The paper solves optimal control problems for stochastic delay equations.
problem Optimal control of stochastic delay differential equations.
method Rewriting the problem in an infinite-dimensional Hilbert space, using dynamic programming and viscosity solutions.
result Characterizes the value function as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.
Derives a Feynman-Kac formula for a fixed delay CIR model.
problem Modeling financial processes with fixed delay.
method Proves existence and uniqueness of a strong solution for a specific SDDE.
result Derives a Feynman-Kac type formula leading to an affine bond pricing formula.
We consider that the price of a firm follows a non linear stochastic delay differential equation. We also assume that any claim value whose value depends on firm value and time follows a non linear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a Random Partia…
In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold M endowed with a connection ∇. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise…
The paper models financial asset prices with jumps and evaluates European option prices using numerical methods.
problem Modeling and pricing European options with jumps in delayed stochastic systems.
method Existence, uniqueness, and positivity of solutions to delayed stochastic differential equations with jumps. Application of Fourier transformation for analytical pricing and Monte-Carlo simulation with a logarithmic Euler-Maruyama scheme for numerical approximation.
result The logarithmic Euler-Maruyama scheme provides a positive and convergent method for approximating the solution to the delayed stochastic differential equations with jumps.
Paper develops Euler scheme for fractional delay diff. eqs with additive noise.
problem Developing a consistent Euler-Maruyama scheme for fractional stochastic delay diff. eqs.
method Euler-Maruyama scheme for fractional Brownian motion with additive noise.
result Achieved convergence rate of H+1/2 for smooth delays when H>1/2.
New method solves stochastic control problems with delays using deep learning.
problem Stochastic control problems with delayed control in drift and diffusion.
method Characterization via Riccati PDEs and deep learning scheme.
result Illustrates effect of delay on Markowitz portfolio allocation problem.
In this paper we investigate novel applications of a new class of equations which we call time-delayed backward stochastic differential equations. Time-delayed BSDEs may arise in finance when we want to find an investment strategy and an investment portfolio which should replicate a liability or meet a target depending…
DiffEqFlux.jl integrates neural networks with differential equations.
problem Combining machine learning and differential equations for modeling complex systems.
method Fusing neural networks and differential equations using DiffEqFlux.jl.
result Demonstrates the integration of differential equations into neural networks and vice versa.
In this paper we show that there are applications that transform the movement of a pendulum into movements in R3. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in…
BayTiDe discovers time-delayed differential equations from noisy data.
problem Discovering time-delayed differential equations from data with large delays and noise.
method Bayesian inference with a sparsity-promoting prior.
result BayTiDe accurately identifies time-delayed differential equations with accuracy proportional to data resolution.
This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochasti…
We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of N banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into a…
Delay-SDE-net models time series with memory and uncertainty, outperforming other models.
problem Accurately modeling time series with memory and uncertainty.
method Stochastic delay differential equations (SDDEs) neural network model with aleatoric and epistemic uncertainty.
result The Delay-SDE-net consistently outperforms other models in predicting time series values and uncertainties.
Study on synchronization in financial markets with time delays.
problem Understanding market dynamics and synchronization in financial systems with time delays.
method Examined a system of coupled non-linear delay-differential equations, linearized for small delays, and analyzed collective dynamics using bifurcation diagrams and numerical solutions.
result Demonstrated that limit cycles can be maintained in coupled N-asset models with appropriate parameterization, leading to market synchronization.
In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form represe…
In this paper we consider backward stochastic differential equations with time-delayed generators of a moving average type. The classical framework with linear generators depending on (Y(t),Z(t)) is extended and we investigate linear generators depending on (t1∫0tY(s)ds,t1∫0tZ(s)ds). We…
Proposes neural delay differential equations for stable system identification with partially observed states.
problem Learning stable models for systems with partial or delayed observations.
method Augments states with history, uses neural delay differential equations, and ensures stability through time delay analysis.
result The approach ensures stability of learned models for partially observed systems.
