New method reduces errors in pricing and sensitivities for discontinuous payoffs.
problem Errors in pricing and sensitivities for discontinuous payoffs in digital and barrier options.
method Alternative methods for estimating sensitivities, including likelihood ratio and hybrid methods.
result New methods substantially reduce test errors in prices and sensitivities.
Differential ML combines AAD with ML for fast, accurate financial derivatives pricing and risk management.
problem Computational bottlenecks in financial derivatives risk management.
method Novel algorithms using automatic adjoint differentiation (AAD) for training fast, accurate approximations in real-time.
result Convergence guarantees for fast, accurate pricing and risk approximations for arbitrary derivatives instruments.
A new framework models uncertainty in structured temporal data using SDEs and neural networks.
problem Uncertainty quantification in machine learning applications involving structured and temporal data.
method Integrates stochastic differential equations (SDEs) with deep generative models in a variational autoencoder framework.
result Improves uncertainty quantification in machine learning applications involving structured and temporal data.
Efficient estimators for smooth Hilbert-valued parameters with theoretical guarantees.
problem Estimating smooth Hilbert-valued parameters with theoretical guarantees.
method Pathwise differentiable Hilbert-valued parameters, efficient influence functions, regularized one-step estimators.
result Theoretical guarantees for efficient estimators even when nuisance functions are arbitrary.
This paper simplifies hedge ratios in financial models using pathwise algorithmic differentiation.
problem Expensive and unstable computation of hedge ratios from pathwise sensitivities.
method Develops reduced stochastic hedge ratios of the form φ_j^r = Σ_j^r ξ_j^q X_q, retaining sensitivity tensor through empirical averages.
result Two coefficient criteria are introduced to minimize pathwise residuals and satisfy moment equations.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
The Monte Carlo pathwise sensitivities approach is well established for smooth payoff functions. In this work, we present a new Monte Carlo algorithm that is able to calculate the pathwise sensitivities for discontinuous payoff functions. Our main tool is to combine the one-step survival idea of Glasserman and Staum wi…
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
We study the use of the multilevel Monte Carlo technique in the context of the calculation of Greeks. The pathwise sensitivity analysis differentiates the path evolution and reduces the payoff's smoothness. This leads to new challenges: the inapplicability of pathwise sensitivities to non-Lipschitz payoffs often makes …
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
Study cash-flow forecasting for derivatives, aligning with replication strategy and addressing timing frictions.
problem Inconsistencies in cash-flow forecasting under different measures and stochastic payment times.
method Use discounting sensitivities (funding-curve hedge ratios) for replication and propose a liquidity valuation adjustment.
result Aligns forecasting with replication strategy and avoids measure-mixing issues.
Derives adjoint formulas for matrix operations and applies them to specific cases.
problem Computing adjoints for matrix operations and specific matrix types.
method Derives adjoint formulas for matrix operations and applies them to specific cases.
result Closed-form expressions for adjoints in specific matrix types.
A new method in finance without probabilities or integrals.
problem Creating a model-free approach to continuous-time finance.
method Pathwise approach using causal functional calculus and transition principle of Isaacs.
result A fully non-linear path-dependent equation characterizes optimal solutions.
The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.
problem Characterizing self-adjoint extensions of the Laplace-Beltrami operator on α-Grushin manifolds. method Introducing an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity.
result Criterion for essential self-adjointness and determination of several self-adjoint extensions.
We use AD to compute gradients for complex functionals in stochastic model calibration.
problem Computing gradients for functions involving expectations in stochastic models.
method Automatic Adjoint Differentiation and parallelization.
result Faster and easier to implement approaches for gradient computation.
Let Δ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold M. One can consider a pencil of operators $\hPi(Δ)=\{Δ_ł\}$ passing through the operator Δ such that any Δł is a linear differential operator acting on densities of weight ł. This pencil can be iden…
Let M be a complete Riemannian manifold and let Ω∗(M) denote the space of differential forms on M. Let d:Ω∗(M)→Ω∗+1(M) be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V…
New estimator for SDEs is shown to be an adjoint state method.
problem Estimating gradients for overparameterized SDEs efficiently.
method Demonstrates generator gradient estimator as an adjoint state method.
result Generator gradient estimator is an adjoint state method for SDEs.
The paper discovers the family of identically-derived Euclidean one-parameter even-dimensional differential linear operators with unique eigenproperties, which prove to be inherently related to the emergent characterizations of fundamental building blocks of embedded minimal surfaces and the Nitsche conjecture proof.
Quantum machine learning solves high-dimensional PDEs with lower variance and improved accuracy.
problem Approximating solutions to high-dimensional parabolic PDEs.
method Pure Variational Quantum Circuit (VQC) for BSDE approximation, using temporal discretization and Monte Carlo simulation.
result VQC achieves lower variance and improved accuracy in most cases, particularly in highly nonlinear regimes.
Proposes a new method for estimating non-pathwise differentiable functional parameters.
problem Estimating dose-response curves for continuous exposure.
method Targeted Highly Adaptive Lasso (HAL) for non-pathwise differentiable functional parameters.
result The Targeted HAL-MLE achieves dimension-free rates up to log(n) factors and outperforms other methods in simulations.
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…
We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itō calculus. Price trajectories are d-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategie…
Abstracts a construction of boundary triplets for self-adjoint elliptic problems.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
The paper presents a method to infer unknown forcing functions in differential equations using Gaussian processes and adjoints.
problem Inferring unknown forcing functions in differential equations from noisy observations.
method Using adjoint methods to efficiently infer Gaussian process (GP) driven differential equations, with truncated basis expansions of the GP kernel.
result Efficient Bayesian inference of forcing functions modeled as GPs using adjoints, with lower computation than MCMC methods.
