549 research outputs found
Stochastic partial differential equations driven by Levy space-time white noise
In this paper we develop a white noise framework for the study of stochastic
partial differential equations driven by a d-parameter (pure jump) Levy white
noise. As an example we use this theory to solve the stochastic Poisson
equation with respect to Levy white noise for any dimension d. The solution is
a stochastic distribution process given explicitly. We also show that if d\leq
3, then this solution can be represented as a classical random field in L2(\mu
), where \mu is the probability law of the Levy process.
The starting point of our theory is a chaos expansion in terms of generalized
Charlier polynomials. Based on this expansion we define Kondratiev spaces and
the Levy Hermite transform
A stochastic maximum principle via Malliavin calculus
This paper considers a controlled It\^o-L\'evy process where the information
available to the controller is possibly less than the overall information. All
the system coefficients and the objective performance functional are allowed to
be random, possibly non-Markovian. Malliavin calculus is employed to derive a
maximum principle for the optimal control of such a system where the adjoint
process is explicitly expressed
Symmetries of the ratchet current
Recent advances in nonequilibrium statistical mechanics shed new light on the
ratchet effect. The ratchet motion can thus be understood in terms of symmetry
(breaking) considerations. We introduce an additional symmetry operation
besides time-reversal, that effectively reverses the nonequilibrium driving.
That operation of field-reversal combined with time-reversal decomposes the
nonequilibrium action so to clarify under what circumstances the ratchet
current is a second order effect around equilibrium, what is the direction of
the ratchet current and what are possibly the symmetries in its fluctuations.Comment: 13 pages, heavily extended versio
Fokker-Planck PIDE for McKean-Vlasov jump diffusions and applications to HJB equations and optimal control
The purpose of this paper is to study optimal control of McKean-Vlasov
(mean-field) stochastic differential equations with jumps (McKean-Vlasov jump
diffusions, for short). To this end, we first prove a Fokker-Planck equation
for the law of the solution of such equations. Then we study the situation when
the law is absolutely continuous with respect to Lebesgue measure. In that case
the Fokker-Planck equation reduces to a deterministic integro-differential
equation (PIDE) for the Radon-Nikodym derivative of the law. Combining this
equation with the original state equation, we obtain a Markovian system for the
state and its law. Furthermore, we apply this to formulate an
Hamilton-Jacobi-Bellman (HJB) equation for the optimal control of McKean-Vlasov
stochastic differential equations with jumps. Finally we apply these results to
solve explicitly the following problems:
Linear-quadratic optimal control of stochastic McKean-Vlasov jump diffusions.
Optimal consumption from a cash flow modelled as a stochastic McKean-Vlasov
differential equation with jumps.Comment: 2
Optimal Consumption and Portfolio in a Jump Diffusion Market with Proportional Transaction Costs
We study the optimal consumption and portfolio in a jump diffusion market with proportional transaction costs. We show that the solution in the jump diffusion case has the same form as in the pure diffusion case; in particular, (under some assumptions) there is a transaction cone D such that it is optimal to make no transactions as long as the wealth position remains in D and to sell/buy stocks according to local time on the boundary of D. The associated integro-differential variational inequality is studied by using the theory of viscosity solutions
Langevin Equation for the Density of a System of Interacting Langevin Processes
We present a simple derivation of the stochastic equation obeyed by the
density function for a system of Langevin processes interacting via a pairwise
potential. The resulting equation is considerably different from the
phenomenological equations usually used to describe the dynamics of non
conserved (Model A) and conserved (Model B) particle systems. The major feature
is that the spatial white noise for this system appears not additively but
multiplicatively. This simply expresses the fact that the density cannot
fluctuate in regions devoid of particles. The steady state for the density
function may however still be recovered formally as a functional integral over
the coursed grained free energy of the system as in Models A and B.Comment: 6 pages, latex, no figure
Distribution of the Oscillation Period in the Underdamped One Dimensional Sinai Model
We consider the Newtonian dynamics of a massive particle in a one dimemsional
random potential which is a Brownian motion in space. This is the zero
temperature nondamped Sinai model. As there is no dissipation the particle
oscillates between two turning points where its kinetic energy becomes zero.
The period of oscillation is a random variable fluctuating from sample to
sample of the random potential. We compute the probability distribution of this
period exactly and show that it has a power law tail for large period, P(T)\sim
T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact
results are confirmed by numerical simulations and also via a simple scaling
argument.Comment: 9 pages LateX, 2 .eps figure
Control of Multi-level Voltage States in a Hysteretic SQUID Ring-Resonator System
In this paper we study numerical solutions to the quasi-classical equations
of motion for a SQUID ring-radio frequency (rf) resonator system in the regime
where the ring is highly hysteretic. In line with experiment, we show that for
a suitable choice of of ring circuit parameters the solutions to these
equations of motion comprise sets of levels in the rf voltage-current dynamics
of the coupled system. We further demonstrate that transitions, both up and
down, between these levels can be controlled by voltage pulses applied to the
system, thus opening up the possibility of high order (e.g. 10 state),
multi-level logic and memory.Comment: 8 pages, 9 figure
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