Paper uses stochastic Perron's method for jump diffusion problems.
problem Analyzing stochastic target problems with unbounded controls.
method Adapting stochastic Perron's method to HJB equations.
result Uniqueness of viscosity solutions and value function.
Study shows how market firm capitalization models converge to stochastic PDE solutions.
problem Understanding convergence of rank-based models with common noise to stochastic PDE solutions.
method Analysis of mean field limit, martingale problem, and pathwise entropy solutions.
result Empirical cumulative distribution function converges to solution of a stochastic PDE under certain conditions.
The paper ensures positivity of solutions to stochastic equations with positive initial data.
problem Ensuring positivity of solutions to stochastic equations with positive initial data.
method Providing sufficient conditions on coefficients for positivity of mild solutions.
result Sufficient conditions for positivity of solutions to stochastic equations.
Exact solutions found for a new SV model with stationary volatility.
problem Finding exact solutions for a new SV model.
method Analytical solutions for transition probability density, option values, and martingale defect.
result First example of an SV model with exact solutions, GBM volatility, and stationary volatility.
We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent δ∈(0,1], the stochastic solution, which represents the price of a European option, is show…
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.
New methods reduce variance in stochastic dual averaging for sparse solutions.
problem Regularized empirical risk minimization problems in machine learning.
method Stochastic dual averaging with variance reduction for sparser solutions.
result Achieve best known convergence rates for both strongly and non-strongly convex regularizers.
New method solves nonseparable stochastic control problems.
problem Nonseparable and non-monotonic stochastic control problems.
method Scenario-decomposition solution framework using progressive hedging algorithm.
result Extends reach of stochastic optimal control.
SON learns SPDE solutions and uncertainty from noisy data.
problem Uncertainty quantification in SPDEs with unknown model uncertainties.
method Combining DeepONet and SNNs, SON models stochasticity and predicts uncertainty.
result SON accurately captures solution structure and quantifies predictive uncertainty.
Optimizes storage valuation using variational analysis.
problem Optimizing intrinsic storage valuation.
method Variational analysis to solve the problem.
result The optimal exercise strategy is revealed, and the solution depends on different constraint types.
Paper develops methods for solving complex stochastic equations using Malliavin calculus.
problem Existence, uniqueness, and regularity of solutions to BSVIEs.
method Malliavin calculus for tackling diagonal processes and nonlinear dependence.
result Developed well-posedness results for BSVIEs, including probabilistic interpretation of PDEs and portfolio optimization.
We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solut…
Study proves convergence of interest rate model approximations.
problem Investigating convergence of stochastic interest rate models.
method Developed analytical tools for true and truncated EM solutions, proving convergence in probability.
result True solution converges in probability to truncated EM solution as step size approaches zero.
Paper uses NMT to predict solutions to stochastic optimization problems quickly.
problem Predicting solutions to stochastic discrete optimization problems under uncertainty.
method Applied a state-of-the-art NMT algorithm with minimal adaptations and hyperparameter tuning.
result NMT can produce accurate solutions in milliseconds with less variability.
Extends XVA valuation under stochastic volatility, characterizing value processes via mild solutions.
problem Valuation of contingent claims in presence of default, collateral, and funding under stochastic volatility.
method Characterizes pre-default value processes via mild solutions to parabolic semilinear PDEs under stochastic volatility.
result Characterizes pre-default value processes via mild solutions to parabolic semilinear PDEs under stochastic volatility, providing sufficient conditions for existence and uniqueness.
Developed a simulation method for 3/2 stochastic volatility model.
problem Pricing options in the 3/2 stochastic volatility model.
method Explicit weak solution for the 3/2 model, using inverse CIR process property.
result Simulation algorithm performance comparable to other methods.
Stochastic binary synapses in neural networks lead to robust solutions.
problem Training low-precision neural networks with stochastic weights.
method Gradient descent on a probability distribution of binary synapses.
result Binary solutions are robust and generalize well, while typical solutions are isolated.
