We study the Schr\"{o}dinger bridge problem (SBP) with nonlinear prior
dynamics. In control-theoretic language, this is a problem of minimum effort
steering of a given joint state probability density function (PDF) to another
over a finite time horizon, subject to a controlled stochastic differential
evolution of the state vector. For generic nonlinear drift, we reduce the SBP
to solving a system of forward and backward Kolmogorov partial differential
equations (PDEs) that are coupled through the boundary conditions, with
unknowns being the "Schr\"{o}dinger factors" -- so named since their product at
any time yields the optimal controlled joint state PDF at that time. We show
that if the drift is a gradient vector field, or is of mixed
conservative-dissipative nature, then it is possible to transform these PDEs
into a pair of initial value problems (IVPs) involving the same forward
Kolmogorov operator. Combined with a recently proposed fixed point recursion
that is contractive in the Hilbert metric, this opens up the possibility to
numerically solve the SBPs in these cases by computing the Schr\"{o}dinger
factors via a single IVP solver for the corresponding (uncontrolled) forward
Kolmogorov PDE. The flows generated by such forward Kolmogorov PDEs, for the
two aforementioned types of drift, in turn, enjoy gradient descent structures
on the manifold of joint PDFs with respect to suitable distance functionals. We
employ a proximal algorithm developed in our prior work, that exploits this
geometric viewpoint, to solve these IVPs and compute the Schr\"{o}dinger
factors via weighted scattered point cloud evolution in the state space. We
provide the algorithmic details and illustrate the proposed framework of
solving the SBPs with nonlinear prior dynamics by numerical examples.

通过使用 Kolmogorov 偏微分方程和近端算法，研究非线性动力学中的 Schr"{o} dinger 桥问题，并提供了数值例子。

针对 Schrödinger 桥问题的 Wasserstein 近端算法：非线性漂移下的密度控制

Wasserstein Proximal Algorithms for the Schrödinger Bridge Problem:  Density Control with Nonlinear Drift

Stochastic differential equations (SDEs) and the Kolmogorov partial
differential equations (PDEs) associated to them have been widely used in
models from engineering, finance, and the natural sciences. In particular, SDEs
and Kolmogorov PDEs, respectively, are highly employed in models for the
approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs,
respectively, can typically not be solved explicitly and it has been and still
is an active topic of research to design and analyze numerical methods which
are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly
all approximation methods for Kolmogorov PDEs in the literature suffer under
the curse of dimensionality or only provide approximations of the solution of
the PDE at a single fixed space-time point. In this paper we derive and propose
a numerical approximation method which aims to overcome both of the above
mentioned drawbacks and intends to deliver a numerical approximation of the
Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse
of dimensionality. Numerical results on examples including the heat equation,
the Black-Scholes model, the stochastic Lorenz equation, and the Heston model
suggest that the proposed approximation algorithm is quite effective in high
dimensions in terms of both accuracy and speed.

本文提出一种针对 Kolmogorov PDEs 的数值逼近方法，旨在克服高维情况下维数诅咒和变量精确性缺乏的问题，且适用于金融衍生品的定价模型。在研究的示例中包括热方程、Black-Scholes 模型、随机 Lorenz 方程和 Heston 模型，实现了高维情况下准确性和速度方面的有效逼近。