We extend Dupire's formula for stochastic interest rates and local volatility.
problem Deriving formulas for stochastic interest rates and local volatility.
method Generalizations of Dupire's formula for stochastic drift and local volatility.
result Validated the limits of the generalized Dupire formulae for specific cases.
Proposes new Monte Carlo methods for calibrating local volatility models with stochastic components.
problem Calibrating local volatility models with stochastic drift and diffusion.
method Developed Monte Carlo algorithms for three models: local volatility with stochastic interest rates, stochastic local volatility with deterministic interest rates, and stochastic local volatility with stochastic interest rates.
result Conditions for the existence of local volatility given European option prices, stochastic interest rate model parameters, and correlations.
Paper improves learning rates for SGD and NAG.
problem Generalization performance of stochastic optimization algorithms.
method Establishes new learning rates for SGD and NAG.
result Improved guarantees in some settings or comparable rates under weaker assumptions.
Developed unbiased estimators for Heston model with stochastic interest rates.
problem Estimating the Heston model with stochastic interest rates.
method Combined unbiased estimators with the Heston model and developed a semi-exact log-Euler scheme.
result Convergence rate of O(h) in the L2 norm for a wide range of models. Improves understanding of stochastic NGVI convergence rates.
problem Lack of knowledge about non-asymptotic convergence rates in stochastic NGVI.
method Proved non-asymptotic convergence rates for conjugate likelihoods and showed implicit optimization for non-conjugate likelihoods.
result First O(T1) non-asymptotic convergence rate for stochastic NGVI in conjugate likelihoods. New dynamics for SGD in small learning rate regime.
problem Improving stochastic gradient descent in small learning rate regime.
method Introducing stochastic modified flows and distribution dependent stochastic modified flows.
result Captures fluctuating dynamics of SGD in small learning rate - infinite width scaling regime.
SALSA automatically adjusts learning rates in stochastic gradient methods.
problem Automatic adjustment of learning rates in stochastic gradient methods.
method SALSA uses a line-search procedure to gradually increase the learning rate, then a statistical test to decrease it.
result SALSA matches the performance of best hand-tuned learning rate schedules in deep learning tasks.
We introduce a tractable multi-currency model with stochastic volatility and correlated stochastic interest rates that takes into account the smile in the FX market and the evolution of yield curves. The pricing of vanilla options on FX rates can be performed effciently through the FFT methodology thanks to the affinit…
Paper establishes convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
problem Analyzing convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
method Novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing length.
result First almost sure convergence rate and maximal concentration bound with exponential tails for contractive stochastic approximation algorithms with Markovian noise.
The paper analyzes insurance risks using stochastic models.
problem Interest rate and variance risks in unit-linked insurance policies.
method General stochastic volatility models and stochastic interest rates are used to price unit-linked life insurance contracts.
result A perfect hedging strategy is provided and compared with the Black-Scholes model.
The paper analyzes convergence rates for SGD and SHB methods.
problem Analyzing convergence rates for stochastic gradient descent and heavy ball methods.
method Stochastic gradient descent and stochastic heavy ball method for general stochastic approximation problems.
result The last iterate of SHB converges almost surely to a minimizer and has faster convergence rates than SGD.
Learning rate annealing improves robustness in stochastic optimization.
problem Tuning learning rates in large-scale models is costly and prone to errors.
method We analyze and demonstrate the benefits of learning rate annealing schemes.
result Stochastic gradient descent with annealed schedules converges more robustly to the optimal solution.
Method calibrates local volatility and stochastic short rate models for equity-rate dynamics.
problem Joint calibration of local volatility and stochastic short rate models.
method Iterative approach using semimartingale optimal transport.
result Demonstrated performance on market data using European SPX options and cap interest rate options.
This work models overnight rates with jumps and discontinuities, extending classical short-rate models.
problem Capturing the jump behavior and discontinuities in overnight rates for accurate modeling.
method Developed a term structure modeling framework based on overnight rates, accommodating stochastic discontinuities.
result Simple specifications can capture the jump behavior of overnight rates, and explicit valuation formulas are provided.
SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.
problem Optimal convergence rates for non-convex functions under gradient noise.
method Geometric interpretation of the PL-condition to analyze convergence rates.
result Convergence rates of SGD and stochastic gradient descent match those of strongly convex quadratics.
