N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". 
In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.

/pdf/universal-differential-equations-for-scientific-machine-7df1mbim0x.pdf

Universal Differential Equations for Scientific Machine Learning

Optim provides a range of optimization capabilities written in the Julia programming language (Bezanson et al. 2017). Our aim is to enable researchers, users, and other Julia packages to solve optimization problems without writing such algorithms themselves. The package supports optimization on manifolds, functions of complex numbers, and input types such as arbitrary precision vectors and matrices. We have implemented routines for derivative free, first-order, and second-order optimization methods. The user can provide derivatives themselves, or request that they are calculated using automatic differentiation or finite difference methods. The main focus of the package has currently been on unconstrained optimization, however, box-constrained optimization is supported, and a more comprehensive support for constraints is underway. Similar to Optim, the C library NLopt (Johnson 2008) contains a collection of nonlinear optimization routines. In Python, scipy.optimize supports many of the same algorithms as Optim does, and Pymanopt (Townsend, Niklas, and Weichwald 2016) is a toolbox for manifold optimization. Within the Julia community, the packages BlackBoxOptim.jl and Optimize.jl provide optimization capabilities focusing on derivative-free and large-scale smooth problems respectively. The packages Convex.jl and JuMP.jl (Dunning, Huchette, and Lubin 2017) define modelling languages for which users can formulate optimization problems. In contrast to the previously mentioned optimization codes, Convex and JuMP work as abstraction layers between the user and solvers from a other packages.

/pdf/optim-a-mathematical-optimization-package-for-julia-5fe8r8dah3.pdf

Optim: A mathematical optimization package for Julia

Machine learning plays a role in many deployed decision systems, often in ways that are difficult or impossible to understand by human stakeholders. Explaining, in a human-understandable way, the relationship between the input and output of machine learning models is essential to the development of trustworthy machine-learning-based systems. A burgeoning body of research seeks to define the goals and methods of explainability in machine learning. In this paper, we seek to review and categorize research on counterfactual explanations, a specific class of explanation that provides a link between what could have happened had input to a model been changed in a particular way. Modern approaches to counterfactual explainability in machine learning draw connections to the established legal doctrine in many countries, making them appealing to fielded systems in high-impact areas such as finance and healthcare. Thus, we design a rubric with desirable properties of counterfactual explanation algorithms and comprehensively evaluate all currently-proposed algorithms against that rubric. Our rubric provides easy comparison and comprehension of the advantages and disadvantages of different approaches and serves as an introduction to major research themes in this field. We also identify gaps and discuss promising research directions in the space of counterfactual explainability.

Counterfactual Explanations for Machine Learning: A Review.

We investigate algorithms for different steps in the decision making process, focusing on systems where we are uncertain about the outcomes but can quantify how probable they are using random variables. Any decision one makes in such a situation leads to a distribution of outcomes and requires a way to evaluate a decision. The standard approach is to marginalise the distribution of outcomes into a single number that tries in some way to summarise the value of each decision. After selecting a marginalisation approach, mathematicians and decision makers focus their analysis on the marginalised value but ignore the distribution. We argue that we should also be investigating the implications of the chosen mathematical approach for the whole distribution of outcomes. We illustrate the effect different mathematical formulations have on the distribution with one-stage and sequential decision problems. We show that different ways to marginalise the distributions can result in very similar decisions but each way has a different complexity and computational cost. It is often computationally intractable to approximate optimal decisions to high precision and much research goes into developing algorithms that are suboptimal in the marginalised sense, but work within the computational budget available. If the performance of these algorithms is evaluated they are mainly judged based on the marginalised values, however, comparing the performance using the full distribution provides interesting information: We provide numerical examples from dynamic pricing applications where the suboptimal algorithm results in higher profit than the optimal algorithm in more than half of the realisations, which is paid for with a more significant underperformance in the remaining realisations. All the problems discussed in this thesis lead to continuous optimisation problems. We develop a new algorithm that can be used on top of existing optimisation algorithms to reduce the cost of approximating solutions. The algorithm is tested on a range of optimisation problems and is shown to be competitive with existing methods.