Proposes a deep learning method for solving complex financial games with delays.
problem Financial modeling with multi-agent interactions and delayed effects.
method Parameterizes controls using recurrent neural networks and trains them with modified fictitious play.
result Demonstrates effectiveness on finance problems with known solutions and new problems with derived Nash equilibria.
We propose an optimal portfolio problem in the incomplete market where the underlying assets depend on economic factors with delayed effects, such models can describe the short term forecasting and the interaction with time lag among different financial markets. The delay phenomenon can be recognized as the integral ty…
Improved GRU model with weighted time-delay feedback for long-term dependencies.
problem Modeling long-term dependencies in sequential data.
method Introducing a gated recurrent unit (GRU) with a weighted time-delay feedback mechanism.
result τ-GRU outperforms state-of-the-art models on various tasks.
A new transform links rotating calorons to solutions of a differential equation.
problem Existence and characterization of rotating calorons.
method Formulated a Nahm transform to relate rotating calorons to solutions of a delayed-differential equation.
result Existence of an eight-parameter family of rotating calorons with nontrivial holonomy.
Proposes a new method combining Reservoir Computing and Normalizing Flow for predicting stochastic dynamical systems.
problem Predicting and capturing long-term behaviors of stochastic dynamical systems.
method Data-driven framework combining Reservoir Computing and Normalizing Flow, integrating error modeling and both approaches virtues.
result Successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors.
Neural Laplace models diverse DEs in the Laplace domain for better dynamics.
problem Inadequate ODEs for long-range dependencies and discontinuities.
method Unified framework in Laplace domain, using stereographic map for smoothness.
result Superior performance in diverse DEs, including complex history dependency and abrupt changes.
Asynchronous stochastic gradient descent (ASGD) is a popular parallel optimization algorithm in machine learning. Most theoretical analysis on ASGD take a discrete view and prove upper bounds for their convergence rates. However, the discrete view has its intrinsic limitations: there is no characterization of the optim…
This paper analyzes a hybrid reinsurance and investment game with bounded memory.
problem A hybrid stochastic differential reinsurance and investment game between reinsurer and insurers.
method Stochastic Stackelberg differential subgame and non-zero-sum stochastic differential subgame, using backward induction and dynamic programming.
result Derive equilibrium strategy and value functions explicitly, showing how delay and competition affect strategies.
Unified framework connects physical laws and machine learning.
problem Combining physical laws and machine learning for scientific applications.
method Universal Differential Equations (UDEs) as a unifying framework.
result Wide variety of applications can be efficiently handled through UDE formalism.
HS-FNO models non-Markovian PDEs by learning history and future states.
problem Non-Markovian dynamics where future states depend on past history.
method History-Space Fourier Neural Operator (HS-FNO) for delay and memory-driven PDEs.
result HS-FNO achieves lowest aggregate errors across various PDE families.
The paper introduces a new short rate model with memory components.
problem Modeling short rate dynamics with past values.
method Integrates memory (delay) components into Merton or Vasiček models.
result Analytical solutions for bond prices and forward rates.
This article is an extension of the work of one of us (Coopersmith, 2011) in deriving the relationship between certain interest rates and the inflation rate of a two component economic system. We use the well-known Fisher relation between the difference of the nominal interest rate and its inflation adjusted value to e…
We present a stochastic analysis of a data set consisiting of 10^6 quotes of the US Doller - German Mark exchange rate. Evidence is given that the price changes x(tau) upon different delay times tau can be described as a Markov process evolving in tau. Thus, the tau-dependence of the probability density function (pdf) …
Paper optimizes insurer's investment strategy in a fluctuating market with memory effects.
problem Optimizing insurer's investment in a market with regime switching and noisy memory.
method Formulated as a stochastic differential delay game, solved using BSDE approach.
result Derives analytical solutions for a specific case of a quadratic penalty function.
