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In a recent paper [7] , a hybrid supercomputing algorithm for elliptic equations has been proposed. The idea is that the interfacial nodal solutions solve a linear system, whose coefficients are expectations of functionals of stochastic differential equations confined within patches of about subdomain size. Compared to standard substructuring techniques, such as the Schur complement method for the skeleton, the hybrid approach produces an explicit and sparse shrunken matrix—hence suitable for substructuring again. The ultimate goal is to push strong scalability beyond the state of the art by leveraging the potential for parallelisation of stochastic calculus. Here, we present a major revamping of that framework, based on the insight of embedding the domain in a cover of overlapping circles (in two dimensions). This allows for efficient Fourier interpolation along the interfaces (now circumferences) and—crucially—for the evaluation of most of the interfacial system entries as the solution of small boundary value problems on a circle. This is both extremely efficient (as they can be solved in parallel and by the pseudospectral method) and free of Monte Carlo error. Stochastic numerics are only needed on the relatively few circles intersecting the domain boundary. In sum, the new formulation is significantly faster, simpler, and more accurate while retaining all of the advantageous properties of PDDSparse. Numerical experiments are included for the purpose of illustration. [ABSTRACT FROM AUTHOR]

Brownian motion in a Euclidean space with a membrane located on a hyperplane and acting in the normal direction is constructed such that its so-called permeability coefficient can be given by an arbitrary Borel measurable function defined on that hyperplane and taking on its values from the interval [ − 1 , 1 ]. In all the publications on the topic, that coefficient was supposed to be a continuous function. A certain limit theorem for the number of crossings through the membrane by the consecutive values of the process constructed at the instants of time 0, 1 / n , 2 / n , ..., [ n t ] / n (for fixed t > 0) is proved under the assumption that n → ∞. The limit distribution in that theorem can be curiously interpreted in the case of a membrane whose permeability coefficient coincides with the indicator of a measurable subset of the hyperplane. [ABSTRACT FROM AUTHOR]

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton–Watson tree conditioned to survive. We prove that the solution at time t is concentrated at a single site with high probability and at two sites almost surely as t → ∞. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution u (t , v) at a vertex v can be well-approximated by a certain functional of v. The main difference with earlier results on Z d is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise. [ABSTRACT FROM AUTHOR]

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively. [ABSTRACT FROM AUTHOR]

This paper shows the nonlinear stochastic Feynman–Kac formula holds under quadratic growth. For this, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, uniqueness, and comparison theorem for one-dimensional BDSDEs are proved when the generator f (t , Y , Z) grows in Z quadratically and the terminal value is bounded, by introducing innovative approaches. Furthermore, in this framework, we utilize BDSDEs to provide a probabilistic representation of solutions to semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs, thereby extending the nonlinear stochastic Feynman–Kac formula for linear growth introduced by Pardoux and Peng (1994). [ABSTRACT FROM AUTHOR]

We consider the continuous parabolic Anderson model with the Gaussian fields under the measure-valued initial conditions, the covariances of which are homogeneous or nonhomogeneous in time and fractional rough in space. We mainly study the spatial behaviors for the Feynman–Kac formulas in the Stratonovich sense. Benefited from the application of Feynman–Kac formula based on Brownian bridge, the precise spatial asymptotics can be obtained under more general conditions than it in the previous literature. [ABSTRACT FROM AUTHOR]

We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator g allowing a general growth in the state variable y and a sub-quadratic growth in the state variable z , when the terminal condition satisfies a sub-exponential moment integrability condition, which is weaker than the usual exp (μ L) -integrability and stronger than L p (p > 1) -integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under some additional assumptions and establish a general stability result for the unbounded solutions. Finally, we derive the nonlinear Feynman–Kac formula in this context. [ABSTRACT FROM AUTHOR]

In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise W ̇ in space. We consider the case H < 1 2 and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman–Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form 1 2 Δ + W ̇ . [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive a stochastic representation of the solution to a nonlocal-in-time evolution equation (with a historical initial condition), which serves a bridge between normal diffusion and anomalous diffusion. We first derive the Feynman–Kac formula by reformulating the original model into an auxiliary Caputo-type evolution equation with a specific forcing term subject to certain smoothness and compatibility conditions. After that, we confirm that the stochastic formula also provides the solution in the weak sense even though the problem data is nonsmooth. Finally, numerical experiments are presented to illustrate the theoretical results and the application of the stochastic formula. [ABSTRACT FROM AUTHOR]

