Study BSΔE on lattices for asset price analysis.
problem Optimal investment and market equilibrium analysis in asset price models.
method Backward stochastic difference equations on lattices.
result Applications to optimal investment and market equilibrium analysis.
The study analyzes implicit biases in neural networks using backward error analysis.
problem Analyzing implicit biases in multitask and continual learning settings.
method Backward error analysis to compute implicit training biases, deriving modified losses with three terms.
result The conflict term, measuring gradient alignment, is a new quantity in continual learning.
We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast F…
SGD converges with perturbed forward-backward passes, explained by geometric amplification.
problem Analyzing convergence of SGD with perturbed forward-backward passes in composite optimization.
method Characterized propagation and amplification of perturbations, derived convergence guarantees for non-convex and PL objectives.
result Perturbations cascade through the computational graph, affecting convergence order under specific conditions.
Solves wealth maximization problem using variational analysis.
problem Maximizing expected utility of terminal wealth.
method Variational analysis, forward-backward stochastic differential equation (FBSDE).
result Characterization and solutions for various utility functions.
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
We establish existence, uniqueness and regularity of solution results for a class of backward stochastic partial differential equations with singular terminal condition. The equation describes the value function of non-Markovian stochastic optimal control problem in which the terminal state of the controlled process is…
The detrending moving average (DMA) algorithm is a widely used technique to quantify the long-term correlations of non-stationary time series and the long-range correlations of fractal surfaces, which contains a parameter θ determining the position of the detrending window. We develop multifractal detrending moving a…
Kernel learning FBSDE filter improves nonlinear filtering efficiency.
problem Nonlinear filtering problem in high-dimensional systems.
method Iterative and adaptive meshfree approach using forward backward SDE and KDE.
result Rigorous convergence analysis provided, supporting empirical results.
The study examines backward compatibility issues in ML systems, especially with noisy data.
problem Backward compatibility challenges in ML systems, especially with noisy data.
method Empirical analysis of ML systems across different architectures and datasets, focusing on data shifts and noise.
result Backward compatibility issues arise even without data shift due to optimization stochasticity and training on large-scale noisy datasets can significantly decrease compatibility.
A new method detects changes in data sequences by comparing backward and forward confidence sequences.
problem Detecting changes in data sequences over time.
method Constructing and comparing backward and forward confidence sequences.
result The method successfully detects changes with strong nonasymptotic guarantees.
Unified framework for inference in complex nonlinear processes.
problem Challenges in inferring nonlinear continuous stochastic processes with sparse observations and complex topologies.
method Neural Backward Filtering Forward Guiding (NBFFG) framework that constructs a variational posterior using a proxy linear-Gaussian process.
result Empirical results show NBFFG outperforms baselines on synthetic benchmarks and high-dimensional phylogenetic analysis tasks.
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of refining time-grids to reduce statistical approximation errors in an adaptive and…
FBSJNN solves PIDEs and FBSDEJs with deep learning, offering theoretical and numerical efficiency.
problem Solving Partial Integro-Differential Equations and Forward-Backward Stochastic Differential Equations with Jumps.
method FBSJNN framework using a single neural network for both solution approximation and non-local integral.
result FBSJNN achieves numerical solutions with a relative error of 10−3, demonstrating efficiency. In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimisation algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.
In this paper, we consider solutions of the backward heat equation with Ricci flow on manifolds as a type of infinite dimensional limit of solutions of a wave equation on a larger manifold with an analysis of wavefront set. Specifically, the projection of the solution of the wave equation $ \left(\frac{2t}{N} \cdot \fr…
This paper proposes BRIEF, a backward reduction algorithm that explores compact CNN-model designs from the information flow perspective. This algorithm can remove substantial non-zero weighting parameters (redundant neural channels) of a network by considering its dynamic behavior, which traditional model-compaction te…
Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.
problem Nonlinear Forward Backward Stochastic Differential Equations (FBSDE) with terminal conditions.
method Backward deep BSDE method applied to FBSDE with nonlinear generators and random initial conditions.
result Derives exact and Taylor-based approximations for time-stepping nonlinear BSDEs.
Backpropagation algorithm is indispensable for the training of feedforward neural networks. It requires propagating error gradients sequentially from the output layer all the way back to the input layer. The backward locking in backpropagation algorithm constrains us from updating network layers in parallel and fully l…
Deep signature algorithm for pricing path-dependent options.
problem Pricing path-dependent options with complex payoff functions.
method Extended backward scheme for state-dependent FBSDEs with reflections, incorporating signature layer for path-dependent FBSDEs.
result Convergence analysis of the algorithm with explicit dependence on truncation order and neural network approximation errors.
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a…
Effective theory for Transformer initialization improves model performance.
problem Improving performance of Transformers at initialization.
method Effective-theory analysis of signal propagation in wide and deep Transformers.
result Particular width scalings of initialization and training hyperparameters.
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,…
A new algorithm solves high-dimensional nonlinear BSDEs using deep learning.
problem Solving high-dimensional nonlinear backward stochastic differential equations (BSDEs).
method Backward differential deep learning, reformulating BSDEs as differential deep learning problems, using Malliavin calculus, discretizing integrals with Euler-Maruyama method, approximating processes with DNNs, backwardly optimizing DNN parameters.
result The proposed algorithm efficiently approximates solutions and their derivatives for high-dimensional BSDEs.
