This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators…
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48 results for “Feynman--Kac”
The paper generalizes Feynman-Kac formula for volatility uncertainty.
problem Calculating sublinear expectation under volatility uncertainty.
method Generalization of Feynman-Kac formula under different hypotheses.
result G-conditional expectation is a viscosity solution of a nonlinear PDE.
We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, althou…
New methods solve SPDEs for financial derivative pricing.
problem Deriving the price of financial derivatives using SPDEs.
method Developed a conditional Feynman-Kac formula to solve SPDEs.
result Established new numerical methods for mixed Monte-Carlo PDEs.
FKEE estimates expectations without samples, using diffusion bridges and PINNs.
problem Estimating expectations without large sample sizes.
method Diffusion bridge models and Feynman-Kac operator approximation using PINNs.
result Significantly reduces variance and improves efficiency.
New method recovers BSDE from financial data without ergodicity.
problem Discovering probabilistic laws from financial data.
method Stochastic SINDy method under risk-neutral measure.
result Recovery of BSDE from limited financial data.
We prove a Feynman-Kac formula for differential forms satisfying absolute boundary conditions on Riemannian manifolds with boundary and of bounded geometry. We use this to construct harmonic forms out of bounded ones on the universal cover of a compact Riemannian manifold whose geometry displays a positivity prop…
The paper develops a Feynman-Kac formula for perturbations of order ≤ 1 in noncommutative geometry.
problem Analyzing perturbations of order ≤ 1 in noncommutative geometry.
method Develops a Feynman-Kac formula for differential operators of order ≤ 1 on complex metric vector bundles over Riemannian manifolds.
result Explicit Feynman-Kac type formula for holomorphic semigroups generated by .
Functional-analytic method for stochastic parallel transport in bundles.
problem Stochastic parallel transport in Hermitian bundles over Riemannian manifolds.
method Purely functional-analytic construction.
result Obtained a general Feynman-Kac formula in vector bundles.
New method trains partial Bayesian neural networks efficiently.
problem Challenges in approximating multi-modal latent variable distributions in pBNNs.
method Formulates pBNN training as a Feynman--Kac model and uses sequential Monte Carlo samplers.
result Proposed training scheme outperforms state of the art in predictive performance.
New method steers protein design towards desired properties.
problem Challenges in designing proteins with specific structures and properties.
method Feynman-Kac framework applied to RFdiffusion models with guiding potentials.
result Significant improvement in predicted interface energetics and binder designability.
Unified kernel framework extends to stochastic systems, improving numerical stability.
problem Extending kernel methods to stochastic dynamical systems with diffusion.
method Unified kernel framework, Feynman-Kac path-integral representations, collocation-based computational framework.
result Kernel equivalence under uniform ellipticity assumptions and improved numerical stability with moderate diffusion.
Study analyzes derivative-free loss method for solving PDEs and fluid problems.
problem Solving elliptic PDEs and fluid problems using neural networks.
method Derivative-free loss method with Feynman-Kac formulation and stochastic walkers.
result Training loss bias scales with time interval and spatial gradient, inversely with walker size.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet …
Improved diffusion models using energy distillation and sequential Monte Carlo.
problem Training instability and inferior performance in energy parameterized diffusion models.
method Introduced a novel training regime for energy functions through distillation of pre-trained diffusion models, and cast the sampling procedure as a Feynman Kac model.
result Demonstrated improved performance and new sampling techniques.
Study on PDEs in Heston model with unique solution and convergence proof.
problem Analyzing PDEs in the Heston model for financial applications.
method Regularity results, verification theorem, unique viscosity solution, convergence proof.
result Unique viscosity solution for wide initial and source data.
We develop a new model for VIX derivatives with closed-form solutions.
problem VIX derivatives pricing and risk management.
method Data-driven Legendre polynomial model for VIX volatility, deriving analytical series solutions.
result Equal or superior accuracy compared to existing models, offering an efficient alternative.
We prove existence, regularity and a Feynman-Kač representation formula of the strong solution to the free boundary problem arising in the financial problem of the pricing of the American Asian option with arithmetic average.
Study heat profiles and eigenfunctions using Brownian motion.
problem Investigate heat profiles and eigenfunctions of Laplace equations.
method Probabilistic tools based on Brownian motion and Feynman-Kac formulae.
result Supremum norm bounds for ground state Dirichlet eigenfunctions and comparison of maximum temperatures.
Study of mean curvature flows with conical singularities using mathematical techniques.
problem Understanding the dynamics of mean curvature flows near conical singularities.
method Feynman-Kac formula and invariant cone method for noncompact settings.
result Generic initial perturbations avoid conical singularities in mean curvature flows.
The paper develops methods to price and hedge options in path-dependent stock models.
problem Pricing and hedging options under complex stock models.
method Develops a path-dependent PDE for option pricing and differentiability of path-dependent SDE solutions.
result Provides formulas for option Greeks and differentiability of path-dependent SDE solutions.
Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.
problem Time inconsistency in stochastic differential games.
method Proves existence and uniqueness of solutions to nonlocal fully-nonlinear parabolic systems.
result Generalizes stochastic game theory to include time-inconsistent preferences.
The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
problem Gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.
method Establishes Bismut-type formulas and gradient estimates for Feynman--Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature and second fundamental form.
result Derives pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds.
Stochastic delay differential equations (SDDE's) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fix…
Risk-minimization and hedging claims on a jump-diffusion market model, Feynman-Kac Theorem and PIDEq-fin.PR
At first, we solve a problem of finding a risk-minimizing hedging strategy on a general market with ratings. Next, we find a solution to this problem on Markovian market with ratings on which prices are influenced by additional factors and rating, and behavior of this system is described by SDE driven by Wiener process…
Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.
problem Well-posedness of parabolic Anderson model on Riemannian manifolds with rough initial conditions.
method Construct intrinsic Gaussian noises, explore global geometry, use Feynman-Kac formula.
result Show well-posedness with non-positive curvature and conditions on .
