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Interest rate models with Markov chains
Imperial Users onl
A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization
We propose a probabilistic numerical algorithm to solve Backward Stochastic
Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs
introduced in [9] for representing fully nonlinear HJB equations. In
particular, this allows us to numerically solve stochastic control problems
with controlled volatility, possibly degenerate. Our backward scheme, based on
least-squares regressions, takes advantage of high-dimensional properties of
Monte-Carlo methods, and also provides a parametric estimate in feedback form
for the optimal control. A partial analysis of the error of the scheme is
provided, as well as numerical tests on the problem of superreplication of
option with uncertain volatilities and/or correlations, including a detailed
comparison with the numerical results from the alternative scheme proposed in
[7]
Quantum trajectories for time-dependent adiabatic master equations
We develop a quantum trajectories technique for the unraveling of the quantum
adiabatic master equation in Lindblad form. By evolving a complex state vector
of dimension instead of a complex density matrix of dimension ,
simulations of larger system sizes become feasible. The cost of running many
trajectories, which is required to recover the master equation evolution, can
be minimized by running the trajectories in parallel, making this method
suitable for high performance computing clusters. In general, the trajectories
method can provide up to a factor advantage over directly solving the
master equation. In special cases where only the expectation values of certain
observables are desired, an advantage of up to a factor is possible. We
test the method by demonstrating agreement with direct solution of the quantum
adiabatic master equation for -qubit quantum annealing examples. We also
apply the quantum trajectories method to a -qubit example originally
introduced to demonstrate the role of tunneling in quantum annealing, which is
significantly more time consuming to solve directly using the master equation.
The quantum trajectories method provides insight into individual quantum jump
trajectories and their statistics, thus shedding light on open system quantum
adiabatic evolution beyond the master equation.Comment: 17 pages, 7 figure
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