Two of the most important areas in computational finance: Greeks and, respectively, calibration, are based on efficient and accurate computation of a large number of sensitivities. This paper gives an overview of adjoint and automatic differentiation (AD), also known as algorithmic differentiation, techniques to calcul…
This work proposes using zero-variance control variates to reduce variance in pathwise gradient estimators for variational inference.
problem Pathwise gradient estimators in variational inference have high variance, leading to inefficient optimization.
method Apply zero-variance control variates to pathwise gradient estimators.
result Zero-variance control variates can significantly reduce the variance of pathwise gradient estimators without requiring complex assumptions.
Efficient inference for adaptive data with directional stability condition.
problem Efficient inference on scalar targets after adaptive data collection.
method Introduces directional stability, a weaker condition than i.i.d. data, and shows asymptotic normality and efficiency of estimators.
result Estimators remain asymptotically normal and semiparametrically efficient under directional stability.
The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential eq…
Pathwise uniqueness shown for specific stochastic equations.
problem Stochastic Volterra equations with singular kernels and Hölder coefficients.
method Established pathwise uniqueness through Hölder continuity of coefficients.
result Pathwise uniqueness and existence of unique strong solutions.
Efficiently differentiate functions of large matrices using new adjoint systems.
problem Differentiating functions of large matrices in scientific and probabilistic machine learning models.
method Deriving and implementing new adjoint systems for Lanczos and Arnoldi iterations in JAX.
result Efficient differentiation of PDEs, Gaussian process models, and Bayesian neural networks.
Proposes DAM for optimizing discrete generative models.
problem Challenges in optimizing discrete generative models.
method Discrete Adjoint Matching (DAM) for discrete state spaces.
result Demonstrates effectiveness on synthetic and mathematical reasoning tasks.
In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form G=21∑1m(Eyi−Ci)2, which often appear in the calibration of stochastic models. { We demonstrate that it allows a perfect SIMD\footnote{Single Input Multiple Data} parallelization and provide its relative co…
Develops pathwise analysis for log-optimal portfolios using rough paths theory.
problem Analyzing stability and approximation of log-optimal portfolios.
method Pathwise approach based on càdlàg rough paths theory.
result Establishes pathwise stability and error estimates for log-optimal portfolios.
Develops portfolio theory without probabilistic analysis, focusing on pathwise decomposition.
problem Ensuring market viability without probabilistic assumptions.
method Uses pathwise decomposition and trend extractors to replace semimartingale decomposition.
result Growth-numéraire and viability equivalences are similar but not identical in pathwise setting.
Second-order optimization speeds up deep hedging for complex options.
problem Hedging exotic options with market frictions in realistic markets.
method Second-order optimization scheme leveraging pathwise differentiability and Kronecker-factoring.
result Our method optimizes the policy in 1/4 the steps of standard optimization.
New measure captures differences across entire distributions of counterfactual outcomes.
problem Capturing differences across entire distributions of counterfactual outcomes.
method Entropic optimal transport measure, statistical functional, smooth transformation of embeddings.
result Established first-order and second-order pathwise differentiability.
A new approach to continuous-time universal portfolios using pathwise Itô calculus.
problem Continuous-time version of Cover's universal portfolio strategies.
method Pathwise Itô calculus approach to establish existence and properties of universal portfolio strategies.
result The universal portfolio strategy's portfolio value process is the average of all values of constant rebalanced strategies.
SNAPO optimizes policies for complex sequential decisions using differentiable simulation.
problem Optimizing policies for high-dimensional, sequential decisions under uncertainty.
method Embeds neural policy in a differentiable simulator, computes gradients efficiently.
result Produces sensitivities at a cost proportional to one reverse pass, regardless of sensitivity count.
We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; …
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs numerically requires the approximation of nested conditional expectations, i.e., it…
ULFS-KDPE estimates parameters efficiently without influence functions.
problem Estimating pathwise differentiable parameters in nonparametric models.
method Kernel debiased plug-in estimator based on universal least favorable submodel.
result Semiparametric efficiency achieved without influence function derivation.
The pathwise coordinate optimization is one of the most important computational frameworks for high dimensional convex and nonconvex sparse learning problems. It differs from the classical coordinate optimization algorithms in three salient features: {\it warm start initialization}, {\it active set updating}, and {\it …
New machine learning methods solve complex PDEs with improved accuracy.
problem Solving fully nonlinear PDEs with convex Hamiltonian.
method Rewriting PDE in dual stochastic control form, estimating optimal feedback control with neural network, approximating value function with neural networks.
result Improved estimation of PDE solution and its derivatives, especially the second derivative.
Framework for pricing waterfall structures using simulation and uncertainty modeling.
problem Pricing complex structured finance instruments under uncertainty.
method Simulation-based uncertainty modeling, calibrated probability distributions, PyTorch implementation, Adjoint Algorithmic Differentiation (AAD).
result Efficient gradient computation for risk sensitivity analysis and optimization.
We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market port…
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
Researchers create a parametrix for resolvents on manifolds with ends.
problem Essential self-adjointness of elliptic symmetric differential operators on manifolds with ends.
method Introduced semiclasical pseudodifferential operators compatible with the end structure.
result Essential self-adjointness of elliptic symmetric differential operators proved.