Paper introduces a new method for solving complex stochastic equations.
problem Solving forward-backward stochastic differential equations with jumps.
method Linear basis function regression technique.
result The proposed method is convergent and effective as shown by numerical experiments.
Modeling time series with jumps using neural networks and stochastic processes.
problem Capturing the dynamics of time series with both continuous flows and discrete jumps.
method Introducing Neural Jump Stochastic Differential Equations (Neural JSDEs) that extend Neural Ordinary Differential Equations (Neural ODEs) with a stochastic process term.
result Demonstrated the model's predictive capabilities on various datasets, including Hawkes processes, Stack Overflow awards, medical records, and earthquake monitoring.
Abstract reviews Markov processes with jumps on manifolds and Lie groups.
problem Analyzing Markov processes with jumps in geometric settings.
method Stochastic differential equations, Courrège theorem, invariant Markov processes.
result Developments in Lie groups and manifolds under various actions.
Neural jump model improves option pricing accuracy.
problem Jump risk in option pricing.
method Neural jump stochastic differential equation model with Gumbel-Softmax gradient learning.
result Neural jump components significantly improve option pricing accuracy.
Paper proves stability of complex equations under various conditions.
problem Stability of backward stochastic differential equations with jumps.
method General framework for convergent sequences of data and solutions.
result Convergent sequence of solutions for associated data.
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.
Neural networks model financial data with Lévy processes.
problem Forecasting chaotic financial time series with big jumps.
method Lévy-induced stochastic differential equation network approximated by neural networks.
result The method improves prediction accuracy using non-Gaussian Lévy processes.
Study optimizes investment strategies in markets with contagious price jumps.
problem Optimizing portfolios in financial markets with contagious price jumps.
method Applied stochastic maximum principle, backward stochastic differential equations, and linear-quadratic control techniques.
result Obtained efficient strategy and efficient frontier in semi-closed form.
Simplified calculus for stochastic processes simplifies complex financial calculations.
problem Complex stochastic processes in economics and finance.
method Intuitive calculus capturing jumps without explicit measure reference.
result Simplifies calculations involving drifts and expected values.
The paper examines conditions for stochastic invariance of cones in SPDEs with jumps.
problem Stochastic invariance of cones in SPDEs with jumps.
method Sufficient conditions for stochastic invariance of closed convex cones in abstract L2-spaces. result Conditions for stochastic invariance of cones are provided and analyzed.
Study BSDEs with default jump, proving properties and pricing claims.
problem Properties and pricing of BSDEs with default jumps.
method Properties and comparison theorems for BSDEs driven by Brownian motion and martingale measure with default jump.
result Representation of BSDE solutions involving conditional expectation and adjoint exponential semi-martingale.
Extends deep solver to FBSDEs with jumps for option pricing.
problem Solving FBSDEs with jumps for financial applications.
method Discretization, ANN parametrization, reinforcement learning, loss function minimization.
result Successfully applied to option pricing in low and high dimensions.
Proposes a method for approximating transition densities of SDEs driven by gamma processes.
problem Calculating transition densities for SDEs driven by gamma processes.
method Taylor-type approximation and conditional expectation of multiple stochastic integrals.
result Efficiency of the proposed method demonstrated through numerical tests.
The paper provides a formula for pricing volatility swaps with stochastic volatility, jumps, and stochastic intensity.
problem Valuation of volatility swaps in markets with stochastic volatility, jumps, and stochastic intensity.
method The paper uses the stochastic volatility model with jumps and stochastic intensity, and the Feynman-Kac theorem to derive a partial integral differential equation. Discrete and continuous sampled volatility swap pricing formulas are obtained using transform techniques.
result The paper delivers a pricing formula for volatility swaps under stochastic volatility with jumps and stochastic intensity.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
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. Solves optimal stopping problem with Poisson constraints using jumps.
problem Optimal stopping with Poisson constraints and jumps.
method Penalized backward stochastic differential equation (PBSDE) with jumps, decomposition method based on Jacod-Pham, comparison theorem of BSDEs with jumps.
result Solves American option pricing in nonlinear markets with Poisson constraints.
In the present paper we present a finite element approach for option pricing in the framework of a well-known stochastic volatility model with jumps, the Bates model. In this model the asset log-returns are assumed to follow a jump-diffusion model where the jump component consists of a Levy process of compound Poisson …
We consider a generalization of the Heath Jarrow Morton model for the term structure of interest rates where the forward rate is driven by Paretian fluctuations. We derive a generalization of Itô's lemma for the calculation of a differential of a Paretian stochastic variable and use it to derive a Stochastic Differenti…
In this article we extend earlier work on the jump-diffusion risk-sensitive asset management problem [SIAM J. Fin. Math. (2011) 22-54] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the HJB equation is a partial integro-diffe…
Investigates optimal investment strategies in financial markets with jumps.
problem Optimal portfolio selection for investors in multi-asset financial markets with jumps.
method Uses martingale optimality principle and Riccati backward stochastic differential equations with jumps.
result Derives semi-closed form optimal strategies and value function for Merton's problem.
Develops a PIDE framework for option pricing with stochastic volatility and jumps.
problem Option pricing under stochastic volatility and jumps.
method PIDE framework derived from Lévy-type process, implemented via finite-difference discretization with FFT for nonlocal jump operator, calibrated using GMM.
result Stochastic volatility accounts for most pricing improvement, reducing implied-volatility RMSE by 39% compared to Black-Scholes.