Gradient-based methods for games suffer from discrete update steps that cause drift, affecting performance.
problem Gradient-based methods for two-player games suffer from drift due to discrete update steps.
method Derived modified continuous dynamical systems to closely follow the discrete dynamics of games.
result Identified distinct components of discretization drift that can alter or destabilize game performance.
The study introduces backward baselines to distinguish past prediction from future prediction in machine learning models.
problem Differentiating between past and future prediction in machine learning models.
method Theoretical, empirical, and normative arguments support a family of simple and efficient statistical tests called backward baselines.
result The study provides a meaningful backward baseline for auditing black-box prediction systems.
The paper extends NUP representations to factor graphs for better estimation.
problem Nontrivial model-based estimation problems.
method Augmenting factor graphs with convex-dual variables and NUP representations; proposing a new iterative algorithm.
result A new dual algorithm for state space problems.
Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
Scalar dynamic risk measures for univariate positions in continuous time are commonly represented as backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensi…
HA-SME models SGD dynamics with Hessian info for better escaping behaviors.
problem Capturing the escaping behaviors of SGD from stationary points.
method HA-SME, a novel SDE with Hessian info in drift and diffusion.
result HA-SME achieves best approximation error and recovers SGD dynamics for quadratics.
A new method ranks and selects features without model fitting.
problem Feature importance measures algorithm-specific and need improvement.
method Integrates global sensitivity analysis with forward selection and backward elimination.
result Demonstrates clear advantage over state-of-the-art methods.
The paper uses neural networks to price complex life insurance contracts with multiple risk factors.
problem Pricing equity-linked life insurance contracts with various stochastic risk factors.
method Assuming hedging to reduce local variance, the price is expressed as a system of non-linear PDEs. Reformulated as a backward SDE with jumps, solved numerically using neural networks.
result Neural networks provide an efficient numerical solution for pricing these complex contracts.
Paper presents IMRCs for evolving tasks with forward and backward learning.
problem Incremental learning of evolving tasks with few samples per task.
method Incremental minimax risk classifiers (IMRCs) that exploit forward and backward learning.
result IMRCs provide significant performance improvement, especially with reduced sample sizes.
Backwards uniqueness proved for flows with asymptotically conical singularities.
problem Proving uniqueness of mean curvature flows with specific singularities.
method Developed new global tools to handle singularities, asymptotic structure, and smooth parts of flows.
result Backwards uniqueness for mean curvature flows with asymptotically conical singularities proved.
In this note, we will show a backwards uniqueness theorem of the mean curvature flow with bounded second fundamental form in arbitrary codimension.
Model for multi-period carbon market pricing with allowances.
problem Carbon market pricing with multiple trading periods and compliance times.
method Singular forward-backward stochastic differential equations (SDEs).
result Value function convergence to infinite period model under certain conditions.
We propose a simple technique for encouraging generative RNNs to plan ahead. We train a "backward" recurrent network to generate a given sequence in reverse order, and we encourage states of the forward model to predict cotemporal states of the backward model. The backward network is used only during training, and play…
In this introductory paper, we discuss how quantitative finance problems under some common risk factor dynamics for some common instruments and approaches can be formulated as time-continuous or time-discrete forward-backward stochastic differential equations (FBSDE) final-value or control problems, how these final val…
In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the origin…
Backward SDEs help price XVA for OTC derivatives.
problem XVA valuation for OTC derivatives with default risk.
method Review and apply BSDEs with random horizon.
result Explicit formula for XVA correction terms.
Recurrent neural networks' hidden state can be reconstructed from its past, providing a theoretical framework for stability and tracking.
problem Hidden-state stability in RNNs
method Backward coherence analysis
result Almost-sure convergence, rates under mixing, interpretable limiting representation, finite pathwise stopping times, and theoretical framework for time-uniform confidence sequences.
New method uses backward SDEs for deep learning uncertainty.
problem Uncertainty quantification in deep learning models.
method Probabilistic machine learning with stochastic neural networks and stochastic optimal control.
result Effectiveness validated through numerical experiments.
New method preserves convergence rates in gradient-based optimization.
problem How to discretize gradient-based optimization systems while preserving stability and convergence rates.
method Geometric framework for dissipative symplectic integration.
result Dissipative symplectic integrators preserve rates of convergence up to a controlled error.
We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.
problem Solving stochastic Volterra equations with singular kernels.
method Establishing connections between SVEs and SPDEs, using stochastic calculus in Hilbert spaces.
result Solutions of SVEs can be expressed in terms of backward Kolmogorov equations.
Backward propagation rules for warped products under Ricci flow.
problem Understanding how warped product structures behave under Ricci flow.
method Establishing sufficient conditions for backward propagation of warped product structures.
result Asymptotically conical shrinkers are multiply-warped products over Einstein manifolds.
New GFlowNet training framework using policy gradients for combinatorial object generation.
problem Training efficiency and robustness in GFlowNet models.
method Policy-dependent rewards and coupled training strategy for forward and backward policies.
result Advanced RL perspectives for robust gradient estimation improve GFlowNet performance.
In this paper, we prove a unique continuation or ``backwards-uniqueness'' theorem for solutions to the Ricci flow. A particular consequence is that the isometry group of a solution cannot expand within the lifetime of the solution.
The paper defines a frequency for mean curvature flow and proves its monotonicity.
problem Backwards uniqueness for solutions of mean curvature flow.
method Defining and proving monotonicity of a parabolic frequency for mean curvature flows.
result Frequency monotonicity implies backwards uniqueness for mean curvature flow solutions.