18 research outputs found
Exact Simulation of Non-stationary Reflected Brownian Motion
This paper develops the first method for the exact simulation of reflected
Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The
running time of generating exact samples of non-stationary RBM at any time
is uniformly bounded by where is the
average drift of the process. The method can be used as a guide for planning
simulations of complex queueing systems with non-stationary arrival rates
and/or service time
Simulations of some Doubly Stochastic Poisson Point Processes
International audienceComputer simulations of point processes are important either to verify the results of certain theoretical calculations that can be very awkward at times, or to obtain practical results when these calculations become almost impossible. One of the most common methods for the simulation of nonstationary Poisson processes is random thinning. Its extension when the intensity becomes random (doubly stochastic Poisson processes) depends on the structure of this intensity. If the random density takes only discrete values, which is a common situation in many physical problems where quantum mechanics introduces discrete states, it is shown that the thinning method can be applied without error. We study in particular the case of binary density and we present the kind of theoretical calculations that then become possible. The results of various experiments realized with data obtained by simulation show fairly good agreement with the theoretical calculations
An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions
Stochastic differential equations (SDEs) or diffusions are continuous-valued
continuous-time stochastic processes widely used in the applied and
mathematical sciences. Simulating paths from these processes is usually an
intractable problem, and typically involves time-discretization approximations.
We propose an exact Markov chain Monte Carlo sampling algorithm that involves
no such time-discretization error. Our sampler is applicable to the problem of
prior simulation from an SDE, posterior simulation conditioned on noisy
observations, as well as parameter inference given noisy observations. Our work
recasts an existing rejection sampling algorithm for a class of diffusions as a
latent variable model, and then derives an auxiliary variable Gibbs sampling
algorithm that targets the associated joint distribution. At a high level, the
resulting algorithm involves two steps: simulating a random grid of times from
an inhomogeneous Poisson process, and updating the SDE trajectory conditioned
on this grid. Our work allows the vast literature of Monte Carlo sampling
algorithms from the Gaussian process literature to be brought to bear to
applications involving diffusions. We study our method on synthetic and real
datasets, where we demonstrate superior performance over competing methods.Comment: 37 pages, 13 figure
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Robust optimal stopping
This paper studies the optimal stopping problem in the presence of
model uncertainty (ambiguity). We develop a method to practically solve this
problem in a general setting, allowing for general time-consistent ambiguity
averse preferences and general payoff processes driven by jump-diffusions.
Our method consists of three steps. First, we construct a suitable Doob
martingale associated with the solution to the optimal stopping problem using
backward stochastic calculus. Second, we employ this martingale to construct
an approximated upper bound to the solution using duality. Third, we
introduce backward-forward simulation to obtain a genuine upper bound to the
solution, which converges to the true solution asymptotically. We analyze the
asymptotic behavior and convergence properties of our method. We illustrate
the generality and applicability of our method and the potentially
significant impact of ambiguity to optimal stopping in a few examples
Robust optimal stopping
This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem represented by the Snell envelope using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples
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Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options
The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points
Extensions of the Cross-Entropy Method with Applications to Diffusion Processes and Portfolio Losses
Rare event simulation is a crucial part of simulations. In financial mathematics, the study of rare events appear naturally when we consider risk measures such as the conditional value at risk. This thesis is composed of three related papers treating the rare event simulations subject: the first paper addresses rare event simulations using for diffusion processes, the second paper addresses rare event simulations for the normal and the Student t-copula model while the last paper addresses rare event simulations for a portfolio model where there is a correlation structure between the loss-given-default and the probability of default
Efficient simulation of clustering jumps with CIR intensity
We introduce a broad family of generalised self-exciting point processes with CIR-type intensities, and develop associated algorithms for their exact simulation. The underlying models are extensions of the classical Hawkes process, which already has numerous applications in modelling the arrival of events with clustering or contagion effect in finance, economics and many other fields. Interestingly, we find that the CIR-type intensity together with its point process can be sequentially decomposed into simple random variables, which immediately leads to a very efficient simulation scheme. Our algorithms are also pretty accurate and flexible. They can be easily extended to further incorporate externally-excited jumps, or, to a multidimensional framework. Some typical numerical examples and comparisons with other well known schemes are reported in detail. In addition, a simple application for modelling a portfolio loss process is presented