68 research outputs found
Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift
In this paper, we first establish well-posedness results for one-dimensional
McKean-Vlasov stochastic differential equations (SDEs) and related particle
systems with a measure-dependent drift coefficient that is discontinuous in the
spatial component, and a diffusion coefficient which is a Lipschitz function of
the state only. We only require a fairly mild condition on the diffusion
coefficient, namely to be non-zero in a point of discontinuity of the drift,
while we need to impose certain structural assumptions on the
measure-dependence of the drift. Second, we study fully implementable
Euler-Maruyama type schemes for the particle system to approximate the solution
of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong
convergence results in terms of the number of time-steps and number of
particles. Due to the discontinuity of the drift, the convergence analysis is
non-standard and the usual strong convergence order known for the
Lipschitz case cannot be recovered for all schemes.Comment: 33 pages, 4 figures, revised introduction and Section
An Offline Learning Approach to Propagator Models
We consider an offline learning problem for an agent who first estimates an
unknown price impact kernel from a static dataset, and then designs strategies
to liquidate a risky asset while creating transient price impact. We propose a
novel approach for a nonparametric estimation of the propagator from a dataset
containing correlated price trajectories, trading signals and metaorders. We
quantify the accuracy of the estimated propagator using a metric which depends
explicitly on the dataset. We show that a trader who tries to minimise her
execution costs by using a greedy strategy purely based on the estimated
propagator will encounter suboptimality due to so-called spurious correlation
between the trading strategy and the estimator and due to intrinsic uncertainty
resulting from a biased cost functional. By adopting an offline reinforcement
learning approach, we introduce a pessimistic loss functional taking the
uncertainty of the estimated propagator into account, with an optimiser which
eliminates the spurious correlation, and derive an asymptotically optimal bound
on the execution costs even without precise information on the true propagator.
Numerical experiments are included to demonstrate the effectiveness of the
proposed propagator estimator and the pessimistic trading strategy.Comment: 12 figure
A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems
A PDE-based accelerated gradient algorithm is proposed to seek optimal
feedback controls of McKean-Vlasov dynamics subject to nonsmooth costs, whose
coefficients involve mean-field interactions both on the state and action. It
exploits a forward-backward splitting approach and iteratively refines the
approximate controls based on the gradients of smooth costs, the proximal maps
of nonsmooth costs, and dynamically updated momentum parameters. At each step,
the state dynamics is realized via a particle approximation, and the required
gradient is evaluated through a coupled system of nonlocal linear PDEs. The
latter is solved by finite difference approximation or neural network-based
residual approximation, depending on the state dimension. Exhaustive numerical
experiments for low and high-dimensional mean-field control problems, including
sparse stabilization of stochastic Cucker-Smale models, are presented, which
reveal that our algorithm captures important structures of the optimal feedback
control, and achieves a robust performance with respect to parameter
perturbation.Comment: Add Sections 2.3 and 2.4 for theoretical convergence result
Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise
In this paper, we first establish well-posedness of McKean-Vlasov stochastic
differential equations (McKean-Vlasov SDEs) with common noise, possibly with
coefficients having super-linear growth in the state variable. Second, we
present stable time-stepping schemes for this class of McKean-Vlasov SDEs.
Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for
an interacting particle system associated with the McKean-Vlasov equation. We
prove stability and strong convergence of order and , respectively. To
obtain our main results, we employ techniques from calculus on the Wasserstein
space. The proof for the strong convergence of the tamed Milstein scheme only
requires the coefficients to be once continuously differentiable in the state
and measure component. To demonstrate our theoretical findings, we present
several numerical examples, including mean-field versions of the stochastic
volatility model and the stochastic double well dynamics with
multiplicative noise.Comment: 36 pages, 3 figure
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