New algorithm tackles delayed feedback in Lipschitz bandits with sublinear regret.
problem Delayed feedback in Lipschitz bandits.
method Design of algorithms for bounded and unbounded stochastic delays.
result Sublinear regret guarantees for both bounded and unbounded delays.
New method efficiently computes gradients for stochastic differential equations.
problem Computing gradients for stochastic differential equations efficiently.
method Generalized adjoint sensitivity method to stochastic differential equations.
result Time-efficient and memory-efficient computation of gradients with high-order solvers.
Gradient descent with delayed updates converges faster with noise, even when delays are significant.
problem Analyzing convergence of gradient descent with delayed gradients and stochastic noise.
method Novel technique using generating functions for convergence analysis.
result Convergence bounds show that stochastic noise mitigates the negative effects of delays, improving performance.
DSPG improves SPSA for distributed optimization with wireless delays.
problem Optimizing global functions in multi-agent systems with wireless delays and errors.
method Cross-entropy based distributed stochastic approximation algorithm (DSPG) using simultaneous perturbation.
result DSPG reduces biases due to communication delays and maintains convergence rate.
Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
New algorithm tackles stochastic bandits with varying arm-dependent delays.
problem Applying existing algorithms to stochastic delayed bandit settings is restricted by strong assumptions on delay distributions.
method Proposes a simple UCB-based algorithm called PatientBandits that weakens assumptions on delay distributions.
result Provides bounds on regret and performance lower bounds for the PatientBandits algorithm.
Develops a stochastic approach to financial market delays.
problem Modeling delays in financial markets with multiple assets.
method Introduces a general stochastic framework for information and order execution delays.
result Delayed markets maintain fundamental asset pricing theorems and no asymptotic free lunch condition.
We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly …
Approximate Bayesian computation (ABC) using a sequential Monte Carlo method provides a comprehensive platform for parameter estimation, model selection and sensitivity analysis in differential equations. However, this method, like other Monte Carlo methods, incurs a significant computational cost as it requires explic…
DASA speeds up SA with delayed agents, achieving N-fold speedup.
problem Speeding up Stochastic Approximation with asynchronous delays.
method DASA: Delay-Adaptive Multi-Agent Stochastic Approximation algorithm.
result First algorithm with convergence rate dependent on mixing time and average delay.
Extend classical theory of affine processes to path-dependent setting
problem Path-dependent affine processes
method Introduce path-dependent coefficients and provide analytic formulas for their Fourier--Laplace transform
result Define path-dependent affine processes through their exponential-affine Fourier--Laplace transform and establish a characterization theorem
Deep neural networks solve stochastic control problems with delay.
problem Challenges in stochastic control problems with delay due to path-dependence and high dimensions.
method Employing recurrent neural networks (RNNs) to parameterize policies and optimize objectives.
result RNNs, especially LSTMs, efficiently capture path-dependence and outperform feedforward networks in training and performance.
Bayesian inference for stochastic differential equations using Wishart diffusions.
problem Inferring stochastic differential equations for regression and dynamical modeling.
method Bayesian non-parametric approach with semi-parametric Wishart processes.
result Modeling diffusion in stochastic differential equations improves performance and avoids overfitting.
Paper solves complex control problems using novel SDEs.
problem Solving stochastic differential games for nonlinear systems.
method Uses Deep Forward-Backward SDEs with neural networks.
result Numerical solution validated on two example systems.
Clarifies when certain stochastic PDEs have affine solutions.
problem Existence of affine realizations for semilinear SPDEs driven by Lévy processes.
method Analyzes conditions for affine solutions to SPDEs driven by Lévy processes.
result Conditions for the existence of affine realizations are established.
Algorithm samples constrained stochastic differential equations.
problem Sampling stochastic differential equations with complex constraints.
method Pathspace Metropolis-adjusted manifold sampling.
result Demonstrated effectiveness in various constrained conditions.