Random learning rate improves neural network training without extra cost.
problem Optimizing neural networks with stochastic gradient descent.
method Uniformly-distributed random learning rate in SGD.
result Random learning rate yields better regularization without extra computational cost.
A novel approach models rating transitions using Lie groups and Deep Learning.
problem Modeling rating transitions with geometric properties and stochastic processes.
method Introducing Itô-SDEs on Lie groups, using TimeGAN for calibration, and examining rating matrix properties.
result The geometric approach using Lie groups and Deep Learning generates a good fit for rating transitions.
The paper analyzes how learning rate affects SGD and provides insights into optimal rates.
problem Understanding the impact of learning rate on stochastic gradient descent.
method Developed a learning-rate-dependent stochastic differential equation (lr-dependent SDE) to analyze SGD.
result Established a linear rate of convergence for SGD and found the optimal linear rate by analyzing the spectrum of the Witten-Laplacian.
Enhances valuation of variable annuities with stochastic interest rate models.
problem Valuation and optimal surrender strategies for variable annuities in Lévy models.
method Hybrid numerical method combining tree methods for interest rate modeling and finite difference techniques for asset price.
result Influence of stochastic interest rates on surrender decisions and contract design.
Unified model for equity option pricing and interest-rate risk assessment.
problem Pricing short and medium-term equity options and interest-rate risk.
method Developed a stochastic modeling framework using Heston, Bates, and CIR models, calibrated using Fourier inversion and FFT.
result Calibration stability and convergence of parameter sets across models.
Measures financial resilience using BSDEs and their properties.
problem Measuring financial resilience in dynamic risk environments.
method Developed stochastic calculus for BSDEs with jumps, revealing resilience rate as expectation of generator.
result Resilience rate can be represented as expectation of BSDE generator, revealing properties of dynamic risk measures.
In this paper, we consider the problem of pricing discretely-sampled variance swaps based on a hybrid model of stochastic volatility and stochastic interest rate with regime-switching. Our modelling framework extends the Heston stochastic volatility model by including the CIR stochastic interest rate and model paramete…
In 'A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options', Heston proposes a Stochastic Volatility (SV) model with constant interest rate and derives a semi-explicit valuation formula. Heston also describes, in general terms, how the model could be extended to inc…
Deterministic GD can behave stochastically in large learning rates for multiscale functions.
problem Understanding deterministic GD's stochastic behavior in large learning rates for multiscale objectives.
method Established a sufficient condition for deterministic GD to converge to a rescaled Gibbs distribution in large learning rates for multiscale functions.
result Deterministic GD can converge to a statistical distribution in large learning rates for multiscale functions.
Study optimizes dividend payout strategies under fluctuating interest rates.
problem Maximizing dividends under stochastic interest rates with negative values.
method Analytical HJB approach and backward SDEs for analysis.
result Explicit optimal strategies found for both time-dependent and strategy-independent stopping times.
In this survey paper we discuss recent advances on short interest rate models which can be formulated in terms of a stochastic differential equation for the instantaneous interest rate (also called short rate) or a system of such equations in case the short rate is assumed to depend also on other stochastic factors. Ou…
This paper offers a new class of models of the term structure of interest rates. We allow each instantaneous forward rate to be driven by a different stochastic shock, constrained in such a way as to keep the forward rate curve continuous. We term the process followed by the shocks to the forward curve ``stochastic str…
The paper analyzes convergence rates for stochastic approximation and reinforcement learning.
problem Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise.
method A novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the Moreau-envelope smoothing for contractive mappings.
result Almost sure convergence rates for specific learning rates are derived, with rates arbitrarily close to o(n1−2η) and o(n−1). Study approximates weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
problem Approximating weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
method Used Euler type scheme with integrated kernels to study weak convergence rate.
result Obtained weak convergence rate of min(3α−1,1) for discretised rough Ornstein-Uhlenbeck process and stochastic rough volatility model. 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.