Researchers solve optimal investment in Heston model with stochastic volatility.
problem Optimal investment strategy in markets with stochastic volatility.
method Reduction of optimal control problem to a linear parabolic boundary problem, leading to an explicit solution.
result Exact solution for optimal investment in Heston model.
Study on non-negative solutions for stochastic Volterra equations with jumps.
problem Existence and uniqueness of non-negative solutions for stochastic Volterra equations with jumps and non-Lipschitz coefficients.
method Developed a nonnegative approximation approach and used Yamada--Watanabe approximation technique for convergence proof.
result Established conditions for strong existence and pathwise uniqueness of non-negative solutions.
Existence of solutions to heat equation on Riemannian manifolds proven.
problem Existence of solutions to stochastic heat equations in Riemannian manifolds.
method Using Dirichlet forms on path/loop space.
result Existence of martingale solutions proven.
Asynchronous methods converge as well as synchronous ones for convex optimization.
problem Optimizing convex functions efficiently in parallel systems.
method Completely asynchronous stochastic gradient procedures.
result Achieve optimal convergence rates under similar conditions as synchronous methods.
Study optimizes investment decisions with fixed costs using stochastic control methods.
problem Optimizing irreversible investment decisions with fixed adjustment costs.
method Stochastic impulse control approach, viscosity solutions, quasi-variational inequality.
result Characterization of optimal control and sensitivity analysis in linear case.
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
Bayesian method reduces misclassification errors in ranking Pareto-optimal solutions.
problem Identifying true Pareto-optimal solutions in noisy multiobjective optimization.
method Sequential allocation of extra samples using stochastic kriging to build predictive distributions.
result The proposed method outperforms existing algorithms in reducing misclassification errors.
Clarifies when solutions to stochastic PDEs stay near given subsets.
problem Understanding the proximity of solutions to stochastic PDEs to given subsets.
method Analyzes distance between closed sets and solutions to stochastic PDEs.
result Clarifies conditions for solutions to stay near given subsets.
Optimizes control of infectious disease spread using stochastic methods.
problem Optimizing control of highly infectious diseases like COVID-19.
method Reformulated Hamilton-Jacobi-Bellman equation as stochastic minimum principle, leading to forward-backward stochastic differential equations.
result Numerous numerical solutions presented under various scenarios.
In this paper we are interested in term structure models for pricing zero coupon bonds under rapidly oscillating stochastic volatility. We analyze solutions to the generalized Cox-Ingersoll-Ross two factors model describing clustering of interest rate volatilities. The main goal is to derive an asymptotic expansion of …
The paper links stochastic completeness to uniqueness and nonexistence in fast diffusion equations on manifolds.
problem Uniqueness and nonexistence of bounded solutions in fast diffusion equations.
method Equivalence of stochastic completeness to uniqueness of solutions to nonlinear evolution equations and nonexistence of bounded solutions to elliptic equations.
result Explicit criteria for uniqueness and nonexistence of bounded solutions to fast diffusion equations on manifolds.
Deep learning approximates SPDE solutions from noise trajectories.
problem Approximating solutions to stochastic partial differential equations (SPDEs).
method Uses neural networks to approximate SPDE solutions based on noise realizations.
result Accurately estimates SPDE solutions and functionals like mean and variance.
Study non-Markovian singular control problems with probabilistic representation.
problem Non-Markovian singular stochastic control problems.
method Probabilistic representation using Z−constrained BSDEs. result Solution identified with Z−constrained BSDEs for non-singular underlying process. Paper develops a new algorithm for sparse signal recovery.
problem Sparse signal recovery from noisy observations.
method Iterative Stochastic Optimization using Stochastic Mirror Descent.
result Linear convergence during preliminary phase of the routine.
New method models stochastic systems efficiently from data.
problem Modeling complex spatial interactions from data.
method Cox process representation and machine learning.
result Efficient algorithm for parameter inference and model selection.