A new method for pricing options with stochastic volatility and jumps.
problem Pricing options under stochastic volatility and jumps.
method Fourth-order compact finite-difference scheme with implicit-explicit Crank-Nicolson framework.
result The method achieves near-fourth-order spatial accuracy and up to two orders of magnitude lower runtime than quadratic finite elements.
In this paper we discuss the basket options valuation for a jump-diffusion model. The underlying asset prices follow some correlated local volatility diffusion processes with systematic jumps. We derive a forward partial integral differential equation (PIDE) for general stochastic processes and use the asymptotic expan…
ROMs speed up option pricing under stochastic volatility and jump-diffusion models.
problem Efficiently pricing European and American options under complex stochastic models.
method Reduced order modeling using POD and penalty method for early exercise constraints.
result Pricing with ROMs is orders of magnitude faster than full order models.
We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied bet…
Paper develops semi-analytic method for American options in time-dependent jump-diffusion models.
problem Pricing American options in models with time-dependent and exponential jumps.
method Generalizes existing methods for barrier and American options to handle arbitrary time dependencies and solves the problem through algebraic and Fredholm-Volterra equations.
result Presents a semi-analytic solution for American options in time-dependent jump-diffusion models with exponential jumps.
This paper uses entropy to derive stock price dynamics and option valuation.
problem Deriving stock price dynamics and option valuation from information constraints.
method Develops an entropic inference framework to derive stochastic processes from information constraints, representing price changes through two channels: continuous and jump.
result The derived dynamics is the Merton jump diffusion, with Geometric Brownian Motion as the no jump limit.
Develops a new trading strategy for renewable producers to manage price volatility.
problem Price volatility and imbalance risk in power markets due to renewable generation.
method Data-driven continuous-time stochastic optimal control framework using SDEs and diffusion models.
result Trading strategy outperforms benchmarks and reduces profit and loss.
Enhances RL for jump processes using MSBVE algorithm.
problem Challenges in continuous-time RL with jumps and noise.
method Introduces MSBVE algorithm to minimize quadratic variation error.
result MSBVE algorithm outperforms MSTDE in jump processes.
Optimal insurance surplus management under stochastic interest rates and jumps.
problem Managing insurance surplus with stochastic interest rates and jump-driven liabilities.
method Stochastic control techniques and normalized surplus projection method.
result Optimal investment policy with myopic and hedging components.
Develops a new method to discover stochastic systems with non-Gaussian noise.
problem Discovering governing laws from complex systems with non-Gaussian noise.
method Theoretical framework and numerical algorithm to extract stochastic differential equations with Gaussian and non-Gaussian noise.
result Demonstrated the efficacy and accuracy of the approach on various systems.
Cheshire optimizes social network activity by incentivizing users to post.
problem Maximize overall activity in social networks through user incentives.
method Modelled user actions with Hawkes processes and SDEs with jumps; used stochastic optimal control.
result Optimal incentivized actions are linearly related to current activity levels.
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.
The article derives small-time expansions for jump-diffusion models with infinite jump activity.
problem Analyzing state-dependent jump-diffusion models with infinite jump activity.
method Derives a second-order expansion for tail probabilities and option prices.
result Obtains a second-order expansion for out-of-the-money European call option prices.
Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the …
Develops a new method for pricing GMWBs with jumps and stochastic interest rates.
problem Pricing guaranteed minimum withdrawal benefits (GMWBs) with jumps and stochastic interest rates.
method Combines semi-Lagrangian method with Fourier pricing and Green's function.
result Mathematically demonstrates convergence to the viscosity solution of the HJB-QVI.
The paper provides a representation for dynamic risk measures and capital allocations.
problem Representation of dynamic risk measures and capital allocations under Itô-Lévy model.
method Representation theorem for dynamic capital allocation derived from BSDEs with quadratic-exponential growth.
result Derivation of a capital allocation representation for dynamic entropic risk measure and static coherent risk measure.
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.
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.
New methods estimate Asian option prices more efficiently.
problem Estimating the price of discretely monitored Asian options.
method General multilevel Monte Carlo methods.
result Estimates with standard deviation O(ε) in O(m+(1/ε)2) expected time. A machine learning method for short-maturity options with jumps and stochastic volatility.
problem Short-maturity options with jumps and stochastic volatility.
method Differential machine learning method combining supervision and PIDE-residual penalty.
result Improves jump-term approximation and reduces Greeks errors compared to baselines.
A financial swap reduces skew and fat tails in a portfolio's performance.
problem Managing skew and fat tails in portfolio performance.
method Used a third moment variation swap and partial differential equation approach.
result The hedged portfolio returns are more Gaussian-like with thin-tails.
The paper studies ABSDEs with jumps and various growth drivers.
problem Solving ABSDEs with specific growth conditions.
method Proves existence of unique solution for ABSDEs with jumps and various growth drivers.
result Existence of unique solution for ABSDEs with specific growth conditions.
New method extracts stochastic laws from data, including Lévy noise.
problem Extracting stochastic laws from data with non-Gaussian noise.
method Using normalizing flows to estimate transition density, then applying nonlocal Kramers-Moyal formulas.
result Can learn stochastic differential equations with Lévy motion.
Study geometric step options with jumps, deriving pricing equations and characterizations.
problem Pricing geometric step options in markets with jumps.
method Symmetry and parity relations, partial integro-differential equations, ordinary integro-differential equations.
result Derive semi-analytical pricing results for geometric step options.
Extends stability approach to BSDEs with jumps, providing criteria for existence and uniqueness.
problem Existence and uniqueness of solutions to BSDEs with jumps.
method Monotone stability approach, non-convex generator, non-global Lipschitz conditions.
result Concrete criteria for existence and uniqueness of solutions, comparison, and bounds.