Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. Our main contributions are: (i) we prove a general tightness theorem for the Skorokhod M 1 -topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) We prove propagation of chaos for a perturbed version of the particle system for general initial conditions. (iii) We prove a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.
We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash in order to limit defaults to a given proportion of entities. We prove that the value of the central agent's control problem converges as the number of defaultable institutions goes to infinity, and that it satisfies a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a forward-backward coupled system of PDEs. Our simulations show that the central agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the terminal losses, the agent optimises a terminal cost function of the equity values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically.
The numerical solution of stochastic partial differential equations and numerical Bayesian estimation is computationally demanding. If the coefficients in a stochastic partial differential equation exhibit symmetries, they can be exploited to reduce the computational effort. To do so, we show that permutation-invariant functions can be approximated by permutation-invariant polynomials in the space of continuous functions as well as in the space of p-integrable functions defined on r0, 1s s for 1 ď p ă 8. We proceed to develop a numerical strategy to compute cubature formulas that exploit permutation-invariance properties related to multisymmetry groups in order to reduce computational work. We show that in a certain sense there is no curse of dimensionality if we restrict ourselves to multisymmetric functions, and we provide error bounds for formulas of this type. Finally, we present numerical results, comparing the proposed formulas to other integration techniques that are frequently applied to high-dimensional problems such as quasi-Monte Carlo rules and sparse grids.
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