Interest rate models with Markov chains

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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 $N$ instead of a complex density matrix of dimension $N^2$, 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 $N$ 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 $N^2$ is possible. We test the method by demonstrating agreement with direct solution of the quantum adiabatic master equation for $8$-qubit quantum annealing examples. We also apply the quantum trajectories method to a $16$-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