10,803 research outputs found
Stochastic path integral formalism for continuous quantum measurement
We generalize and extend the stochastic path integral formalism and action
principle for continuous quantum measurement introduced in [A. Chantasri, J.
Dressel and A. N. Jordan, Phys. Rev. A {\bf 88}, 042110 (2013)], where the
optimal dynamics, such as the most-likely paths, are obtained by extremizing
the action of the path integral. In this work, we apply exact functional
methods as well as develop a perturbative approach to investigate the
statistical behaviour of continuous quantum measurement, with examples given
for the qubit case. For qubit measurement with zero qubit Hamiltonian, we find
analytic solutions for average trajectories and their variances while
conditioning on fixed initial and final states. For qubit measurement with
unitary evolution, we use the perturbation method to compute expectation
values, variances, and multi-time correlation functions of qubit trajectories
in the short-time regime. Moreover, we consider continuous qubit measurement
with feedback control, using the action principle to investigate the global
dynamics of its most-likely paths, and finding that in an ideal case, qubit
state stabilization at any desired pure state is possible with linear feedback.
We also illustrate the power of the functional method by computing correlation
functions for the qubit trajectories with a feedback loop to stabilize the
qubit Rabi frequency.Comment: 24 pages, 4 figures and 1 tabl
Moment Methods for Exotic Volatility Derivatives
The latest generation of volatility derivatives goes beyond variance and
volatility swaps and probes our ability to price realized variance and sojourn
times along bridges for the underlying stock price process. In this paper, we
give an operator algebraic treatment of this problem based on Dyson expansions
and moment methods and discuss applications to exotic volatility derivatives.
The methods are quite flexible and allow for a specification of the underlying
process which is semi-parametric or even non-parametric, including
state-dependent local volatility, jumps, stochastic volatility and regime
switching. We find that volatility derivatives are particularly well suited to
be treated with moment methods, whereby one extrapolates the distribution of
the relevant path functionals on the basis of a few moments. We consider a
number of exotics such as variance knockouts, conditional corridor variance
swaps, gamma swaps and variance swaptions and give valuation formulas in
detail
Characteristic Function of Time-Inhomogeneous L\'evy-Driven Ornstein-Uhlenbeck Processes
Distributional properties -including Laplace transforms- of integrals of
Markov processes received a lot of attention in the literature. In this paper,
we complete existing results in several ways. First, we provide the analytical
solution to the most general form of Gaussian processes (with non-stationary
increments) solving a stochastic differential equation. We further derive the
characteristic function of integrals of L\'evy-processes and L\'evy driven
Ornstein-Uhlenbeck processes with time-inhomogeneous coefficients based on the
characteristic exponent of the corresponding stochastic integral. This yields a
two-dimensional integral which can be solved explicitly in a lot of cases. This
applies to integrals of compound Poisson processes, whose characteristic
function can then be obtained in a much easier way than using joint
conditioning on jump times. Closed form expressions are given for
gamma-distributed jump sizes as an example.Comment: 15 pages, 26 pages, to appear in Statistics and Probability Letter
OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC CONDITIONING
A mathematical framework for Continuous Time Finance based on operator algebraic
methods oers a new direct and entirely constructive perspective on the field. It also
leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations.
This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can eectively use block-factorization algorithms. Finally, we discuss
the method of dynamic conditioning that allows one to dynamically correlate over possibly
even hundreds of processes in a numerically noiseless framework while preserving marginal
distributions
G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion
The present paper is devoted to the study of sample paths of G-Brownian
motion and stochastic differential equations (SDEs) driven by G-Brownian motion
from the view of rough path theory. As the starting point, we show that
quasi-surely, sample paths of G-Brownian motion can be enhanced to the second
level in a canonical way so that they become geometric rough paths of roughness
2 < p < 3. This result enables us to introduce the notion of rough differential
equations (RDEs) driven by G-Brownian motion in the pathwise sense under the
general framework of rough paths. Next we establish the fundamental relation
between SDEs and RDEs driven by G-Brownian motion. As an application, we
introduce the notion of SDEs on a differentiable manifold driven by GBrownian
motion and construct solutions from the RDE point of view by using pathwise
localization technique. This is the starting point of introducing G-Brownian
motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin.
The last part of this paper is devoted to such construction for a wide and
interesting class of G-functions whose invariant group is the orthogonal group.
We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian
motion of independent interest
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