23,583 research outputs found
Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing
Branching processes are a class of continuous-time Markov chains (CTMCs) with
ubiquitous applications. A general difficulty in statistical inference under
partially observed CTMC models arises in computing transition probabilities
when the discrete state space is large or uncountable. Classical methods such
as matrix exponentiation are infeasible for large or countably infinite state
spaces, and sampling-based alternatives are computationally intensive,
requiring a large integration step to impute over all possible hidden events.
Recent work has successfully applied generating function techniques to
computing transition probabilities for linear multitype branching processes.
While these techniques often require significantly fewer computations than
matrix exponentiation, they also become prohibitive in applications with large
populations. We propose a compressed sensing framework that significantly
accelerates the generating function method, decreasing computational cost up to
a logarithmic factor by only assuming the probability mass of transitions is
sparse. We demonstrate accurate and efficient transition probability
computations in branching process models for hematopoiesis and transposable
element evolution.Comment: 18 pages, 4 figures, 2 table
A sparse grid approach to balance sheet risk measurement
In this work, we present a numerical method based on a sparse grid
approximation to compute the loss distribution of the balance sheet of a
financial or an insurance company. We first describe, in a stylised way, the
assets and liabilities dynamics that are used for the numerical estimation of
the balance sheet distribution. For the pricing and hedging model, we chose a
classical Black & Scholes model with a stochastic interest rate following a
Hull & White model. The risk management model describing the evolution of the
parameters of the pricing and hedging model is a Gaussian model. The new
numerical method is compared with the traditional nested simulation approach.
We review the convergence of both methods to estimate the risk indicators under
consideration. Finally, we provide numerical results showing that the sparse
grid approach is extremely competitive for models with moderate dimension.Comment: 27 pages, 7 figures. CEMRACS 201
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