceE and (W t ) t ≧0 is anE-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where β is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear β which is based on our general results and second quantization. We also prove new results on general diffusion Dirichlet forms in infinite dimensions, in particular that the Fukushima decomposition holds in this case.

In this paper we systematically develop, as a technical tool for our main application below, a stochastic calculus for generalized Dirichlet forms (cf. [15]). In particular, we show Fukushima's decomposition of additive functional including its extended version to functions not necessarily in the domain f of the generalized Dirichlet form, cf. Theorem 4.5 (ii), and an Itό-type formula in this framework. The extended version of the Fukushima decomposition will be applied in combination with the Itό-type formula in Subsection 6.3. The class of generalized Dirichlet forms is much larger than the well-studied class of symmetric and coercive Dirichlet forms as in [6] resp. [10] and time dependent Dirichlet forms as in [12]. It contains examples of an entirely new kind (cf. Section 6, [15]). Therefore, the results obtained in this paper lead to extensions of the corresponding results in the theories. In particular the proofs are locally completely different (cf. e.g. Theorem 2.3 and Theorem 2.5; though for the reader's convenience we tried to follow the line of argument in [6] as closely as possible). This difference has several reasons: First of all we do not assume any sector condition', in certain cases we have to handle £-quasi-lower-semicontinuous functions instead of ^-quasi-continuous functions (cf. e.g. Remark 2.6 (i)), and finally, the Dirichlet space of the classical situation generalizes to a space T which is not necessarily stable under composition with normal contractions. In contrast to the classical theory it is not known whether regularity or quasi-regularity alone imply the existence of an associated process. An additional structural assumption on T is made in [15, IV.2, D3] (i.e. the existence of a nice intermediate space y has to be assumed) in order to construct explicitly an associated m -tight special standard process M. In addition to the new theoretical results described above we also present new applications. In Section 6 we construct weak solutions to stochastic differential equations in infinite dimensions of the type

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closed-loop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.

A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.

This paper is concerned with optimal control of linear backward stochastic differential equations (BSDE) with a quadratic cost criteria, or backward linear-quadratic (BLQ) control. The solution of this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. Two alternative, though equivalent, expressions for the optimal control are obtained. The first of these involves a pair of Riccati type equations, an uncontrolled BSDE and an uncontrolled forward stochastic differential equation (SDE), while the second is in terms of a Hamiltonian system. A key step in our derivation is a proof of global solvability of the aforementioned Riccati equations. Although of independent interest, this issue has particular relevance to the BLQ problem since these Riccati equations play a central role in our solution. Last but not least, it is demonstrated that the optimal control obtained coincides with the solution of a certain forward linear-quadratic (LQ) problem. This, in turn, reveals the origin of the Riccati equations introduced.

%\baselineskip 24pt We discuss the existence of solutions of linear stochastic differential equations in~$\bf{R}^\infty$. The existence of strong continuous (respectively, \hbox{$L^p$-con}\-tinuous) solutions of autonomous linear stochastic differential equations in $\bf{R}^\infty$ with continuous (respectively, \hbox{$L^p$-con}\-tinuous) right-hand sides is proved. Uniqueness conditions are obtained.

Sufficient conditions for the exponential stability of a class ofnonlinear, non-autonomous stochastic differential equations in infinitedimensions are studied. The analysis consists of introducing a suitableapproximating solution systems and using a limiting argument to pass onstability of strong solutions to mild ones. As a consequence, the classicalcriteriaof stability in A. Ichikawa [8] are improved and extended to cover a class ofnon-autonomous stochastic evolution equations.Two examples are investigated to illustrate our theory.

Moment boundedness of linear stochastic delay differential equations with distributed delay

In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean square finite-approximate controllability of linear stochastic evolution systems. A control is also found that provides finite-dimensional mean exact controllability in addition to the requirement of approximate mean square controllability. Under the assumption of approximate mean square controllability of the associated linear stochastic system, we obtain sufficient conditions for the mean square finite-approximate controllability of a semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients using the Picard-type iterations. An application to stochastic heat conduction equations is considered.

