Study deep neural nets for solving complex insurance equations.
problem Solving linear and semilinear parabolic PIDEs in high dimensions.
method Deep neural network algorithms for integro-differential equations.
result Viability of deep learning for solving high-dimensional PIDEs.
Method uses neural networks to fit nonlinear operators from data.
problem Finding nonlinear integro-differential operators from data.
method Parametrizes spatial operator with neural networks and Fourier transforms.
result Can recover spatial operators in fractional heat and Kuramoto-Sivashinsky equations.
Study on survival probability of insurance companies using integro-differential equations.
problem Survival probability of insurance companies over infinite time.
method Analytical and numerical methods for solving integro-differential equations with singularities.
result Existence and uniqueness of solutions to the integro-differential equation.
An unsupervised deep learning method solves PIDEs for option pricing.
problem Solving partial integro-differential equations for financial option pricing.
method Employing unsupervised deep learning to directly solve PIDEs without requiring labeled data.
result An unsupervised neural network accurately solves PIDEs and calculates derivatives and integrals.
Physics-informed neural networks approximate diffusion process pdfs efficiently.
problem Approximating the probability density function of diffusion processes.
method Physics-informed neural networks solving Fokker-Planck or integro-differential equations.
result Neural network solutions approximate target solutions for various types of differential equations.
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. This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
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.
Study of financial models using PIDEs with and without market liquidity.
problem Financial models under illiquid markets and their PIDEs.
method Investigation of linear and nonlinear PIDEs, including Lévy processes, using abstract semilinear parabolic equation theory.
result Existence and uniqueness of solutions to PIDEs for admissible Lévy measures.
New method uses PINNs to efficiently compute Gerber-Shiu functions.
problem Calculating the Gerber-Shiu function efficiently.
method Physics-informed neural networks (PINNs) embedded with differential equations.
result Demonstrates good performance in approximating Gerber-Shiu functions.
Mean-field approximations simplify insurance liability calculations.
problem High-dimensional system of equations makes insurance liability calculation infeasible.
method Use mean-field model to replace high-dimensional system with a low-dimensional non-linear system.
result Insurance liability converges to mean-field approximation as cohort size increases.
We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. A uniqueness theorem is given for the solutions of this equation. This result generalizes Dupire…
Isogeometric analysis simplifies option pricing with NURBS surfaces.
problem Solving complex option pricing equations.
method Isogeometric analysis using NURBS for numerical solution.
result Small discretization steps yield accurate results.
Study on Langevin dynamics for recovering planted signals in spiked matrix models.
problem Recovering a planted signal in spiked matrix models.
method Path-wise characterization of overlap using integro-differential equations and explicit formula derivation.
result Sharp phase transition in limiting overlap: positive in one regime, zero in another due to injected noise.
New model for electricity pricing captures mean reversion and jumps.
problem Capturing mean reversion and jumps in electricity market prices.
method Exponential functional of a jump Lévy process, partial integro-differential equation (PIDE), finite differences method.
result European option value is the unique viscosity solution of a PIDE.
New method infers hidden states in continuous-time phenomena better than traditional models.
problem Traditional HSMM's are limited to discrete time grids and cannot handle irregularly spaced data.
method Formulated integro-differential forward and backward equations for CTSMC's, introduced scalable Viterbi-type algorithm.
result Efficiently solved equations for posterior marginals and path estimates.
A new method calibrates jump-diffusion models from option prices.
problem Calibrating jump-diffusion models from market data.
method Forward Dupire-type PIDE, Tikhonov regularization.
result Robust method for identifying local volatility and jump size.
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 calculates fair strike for variance swaps on time-changed Markov processes.
problem Calculating fair strike for variance swaps on time-changed Markov processes.
method Proving the fair strike equals the price of a European contract and solving the integro-differential equation.
result The fair strike for variance swaps can be computed explicitly for certain Markov processes.
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Lévy models by solving partial integro differential equations…
In this paper we derive an effective equation for derivative pricing which accounts for the presence of virtual arbitrage opportunities and their elimination by the market. We model the arbitrage return by a stochastic process and find an equation for the average derivative price. This is an integro-differential equati…
We consider an integro-differential equation derived from a system of coupled parabolic PDE and an ODE which describes an European option pricing with liquidity shocks. We study the well-posedness and prove comparison principle for the corresponding initial value problem.
We analyze deep neural networks in the large size and iteration limit, revealing a deterministic system of equations.
problem Understanding the behavior of deep neural networks in the asymptotic regime of large network sizes and iterations.
method Sequential limit of each hidden layer and characterization of parameter evolution, using weak convergence and stochastic analysis.
result The limit neural network recovers a global minimum with zero loss for the objective function.
Generalizes Hasimoto transformation to arbitrary flows on space curves.
problem Modeling and analyzing wave motions on space curves.
method Develops a mapping between curve evolution and scalar equations, transforming general vector fields in the Frenet frame.
result Binormal flows are generally length preserving but bending energy is fragile.