New method improves training of PINNs for PDEs by adding noisy supervision terms.
problem Slow or failed convergence of PINNs on challenging PDEs.
method Operator preconditioning using Feynman-Kac supervision and non-asymptotic error bounds.
result Non-asymptotic error bounds for FK-PINNs, showing improved performance over standard PINNs.
Consistency conditions for affine term structure modelscond-mat.other
ATSM are widely applied for pricing of bonds and interest rate derivatives but the consistency of ATSM when the short rate, r, is unbounded from below remains essentially an open question. First, the standard approach to ATSM uses the Feynman-Kac theorem which is easily applicable only when r is bounded from below. Sec…
Study uses G-BSDEs to decompose pricing kernels under robust G-expectation.
problem Long-term decomposition of robust pricing kernels under G-expectation.
method Proposes and analyzes three types of quadratic G-BSDEs to decompose pricing kernels.
result Pricing kernels decomposed into four components: discounting, transitory, symmetric martingale, and volatility uncertainty.
This paper models short rates with jumps using PDEs.
problem Capturing jumps and spikes in interest rates.
method PDE approach for pricing interest rate derivatives.
result Established Feynman-Kač representation and derived solutions.
Deep learning solves high-dimensional PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs numerically.
method Reformulating as a statistical learning problem using Feynman-Kac formula.
result Single neural network learns entire family of PDEs.
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…
Deep neural nets (DNNs) compression is crucial for adaptation to mobile devices. Though many successful algorithms exist to compress naturally trained DNNs, developing efficient and stable compression algorithms for robustly trained DNNs remains widely open. In this paper, we focus on a co-design of efficient DNN compr…
General Equilibrium Amplification and Crisis Vulnerability: Cross-Crisis Evidence from Global Banksecon.EM
This study quantifies systemic importance in global banks using a continuous framework that amplifies localized shocks.
problem Analyzing financial contagion and systemic risk in global banks.
method Developed a continuous framework incorporating geographic proximity and interbank network linkages, using a master equation and Feynman-Kac representation.
result The amplification factor correctly identifies systemically important institutions and predicts crisis outcomes.
Study indifference pricing for insurance policies in a regime-switching market model.
problem Indifference pricing of pure endowment policies in a stochastic-factor model with different economic regimes.
method Stochastic control approach based on Hamilton-Jacobi-Bellman equation, Feynman-Kac formula, and sensitivity analysis.
result Characterization of indifference price as a solution to a linear PDE and a backward PDE.
AFT combines AIS, SMC, and NFs for better Monte Carlo estimates.
problem Estimating normalizing constants of complex probability distributions.
method Annealed Flow Transport (AFT) integrates AIS, SMC, and normalizing flows.
result AFT improves Monte Carlo estimates of normalizing constants and expectations.
We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as ass…
This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman-Kac formula. In particular, it is proved that the complexi…
Deep density methods improve filtering in high-dimensional systems.
problem Nonlinear filtering in high-dimensional systems.
method Two deep density methods based on Feynman-Kac formulas and neural networks.
result Logarithmic deep backward stochastic differential equation filter outperforms classical methods in high dimensions.
FlowKac solves high-dimensional Fokker-Planck equations efficiently.
problem Intractability of Fokker-Planck equation solutions in high dimensions.
method Reformulates Fokker-Planck using Feynman-Kac, adaptive stochastic sampling, and normalizing flows.
result Significant computational efficiency and accuracy improvements over existing methods.
Three little arbitrage theoremsq-fin.MF
Three theorems about arbitrage bubbles in financial equations.
problem Characterizing and solving generalized Black-Scholes equations with arbitrage bubbles.
method Analytical proofs of three theorems using the Feynman-Kac theorem.
result Exact solutions for Call contracts with arbitrage bubbles.
GLASS Flows improves flow and diffusion model performance by optimizing sampling efficiency.
problem Efficiency bottleneck in sampling Markov transitions for flow and diffusion models.
method Introduces GLASS Flows, a new sampling paradigm that simulates a 'flow matching model within a flow matching model' to sample Markov transitions efficiently.
result Eliminates the trade-off between stochastic evolution and efficiency in large-scale text-to-image models.
Path Integral and Asian Optionsq-fin.PR
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …
Exchange uses incentives to optimize limit order book dynamics.
problem Optimizing market liquidity in fragmented electronic markets.
method Modeling limit order book as SPDE and using control theory to design incentives.
result Exchange can design incentives to modify order book shape and increase liquidity.
New method prices interest rate derivatives without Monte Carlo, achieving high accuracy and speed.
problem Arbitrage-free pricing of path-dependent interest rate derivatives using infinite-dimensional models.
method Casting the stochastic pricing problem as a deterministic PDE solved by FINNs, which minimize violations of the PDE and boundary conditions.
result FINNs achieve pricing accuracy within 0.04 to 0.07 cents per dollar of contract value compared to Monte Carlo benchmarks.
In this paper, a pricing formula for volatility swaps is delivered when the underlying asset follows the stochastic volatility model with jumps and stochastic intensity. By using Feynman-Kac theorem, a partial integral differential equation is obtained to derive the joint moment generating function of the previous mode…
Second Order BSDEs with Jumps: Existence and probabilistic representation for fully-nonlinear PIDEsmath.PR
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…
Empirical adversarial risk minimization (EARM) is a widely used mathematical framework to robustly train deep neural nets (DNNs) that are resistant to adversarial attacks. However, both natural and robust accuracies, in classifying clean and adversarial images, respectively, of the trained robust models are far from sa…