We consider the problem of optimal investment in a market with borrowing and random coefficients. We assume that the bond interest rate, the borrowing interest rate, the appreciation rate and the volatility of stock, are random and possibly unbounded. Due to the possibility of borrowing, the formulated optimal investment problem is an optimal stochastic control problem with nonlinear system dynamics and possibly unbounded coefficients. Explicit closed-form solutions in terms of a linear backward stochastic differential equation are obtained for the power and logarithmic utility from terminal wealth. The optimal controls turn out to be of a linear state-feedback form.

We consider the integrability problem of an exponential process with unbounded coefficients. The integrability is established under weaker conditions of Kazamaki type, which complements the results of Yong obtained under a Novikov type condition. As applications, we consider the solvability of linear backward stochastic differential equations (BSDEs) and market completeness, the solvability of a Riccati BSDE and optimal investment, all in the setting of unbounded coefficients.

We study linear backward stochastic partial differential equations (BSPDEs) of parabolic type. We consider a new boundary value problem where a Cauchy condition is replaced by a prescribed average of the solution over time. We establish well-posedness, existence, uniqueness, and regularity, for the solutions of this new problem. In particular, this can be considered as a possibility to recover a solution of a BSPDE in a setting where its values at the terminal time are unknown, and where the average of the solution over time is preselected.

The classical theory of controllability and observability for deterministic systems is extended to linear stochastic time-varying systems defined on infinite dimensional Hilbert spaces. Two types of stochastic controllability (observability) are studied: approximate and complete controllability (observability). Tests for complete and approximate controllability (observability) are proved, and the relation between the controllability (observability) of linear stochastic systems and the controllability (observability) of the corresponding deterministic systems is studied.

Dependence on the boundary condition for linear stochastic differential equations in the plane

Detailed microscale simulation is typically too computationally expensive for the long time simulations necessary to explore macroscale dynamics. Projective integration uses bursts of the microscale simulator, on microscale time steps, and then computes an approximation to the system over a macroscale time step by extrapolation. Projective integration has the potential to be an effective method to compute the long time dynamic behaviour of multiscale systems. However, many multiscale systems are significantly influenced by noise. By a maximum likelihood estimation, we fit a linear stochastic differential equation to short bursts of data. The analytic solution of the linear stochastic differential equation then estimates the solution over a macroscale, projective integration, time step. We explore how the noise affects the projective integration in two different methods. Monte Carlo simulation suggests design parameters offering stability and accuracy for the algorithms. The algorithms developed here may be applied to compute the long time dynamic behaviour of multiscale systems with noise and to exploit parallel computation. References Yacine Ait-Sahalia. Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica, 70(1):223--262, 2002. Yacine Ait-Sahalia. Likelihood inference for diffusions: a survey. In Jianqing Fan and Hira L. Koul, editors, Frontiers in Statistics: in Honor of Peter J. Bickel's 65th Birthday, chapter 17, pages 369--405. Imperial College Press, 2006. Yacine Ait-Sahalia and Robert Kimmel. Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics, 83(2):413--452, 2007. doi:10.1016/j.jfineco.2005.10.006 C. P. Calderon. Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics. Molecular Simulation, 33(9--10):713--731, August 2007. doi:10.1080/08927020701344740 Radek Erban, Ioannis G. Kevrekidis, and Hans G. Othmer. An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal. Physica D: Nonlinear Phenomena, 215(1):1--24, 2006. doi:10.1016/j.physd.2006.01.008 C. W. Gear, I. G. Kevrekidis, and C. Theodoropoulos. `Coarse' integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Computers and Chemical Engrg, 26:941--963, 2002. C. W. Gear and Ioannis G. Kevrekidis. Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum. SIAM Journal on Scientific Computing, 24(4):1091--1106, 2003. doi:10.1137/S1064827501388157 Dror Givon and Ioannis G. Kevrekidis. Coarse-grained projective schemes for certain singularly perturbed stochastic problems. Technical report, Princeton University, 2008. Ioannis G. Kevrekidis and Giovanni Samaey. Equation-free multiscale computation: Algorithms and applications. Annu. Rev. Phys. Chem., 60:321--44, 2009. doi:10.1146/annurev.physchem.59.032607.093610 P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations, volume 23 of Applications of Mathematics. Springer--Verlag, 1992. C. I. Siettos, M. D. Graham, and I. G. Kevrekidis. Coarse brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation. J. Chemical Physics, 118(22):10149--10156, 2003. doi:10.1063/1.1572456