Stochastic VB improves nonlinear model inference speed and accuracy.
problem Bayesian inference of nonlinear models from noisy data.
method Stochastic Variational Bayesian (VB) inference for nonlinear models.
result Stochastic VB achieves comparable parameter recovery to analytical solution but is faster.
The paper characterizes optimal solutions for utility optimization with stochastic elements.
problem Optimal portfolio optimization under uncertainty.
method Characterization of fully coupled FBSDEs in terms of BSDEs.
result Explicit examples and methods to quantify incompleteness and find optimal solutions.
Solves optimal control for stochastic processes with absorbing states.
problem Optimal control of stochastic processes with absorbing states.
method Solves through system of partial differential equations.
result Explicit solution for Merton portfolio problem with default probability.
The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…
New theory accelerates stochastic optimization by leveraging local growth rate.
problem First-order stochastic convex optimization convergence rate.
method Developed two accelerated stochastic subgradient methods.
result Optimal iteration complexity of O(1/ε2(1−θ)) for achieving ε-optimal solution. Develops a new method for financial term structure modeling.
problem Analyzing financial term structures with discontinuities.
method Cylindrical stochastic integration approach.
result Establishes a Heath-Jarrow-Morton framework.
New Brownian motion defined in Minkowski normed spaces.
problem Constructing Brownian motion in non-Euclidean spaces.
method Singular McKean--Vlasov stochastic differential equation.
result Pathwise uniqueness of solutions to the stochastic differential equation.
Maximum principle proves positivity of forward rates in stochastic models.
problem Proving positivity of forward rates in stochastic models.
method Maximum principle for mild solutions to SPDEs with Lipschitz coefficients and Wiener noise.
result Sufficient conditions for positivity of forward rates in the Heath-Jarrow-Morton model.
Stochastic cutting planes improve data-driven optimization speed.
problem Data-driven Mixed-Integer Nonlinear Optimization problems.
method Stochastic version of cutting-plane method.
result Stochastic algorithm converges to ε-optimal solution with high probability.
Study proposes a perturbation method for constructing global solutions to DSGE models.
problem Constructing global solutions to dynamic stochastic general equilibrium models (DSGE).
method Perturbation technique around a deterministic path, solving linear rational expectations models with time-varying parameters.
result Global solutions to stochastic models can be constructed if a deterministic path is global.
Improved simulation for path-dependent options without matrix inversion.
problem Evaluating path-dependent options efficiently and accurately.
method Stochastic approximation, explicit solutions to Heston model, importance sampling.
result Up to two orders of magnitude speed improvement over standard Monte Carlo methods.
Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.
problem Existence and qualitative properties of radial solutions on Cartan-Hadamard manifolds.
method Analytical and asymptotic analysis of radial solutions, focusing on critical and supercritical exponents.
result Existence of one-parameter family of radial solutions for critical or supercritical exponents, with different dimensions of existence regions based on stochastic completeness.
Paper finds unique viscosity solution to complex control problems.
problem Complex stochastic control problems with singular terminal state constraints.
method Establishes existence of unique nonnegative continuous viscosity solution using novel comparison principle.
result Unique viscosity solution to HJB equation for linear-quadratic control problems.
Large deviation principles for multivariate stochastic volatility models.
problem Understanding the behavior of log-processes in multivariate stochastic volatility models.
method Establishing a comprehensive sample path large deviation principle for log-processes.
result Asymptotic formulas for first exit times and barrier option prices derived from the LDP.
The article models illiquid stocks using quantum calculus with asymptotic methods.
problem Modeling illiquid financial markets.
method Application of quantum stochastic calculus and asymptotic methods.
result Power series solutions can approximate quantum stochastic processes for longer time frames.
New methods optimize complex optimization problems with improved efficiency.
problem Optimizing complex problems with a convex lower-level objective.
method Uses stochastic cutting planes and conditional gradient updates.
result Improves complexity for both convex and non-convex upper-level functions.