Algorithms for decision making

Estimation of parameters is a crucial part of model development. When models are deterministic, one can minimise the fitting error; for stochastic systems one must be more careful. Broadly parameterisation methods for stochastic dynamical systems fit into maximum likelihood estimation- and method of moment-inspired techniques. We propose a method where one matches a finite dimensional approximation of the Koopman operator with the implied Koopman operator as generated by an extended dynamic mode decomposition approximation. One advantage of this approach is that the objective evaluation cost can be independent the number of samples for some dynamical systems. We test our approach on two simple systems in the form of stochastic differential equations, compare to benchmark techniques, and consider limited eigen-expansions of the operators being approximated. Other small variations on the technique are also considered, and we discuss the advantages to our formulation.

Operator Fitting for Parameter Estimation of Stochastic Differential Equations

When randomness in demand affects the sales of a product, retailers use dynamic pricing strategies to maximize their profits. In this article, we formulate the pricing problem as a continuous-time stochastic optimal control problem and find the optimal policy by solving the associated Hamilton-Jacobi-Bellman (HJB) equation. We propose a new approach to modelling the randomness in the dynamics of sales based on diffusion processes. The model assumes a continuum approximation to the stock levels of the retailer which should scale much better to large-inventory problems than the existing Poisson process models in the revenue management literature. The diffusion process approach also enables modelling of the demand volatility, whereas Poisson process models do not. 
We present closed-form solutions to the HJB equation when there is no randomness in the system. It turns out that the deterministic pricing policy is near-optimal for systems with demand uncertainty. Numerical errors in calculating the optimal pricing policy may, in fact, result in a lower profit on average than with the heuristic pricing policy.

Dynamic pricing in retail with diffusion process demand

Recent high-dimensional single-cell technologies such as mass cytometry are enabling time series experiments to monitor the temporal evolution of cell state distributions and to identify dynamically important cell states, such as fate decision states in differentiation. However, these technologies are destructive, and require analysis approaches that temporally map between cell state distributions across time points. Current approaches to approximate the single-cell time series as a dynamical system suffer from too restrictive assumptions about the type of kinetics, or link together pairs of sequential measurements in a discontinuous fashion. We propose Dynamic Distribution Decomposition (DDD), an operator approximation approach to infer a continuous distribution map between time points. On the basis of single-cell snapshot time series data, DDD approximates the continuous time Perron-Frobenius operator by means of a finite set of basis functions. This procedure can be interpreted as a continuous time Markov chain over a continuum of states. By only assuming a memoryless Markov (autonomous) process, the types of dynamics represented are more general than those represented by other common models, e.g., chemical reaction networks, stochastic differential equations. Furthermore, we can a posteriori check whether the autonomy assumptions are valid by calculation of prediction error—which we show gives a measure of autonomy within the studied system. The continuity and autonomy assumptions ensure that the same dynamical system maps between all time points, not arbitrarily changing at each time point. We demonstrate the ability of DDD to reconstruct dynamically important cell states and their transitions both on synthetic data, as well as on mass cytometry time series of iPSC reprogramming of a fibroblast system. We use DDD to find previously identified subpopulations of cells and to visualise differentiation trajectories. Dynamic Distribution Decomposition allows interpretation of high-dimensional snapshot time series data as a low-dimensional Markov process, thereby enabling an interpretable dynamics analysis for a variety of biological processes by means of identifying their dynamically important cell states.

/pdf/dynamic-distribution-decomposition-for-single-cell-snapshot-gq5hfpvfae.pdf

Asbjørn Nilsen Riseth

Papers

Optim: A mathematical optimization package for Julia

Algorithms for decision making

Operator Fitting for Parameter Estimation of Stochastic Differential Equations

Dynamic pricing in retail with diffusion process demand

Dynamic distribution decomposition for single-cell snapshot time series identifies subpopulations and trajectories during iPSC reprogramming.