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: where = C ( 0 , T ; R d) , (u (⋅ , ϕ)) t ≔ (u (t + θ , ϕ)) θ ∈ − δ , 0 and L u (t , ϕ) ≔ 〈 b (t , ϕ) , ∂ x u (t , ϕ) 〉 + 1 2 Tr σ (t , ϕ) σ ∗ (t , ϕ) ∂ x x 2 u (t , ϕ) . The result is obtained by a stochastic approach. More precisely, we prove a new type of nonlinear Feynman–Kac representation formula associated to a backward stochastic differential equation with time-delayed generator, which is of non-Markovian type. Applications to the large investor problem and risk measures via g –expectations are also provided. [ABSTRACT FROM AUTHOR]

In this paper, a new mathematical model related to a mining extraction project under uncertainty is proposed. The underlying stochastic factors are the commodity price and the remaining resource, the dynamics of which are introduced. In the stochastic differential equation satisfied by the remaining resource, the extraction rate is involved. The main innovative modelling feature, comes from considering the extraction rate to be proportional to the commodity price, which is more realistic. In this way, an ultraparabolic hypoelliptic differential operator governs the associated PDE of the mathematical model. The mathematical analysis allows to obtain the existence and uniqueness of a classical solution. Uniqueness follows from a suitable Feynman–Kac representation formula. Existence of a classical solution is obtained after a suitable change of variables, the determination of sub and supersolutions and passing to the limit from problems in bounded domains to the unbounded one. For the numerical solution, after justifying the required boundary conditions on the computational bounded domain, the proposed numerical techniques mainly consist of a Crank–Nicolson characteristics method for the time discretization to cope with the convection dominating setting and Lagrange finite elements for the discretization in the commodity and resource variables. Finally, some numerical examples are discussed to illustrate the good performance of the new model and the proposed numerical methods. [ABSTRACT FROM AUTHOR]

We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman–Kac formula, where the underlying stochastic processes satisfy variational inequalities modeling elasto-plastic and obstacle oscillators. We then focus on solving them numerically The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities. • PDEs related to the Feynman–Kac formula are derived for stochastic nonsmooth systems. • The PDEs incorporate non-standard boundary conditions. • A finite difference scheme is compared with probabilistic simulations. • Statistics of stochastic elasto-plastic and obstacle oscillators are computed. • Asymptotic formula for probabilities of crossing threshold is computed. [ABSTRACT FROM AUTHOR]

This paper presents a tutorial overview of path integral (PI) approaches for stochastic optimal control and trajectory optimization. We concisely summarize the theoretical development of path integral control to compute a solution for stochastic optimal control and provide algorithmic descriptions of the cross-entropy (CE) method, an open-loop controller using the receding horizon scheme known as the model predictive path integral (MPPI), and a parameterized state feedback controller based on the path integral control theory. We discuss policy search methods based on path integral control, efficient and stable sampling strategies, extensions to multi-agent decision-making, and MPPI for the trajectory optimization on manifolds. For tutorial demonstrations, some PI-based controllers are implemented in Python, MATLAB and ROS2/Gazebo simulations for trajectory optimization. The simulation frameworks and source codes are publicly available at the github page. [ABSTRACT FROM AUTHOR]

Abstract We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set. [ABSTRACT FROM AUTHOR]

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge. [ABSTRACT FROM AUTHOR]

We study the classical bargaining problem and its two canonical solutions, ( Nash and Kalai–Smorodinsky ), from a novel point of view: we ask for stability of the solution if both players are able distort the underlying bargaining process by reference to a third party (e.g. a court). By exploring the simplest case, where decisions of the third party are made randomly we obtain a stable solution, where players do not have any incentive to refer to such a third party. While neither the Nash nor the Kalai–Smorodinsky solution are able to ensure stability in case reference to a third party is possible, we found that the Kalai–Smorodinsky solution seems to always dominate the stable allocation which constitutes novel support in favor of the latter. [ABSTRACT FROM AUTHOR]

This paper concerns the stochastic partial differential equation with multiplicative noise ∂ u ∂ t = L u + u W ̇ , where L is the generator of a symmetric Lévy process X , W ̇ is a Gaussian noise and u W ̇ is understood both in the senses of Stratonovich and Skorohod. The Feynman–Kac type of representations for the solutions and the moments of the solutions are obtained, and the Hölder continuity of the solutions is also studied. As a byproduct, when γ ( x ) is a nonnegative and nonnegative-definite function, a sufficient and necessary condition for ∫ 0 t ∫ 0 t | r − s | − β 0 γ ( X r − X s ) d r d s to be exponentially integrable is obtained. [ABSTRACT FROM AUTHOR]

We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates—resulting in an overall acceleration of PDD for a given error tolerance. The key feature of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement between one and two orders of magnitude over the previous version of PDD. [ABSTRACT FROM AUTHOR]