In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate O(ε21) improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of Hazan an…
We formulate and study a general family of (continuous-time) stochastic dynamics for accelerated first-order minimization of smooth convex functions. Building on an averaging formulation of accelerated mirror descent, we propose a stochastic variant in which the gradient is contaminated by noise, and study the resultin…
Stochastic gradient descent (SGD) gives an optimal convergence rate when minimizing convex stochastic objectives f(x). However, in terms of making the gradients small, the original SGD does not give an optimal rate, even when f(x) is convex. If f(x) is convex, to find a point with gradient norm ε, we …
New algorithm optimally identifies best arm in both stochastic and adversarial settings.
problem Best arm identification in stochastic and adversarial reward scenarios.
method Parameter-free algorithm designed to be robust to adversarial rewards and optimal in stochastic problems.
result Algorithm's error rate matches optimal bounds in stochastic problems and is robust to adversarial rewards.
Study compares models for pricing multi-strike quanto call options with SV, SC, and SER.
problem Pricing multi-strike quanto call options with stochastic volatility, correlation, and exchange rates.
method Comparative analysis of SV, SC, and SER models; Monte Carlo simulation; Milstein scheme; antithetic variates; correlation risk parameters.
result GARCH-Jump SV, Weibull SC, and Ornstein Uhlenbeck (OU) SER model combination performs best.
In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence ra…
Stochastic algorithm achieves sublinear convergence for bi-objective optimization.
problem Optimizing two conflicting functions using gradient or subgradient descent.
method Stochastic alternating algorithm with varying steps for each objective.
result Achieves sublinear convergence rate of O(1/T) under strong convexity.
Stochastic version of proximal distance algorithm analyzed and validated.
problem Optimization of constrained estimation problems.
method Stochastic proximal distance algorithm, with convergence guarantees and finite error bounds.
result Convergence guarantees and finite error bounds for the first time.
In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these metho…
SGD works well with large learning rates at the edge of stability.
problem Stochasticity at the edge of stability in deep learning.
method Sharp convergence guarantees for SGD with multiclass cross-entropy loss.
result SGD self-stabilizes, ensuring convergence with large learning rates.
Derives effective continuous dynamics for adaptive SGD methods.
problem Analyzing noise in adaptive SGD methods.
method Stochastic modified equations framework and Malladi's scaling rules.
result Sampling-induced noise in SGD limits to independent Brownian motions.
Motivated by machine learning applications in networks of sensors, internet-of-things (IoT) devices, and autonomous agents, we propose techniques for distributed stochastic convex learning from high-rate data streams. The setup involves a network of nodes---each one of which has a stream of data arriving at a constant …
The paper calculates how fast optimal investment strategies approach CRRA strategies in stochastic factor models.
problem Understanding convergence rates of optimal investment strategies in stochastic factor models.
method Analyzes optimal feedback functions in nonlinear and quadratic term structure models, considering decay of bond prices and power-like utility at high wealth levels.
result Convergence rates of optimal investment strategies to CRRA strategies are determined by bond price decay and power-like utility behavior.
Paper derives convergence rates and confidence intervals for LSA with Markovian noise.
problem Analyzing convergence rates and constructing confidence intervals for LSA with Markovian noise.
method Derives non-asymptotic Berry-Esseen bounds and multiplier block bootstrap procedure.
result Provides O(n−1/4) convergence rates and guarantees consistent inference. We propose a novel and generic calibration technique for four-factor foreign-exchange hybrid local-stochastic volatility models with stochastic short rates. We build upon the particle method introduced by Guyon and Labordère [Nonlinear Option Pricing, Chapter 11, Chapman and Hall, 2013] and combine it with new variance…
Despite the development of numerous adaptive optimizers, tuning the learning rate of stochastic gradient methods remains a major roadblock to obtaining good practical performance in machine learning. Rather than changing the learning rate at each iteration, we propose an approach that automates the most common hand-tun…
Modeling climate change costs with stochastic interest rates shows inequality, but funding abatement can reduce this.
problem Evaluating the costs and benefits of climate change mitigation with uncertain discount rates.
method Amended DICE model with stochastic interest rates and funding abatement costs.
result Introducing funding abatement can reduce intergenerational inequality in climate change costs.
We study the convergence rate of stochastic optimization of exact (NP-hard) objectives, for which only biased estimates of the gradient are available. We motivate this problem in the context of learning the structure and parameters of Ising models. We first provide a convergence-rate analysis of deterministic errors fo…