The paper studies value adjustments and dynamic hedging for reinsurance counterparty risk.
problem Reinsurance counterparty credit risk (RCCR) and its impact on insurance companies.
method A novel model accounting for contagion effects, characterized value adjustment via PIDE, derived hedging strategies using quadratic method.
result Dynamic hedging strategies can significantly reduce reinsurance counterparty risk.
Machine learning finds linear equations from noisy data.
problem Discovering conservation laws from noisy data.
method Gaussian process priors modified for differential operators.
result Parameters of linear equations inferred from data.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We apply Gauge Theory of Arbitrage (GTA) {hep-th/9710148} to derivative pricing. We show how the standard results of Black-Scholes analysis appear from GTA and derive correction to the Black-Scholes equation due to a virtual arbitrage and speculators reaction on it. The model accounts for both violation of the no-arbit…
A method for pricing two-asset options using a finite element approach for Levy processes.
problem Pricing two-asset options with Levy process under exponential model.
method Finite element method (FEM) for a partial integro-differential equation (PIDE).
result Good performance of the proposed method for pricing two-asset options.
The paper efficiently solves a complex option valuation equation for two assets.
problem Valuation of European options under a two-asset Kou jump-diffusion model.
method Extends an efficient algorithm for a one-dimensional integral to a two-dimensional one, using operator splitting schemes for time discretization.
result The method achieves optimal computational cost and stable convergence for various operator splitting schemes.
Develops a PD estimation model using Lévy-driven processes for credit risk.
problem Estimating Probability of Default under new IFRS 9 regulations.
method Lévy-driven Ornstein-Uhlenbeck process with multiple latent variables, Integral Equation and PIDE formulation.
result Existence of weak and strong solutions for PD function, numerical schemes developed.
We find approximate solutions of partial integro-differential equations, which arise in financial models when defaultable assets are described by general scalar Lévy-type stochastic processes. We derive rigorous error bounds for the approximate solutions. We also provide numerical examples illustrating the usefulness a…
Paper analyzes multidimensional PIDEs for financial modeling, proving existence and uniqueness in Bessel spaces.
problem Analyzing solutions of non-local nonlinear PIDEs in multidimensional spaces.
method Employing abstract semilinear parabolic equations theory in Bessel potential spaces.
result Existence and uniqueness of solutions for a wide class of Lévy measures in multidimensional spaces.
In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [15]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of som…
Paper proves existence and uniqueness of solutions to PIDEs in Bessel spaces for option pricing.
problem Existence and uniqueness of solutions to PIDEs in Bessel spaces.
method Abstract semilinear parabolic equations and Bessel potential spaces.
result Proves existence and uniqueness of solutions in Bessel potential spaces.
In mathematical finance a popular approach for pricing options under some Levy model is to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution while numerical solution bri…
New framework explains neural network bias in solving differential equations.
problem Understanding and controlling the bias in PINNs for differential equations.
method Deriving an integro-differential equation from PINNs and GPR equivalence.
result PINN predictions are influenced by a kernel term reflecting architecture choices.
In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constrai…
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…
New deep learning method for option pricing in jump-diffusion models.
problem Option pricing in jump-diffusion models with high-dimensional assets.
method Implicit-explicit minimizing movement time-stepping approach using deep ANNs.
result Consistent and asymptotically correct solutions for large underlyings.
We formulate the fractional Ricci flow theory for (pseudo) Riemannian geometries enabled with nonholonomic distributions defining fractional integro-differential structures, for non-integer dimensions. There are constructed fractional analogs of Perelman's functionals and derived the corresponding fractional evolution …
The paper develops and tests operator splitting schemes for American options in a complex model.
problem Efficient numerical solution of American options under a two-asset Merton jump-diffusion model.
method Adaptation of IMEX and ADI operator splitting schemes to solve the two-dimensional PIDCP.
result Investigates and compares the convergence and performance of eight operator splitting methods.
Optimal dividend strategy with ratcheting and capital injection under Cramér-Lundberg model.
problem Optimal dividend payout for an insurance company with ratcheting constraints and capital injections.
method Systematic probabilistic and PDE-based approach to solve HJB equation, constructing strong solution and optimal strategy.
result Existence and uniqueness of strong solution, explicit optimal feedback control strategy.
We model the term structure of the forward default intensity and the default density by using Lévy random fields, which allow us to consider the credit derivatives with an after-default recovery payment. As applications, we study the pricing of a defaultable bond and represent the pricing kernel as the unique solution …
We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential…
For α∈(1,2), we present a generalized central limit theorem for α-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, p…
New model for Knightian uncertainty with jumps.
problem Knightian uncertainty and non-linear jumps.
method Probabilistic construction of non-linear affine processes with jumps.
result Tractable model for Knightian uncertainty with sublinear expectations.
We derive a forward equation for arbitrage-free barrier option prices, in terms of Markovian projections of the stochastic volatility process, in continuous semi-martingale models. This provides a Dupire-type formula for the coefficient derived by Brunick and Shreve for their mimicking diffusion and can be interpreted …