11,573 research outputs found
The H
This paper discusses the state feedback H∞ control problem for a class of bilinear stochastic systems driven by both Brownian motion and Poisson jumps. By completing square method, we obtain the H∞ control by solutions of the corresponding Hamilton-Jacobi equations (HJE). By the tensor power series method, we also shift such HJEs into a kind of Riccati equations, and the H∞ control is represented with the form of tensor power series
General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients
The main purpose of this paper is to discuss detailed the stochastic LQ
control problem with random coefficients where the linear system is a
multidimensional stochastic differential equation driven by a multidimensional
Brownian motion and a Poisson random martingale measure. In the paper, we will
establish the connections of the multidimensional Backward stochastic Riccati
equation with jumps (BSRDEJ in short form) to the stochastic LQ problem and to
the associated Hamilton systems. By the connections, we show the optimal
control have the state feedback representation. Moreover, we will show the
existence and uniqueness result of the multidimensional BSRDEJ for the case
where the generator is bounded linear dependence with respect to the unknowns
martingale term
Marcus versus Stratonovich for Systems with Jump Noise
The famous It\^o-Stratonovich dilemma arises when one examines a dynamical
system with a multiplicative white noise. In physics literature, this dilemma
is often resolved in favour of the Stratonovich prescription because of its two
characteristic properties valid for systems driven by Brownian motion: (i) it
allows physicists to treat stochastic integrals in the same way as conventional
integrals, and (ii) it appears naturally as a result of a small correlation
time limit procedure. On the other hand, the Marcus prescription [IEEE Trans.
Inform. Theory 24, 164 (1978); Stochastics 4, 223 (1981)] should be used to
retain (i) and (ii) for systems driven by a Poisson process, L\'evy flights or
more general jump processes. In present communication we present an in-depth
comparison of the It\^o, Stratonovich, and Marcus equations for systems with
multiplicative jump noise. By the examples of areal-valued linear system and a
complex oscillator with noisy frequency (the Kubo-Anderson oscillator) we
compare solutions obtained with the three prescriptions.Comment: 14 pages, 4 figure
Forward-Backward Doubly Stochastic Differential Equations with Random Jumps and Stochastic Partial Differential-Integral Equations
In this paper, we study forward-backward doubly stochastic differential
equations driven by Brownian motions and Poisson process (FBDSDEP in short).
Both the probabilistic interpretation for the solutions to a class of
quasilinear stochastic partial differential-integral equations (SPDIEs in
short) and stochastic Hamiltonian systems arising in stochastic optimal control
problems with random jumps are treated with FBDSDEP. Under some monotonicity
assumptions, the existence and uniqueness results for measurable solutions of
FBDSDEP are established via a method of continuation. Furthermore, the
continuity and differentiability of the solutions of FBDSDEP depending on
parameters is discussed. Finally, the probabilistic interpretation for the
solutions to a class of quasilinear SPDIEs is given
The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications
This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem andthe linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type
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