In this article we prove the continuity of the deterministic function u : [ 0 , T ] × D ̄ → R , defined by u ( t , x ) : = Y t t , x , where the process ( Y s t , x ) s ∈ [ t , T ] is given by the generalized multivalued backward stochastic differential equation: { − d Y s t , x + ∂ φ ( Y s t , x ) d s + ∂ ψ ( Y s t , x ) d A s t , x ∋ f ( s , X s t , x , Y s t , x ) d s + g ( s , X s t , x , Y s t , x ) d A s t , x − Z s t , x d W s , t ≤ s < T , Y T = h ( X T t , x ) . The process ( X s t , x , A s t , x ) s ≥ t is the solution of a stochastic differential equation with reflecting boundary conditions. [ABSTRACT FROM AUTHOR]

The paper examines measurability conditions for a random process for which a general form of the Feynman–Kac formula is valid. [ABSTRACT FROM AUTHOR]

A coupled forward–backward stochastic differential system (FBSDS) is formulated in spaces of fields for the incompressible Navier–Stokes equation in the whole space. It is shown to have a unique local solution, and further if either the Reynolds number is small or the dimension of the forward stochastic differential equation is equal to two, it can be shown to have a unique global solution. These results are shown with probabilistic arguments to imply the known existence and uniqueness results for the Navier–Stokes equation, and thus provide probabilistic formulas to the latter. Related results and the maximum principle are also addressed for partial differential equations (PDEs) of Burgers’ type. Moreover, from truncating the time interval of the above FBSDS, approximate solution is derived for the Navier–Stokes equation by a new class of FBSDSs and their associated PDEs; our probabilistic formula is also bridged to the probabilistic Lagrangian representations for the velocity field, given by Constantin and Iyer (2008) and Zhang (2010); finally, the solution of the Navier–Stokes equation is shown to be a critical point of controlled forward–backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

We discuss the probabilistic representation of the solutions of the heat equation perturbed by a repulsive point interaction in terms of a perturbation of Brownian motion, via a Feynman–Kac formula involving a local time functional. An application to option pricing is given, interpolating between the extreme cases of classical Black–Scholes options and knockouts having the barrier situated exactly at the exercise price. [ABSTRACT FROM AUTHOR]

• The Feynman-Kac formula reduces truncation error in time direction. • Piecewise quadratic approximation gives the third-order convergence in asset direction. • Unconditional stability is achieved by analytic integration over the lognormal distribution. • Local volatility surface and interest rate term structure can be used as input data. • We use vectorized code to reduce time complexity compared to the Crank-Nicolson. We propose an unconditionally stable numerical algorithm, which uses the Feynman-Kac formula of the Black-Scholes equation to obtain accurate option prices and hedge parameters. We discretize the asset and time using uniform grid points. We approximate the option values by piecewise quadratic polynomials for each time step and integrate them analytically over the log-normal distribution. The piecewise quadratic approximation gives the third-order convergence in the asset direction, and the analytic integration reduces truncation error in the time direction. The estimation errors are propagated backward in time following the convection and diffusion characteristics of the Black-Scholes equation, which assures the unconditional stability of our method. The vectorized code implementation reduces the time complexity. The convergence test shows that our approach outperforms the Crank-Nicolson scheme of the finite difference method in both time and asset directions, and the stability test verifies that our method is stable as the Crank-Nicolson. Furthermore, we show that our algorithm reduces the price errors and hedge parameter errors by more than 50% from the benchmark. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study comparison theorem, nonlinear Feynman–Kac formula and Girsanov transformation of the following BSDE driven by a -Brownian motion: where is a decreasing -martingale. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward–backward stochastic differential equations (FBSDEs) is established, which can be regarded as an extension of the Feynman–Kac formula to the non-Markovian framework. [Copyright &y& Elsevier]

Abstract: Let be the true self-repelling motion (TSRM) constructed by Tóth and Werner (1998) [22], its occupation time density (local time) and the height of the local time profile at the actual position of the motion. The joint distribution of was identified by Tóth (1995) [20] in somewhat implicit terms. Now we give explicit formulas for the densities of the marginal distributions of and . The distribution of has a particularly surprising shape: its density has a sharp local minimum with discontinuous derivative at . As a consequence we also obtain a precise version of the large deviation estimate of Dumaz (2011) [5]. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a version of the Feynman–Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman–Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions. [Copyright &y& Elsevier]

Abstract: This article deals with the existence and the uniqueness of solutions to quadratic and superquadratic Markovian backward stochastic differential equations (BSDEs) with an unbounded terminal condition. Our results are deeply linked with a strong a priori estimate on that takes advantage of the Markovian framework. This estimate allows us to prove the existence of a viscosity solution to a semilinear parabolic partial differential equation with nonlinearity having quadratic or superquadratic growth in the gradient of the solution. This estimate also allows us to give explicit convergence rates for time approximation of quadratic or superquadratic Markovian BSDEs. [Copyright &y& Elsevier]

Abstract: For the one-dimensional telegraph process, we obtain explicitly the distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of subnormal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals. [Copyright &y& Elsevier]

Abstract: Methods from stochastic analysis are combined with functional analytic methods in order to prove a Feynman–Kac formula för Schrödinger type operators with nonnegative locally square integrable potentials on vector bundles over complete Riemannian manifolds. In particular, we obtain a Feynman–Kac–Itô formula on manifolds for Schrödinger operators with magnetic fields. [ABSTRACT FROM AUTHOR]

Abstract: We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann–Dirichlet boundary condition: where and are subdifferential operators and is a second-differential operator given by The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality: where is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman–Kaç representation formula for the viscosity solution of the PVI problem. [Copyright &y& Elsevier]

Abstract: Under the foundation of Duffie & Huang (1996) , this paper integrates the reduced form model and the structure model for a default risk measure, giving rise to a new pricing model of interest rate swap with a bilateral default risk. This model avoids the shortcomings of ignoring the dynamic movements of the firm’s assets of the reduced form model but adds only a little complexity and simplifies the pricing formula significantly when compared with Li (1998) . With the help of the Crank–Nicholson difference method, we give the numerical solutions of the new model to study the default risk effects on the swap rate. We find that for a one year interest rate swap with the coupon paid per quarter, the variance of the default fixed rate payer decreases from 0.1 to 0.01 only causing about a 1.35%’s increase in the swap rate. This is consistent with previous results. [Copyright &y& Elsevier]

Abstract: By using the super Poincaré inequality of a Markov generator on over a σ-finite measure space , the Schrödinger semigroup generated by for a class of (unbounded below) potentials V is proved to be -compact provided for all . This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., under the condition that as . Concrete examples are provided to illustrate the main result. [Copyright &y& Elsevier]

Affine term structural models (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. Second, if the tuple of state variables belongs to the region where r is positive, the bond price should decrease in any state variable for which the corresponding coefficient in the formula for r is positive; the bond price should also decrease as the time to maturity increases. In the paper, sufficient conditions for the application of the Feynman–Kac formula, and monotonicity of the bond price are derived, for wide classes of affine term structure models in the pure diffusion case. Necessary conditions for the monotonicity are obtained as well. The results can be generalized for jump-diffusion processes. [Copyright &y& Elsevier]

A Feynman-Kac formula for Schro¨dinger operators including a one-center point interaction in R3 plus a bounded potential is proved. Functional integration methods on similar Kac's averages with point interactions allow us to construct bounded self-adjoint semigroups in L2(R3), with bounded below Schrodinger generators, when V+ ∊ Lloc2 and V− belongs to a large class of L2 + L−8 potentials. Moreover, a pointwise bound on the range of the semigroup is given. [Copyright &y& Elsevier]

We derive an upper bound on the large-time exponential behavior of the solution to a stochastic partial differential equation on a compact manifold with multiplicative noise potential. The potential is a random field that is white-noise in time, and Ho¨lder-continuous in space. The stochastic PDE is interpreted in its evolution (semigroup) sense. A Feynman–Kac formula is derived for the solution, which is an expectation of an exponential functional of Brownian paths on the manifold. The main analytic technique is to discretize the Brownian paths, replacing them by piecewise-constant paths. The error committed by this replacement is controlled using Gaussian regularity estimates; these are also invoked to calculate the exponential rate of increase for the discretized Feynman–Kac formula. The error is proved to be negligible if the diffusion coefficient in the stochastic PDE is small enough. The main result extends a bound of Carmona and Viens (Stochast. Stochast. Rep. 62 (3–4) (1998) 251) beyond flat space to the case of a manifold. [Copyright &y& Elsevier]

• Quantitative link between WOG and Green's function for groundwater flow is found. • Performance is assessed in five test cases with increased level of complexity. • The method is applicable to heterogeneity, various boundary conditions and forcing. The use of the Walk on Grid (WOG) approach for the reliable evaluation of the Green's function associated with groundwater flow scenarios in heterogeneous geologic media is explored. The study rests on the observation that, while the Green's function method (GFM) is one of the most significant and convenient approaches to tackle groundwater flow, Green's function evaluation is fraught with remarkable difficulties in the presence of realistic groundwater settings taking place in complex heterogeneous geologic formations. Here WOG approach is used to simulate pressure dissipation by lattice random walk and establish a quantitative relationship between space-time distribution of random walkers and the Green's function associated with the underlying flow problem. WOG-based Green's function method is tested (a) in three scenarios where analytical formulations are available for the Green's function and (b) in two groundwater flow systems with increased level of complexity. Our results show that WOG can (a) accurately evaluate the Green's function, being highly efficient when the latter can be analytically expressed in terms of infinite series; and (b) accurately and efficiently evaluate temporal evolutions of hydraulic heads at target locations in the heterogeneous systems. As such, a WOG-based approach can be employed as an efficient surrogate model in scenarios involving groundwater flow in complex heterogeneous domains. [ABSTRACT FROM AUTHOR]