Nonlinear Feynman-Kac Formula Research Articles

Semilinear Feynman–Kac formulae for B-continuous viscosity solutions

. We prove the existence of a B-continuous viscosity solution for a class of infinite dimensional semilinear partial differential equations (PDEs) using probabilistic methods. Our approach also yields a stochastic representation formula for the solution in terms of a scalar-valued backward stochastic differential equation. The uniqueness is proved under additional assumptions using a comparison theorem for viscosity solutions. Our results constitute the first nonlinear Feynman–Kac formula using the notion of B-continuous viscosity solutions and thus introduces a framework allowing for generalizations to the case of fully nonlinear PDEs.

An asymptotic expansion of the solution of a semi-linear partial differential equation implied by a nonlinear Feynman–Kac formula

This paper introduces an asymptotic expansion for the smooth solution of a semi-linear partial differential equation. Our scheme is based on Itô’s formula, Taylor’s expansion, nonlinear Feynman–Kac formula and some algebras.

Power utility maximization problem under rough stochastic local volatility models with continuous-time Markov chain approximation

This paper studies the portfolio optimization problem based on rough stochastic local volatility models. Due to the challenge that this model is neither a semimartingale nor a Markov process, we first introduce the semimartingale approximate optimization problem and present the convergence analysis. We use the classic stochastic control method and continuous-time Markov chain approximation approach to solve the approximate optimization problem respectively. Then the optimal investment strategy and the HJB equation satisfied by the value function are obtained respectively. Furthermore, for the classic stochastic control method, we demonstrate the existence of solutions for the associated partial differential equation via the nonlinear Feynman–Kac formula. For continuous-time Markov chain approximation method, we prove the convergence of the value function. Finally, we provide a numerical analysis to illustrate the above results by using the power utility function. The numerical analysis shows that continuous-time Markov chain approximation method is effective in solving the portfolio optimization problem under rough stochastic local volatility models.

Study of Pricing of High-Dimensional Financial Derivatives Based on Deep Learning

Many problems in the fields of finance and actuarial science can be transformed into the problem of solving backward stochastic differential equations (BSDE) and partial differential equations (PDEs) with jumps, which are often difficult to solve in high-dimensional cases. To solve this problem, this paper applies the deep learning algorithm to solve a class of high-dimensional nonlinear partial differential equations with jump terms and their corresponding backward stochastic differential equations (BSDEs) with jump terms. Using the nonlinear Feynman-Kac formula, the problem of solving this kind of PDE is transformed into the problem of solving the corresponding backward stochastic differential equations with jump terms, and the numerical solution problem is turned into a stochastic control problem. At the same time, the gradient and jump process of the unknown solution are separately regarded as the strategy function, and they are approximated, respectively, by using two multilayer neural networks as function approximators. Thus, the deep learning-based method is used to overcome the “curse of dimensionality” caused by high-dimensional PDE with jump, and the numerical solution is obtained. In addition, this paper proposes a new optimization algorithm based on the existing neural network random optimization algorithm, and compares the results with the traditional optimization algorithm, and achieves good results. Finally, the proposed method is applied to three practical high-dimensional problems: Hamilton-Jacobi-Bellman equation, bond pricing under the jump Vasicek model and option pricing under the jump diffusion model. The proposed numerical method has obtained satisfactory accuracy and efficiency. The method has important application value and practical significance in investment decision-making, option pricing, insurance and other fields.

A fully nonlinear Feynman–Kac formula with derivatives of arbitrary orders

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman–Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided.

Numerical schemes for fully coupled mean-field forward backward stochastic differential equations

In this paper, based on the nonlinear Feynman-Kac formula and Itô-Taylor schemes for solving mean-field forward stochastic differential equations, we propose numerical schemes for solving fully coupled mean-field forward backward stochastic differential equations. Our numerical experiments show that the schemes are stable, effective, and can be of second-order accuracy when weak second-order schemes are used for solving mean-field forward stochastic differential equations.

Deep neural networks based temporal-difference methods for high-dimensional parabolic partial differential equations

• We develop the FBSTD, an NN based temporal-difference learning for PDEs. • The cost of the FBSTD is significantly lower than some popular NN methods for FBSDEs. • The FBSTD can be applied to solve general nonlinear, high order and high-dimensional PDEs. Solving high-dimensional partial differential equations (PDEs) is a long-standing challenge, for which classical numerical methods suffer from the well-known curse of dimensionality . In this paper, we propose deep neural networks (NN) based temporal-difference (TD) learning methods for numerically solving high-dimensional parabolic PDEs. To this end, we approximate the solution of the original PDE with an NN function. To calculate this neural network function, we first transform the deterministic parabolic PDE to a forward-backward stochastic differential equations (FBSDE) system with the nonlinear Feynman-Kac formula, and then transform the forward updating process of FBSDE as a Markov reward process (MRP). On this basis, we approximate the solution of the PDE by training an NN function with a reinforcement learning technique described as below: we first discretize the temporal interval to a finite number of time steps and then at each time step, we generate many trajectories and design the loss function as the mean square of temporal difference error on all trajectories. With this loss function, we update the parameters of the NN function with the stochastic gradient descent (SGD) method. Numerical experiments show that comparing to some other existed deep learning methods, our method not only accelerates the computational speed but also improves the computational accuracy. In particular, the relative errors of our algorithm achieve the order of O ( 10 − 4 ) even the dimension of the problem is as high as 100.

Mass-conserving stochastic partial differential equations and backward doubly stochastic differential equations

In this paper, we first study the connection between mass-conserving SPDEs on a bounded domain and backward doubly stochastic differential equations, which is a new extension of nonlinear Feynman-Kac formula to mass-conserving SPDEs. Then the infinite horizon mass-conserving SPDEs and their stationary solutions are considered without monotonic conditions, while the Poincare inequality plays an important role. Finally, the existence and the stationarity to solutions of non-Lipschitz mass-conserving stochastic Allen-Cahn equations are obtained.

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Uniformly Continuous Coefficients in (y, z)

In this paper, we investigate backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients in (y, z). The existence and uniqueness of solutions are obtained via a method of Picard iteration, a linearization method and a monotone convergence argument. Furthermore, we establish the corresponding comparison theorem and related nonlinear Feynman–Kac formula.

Reflected Quadratic BSDEs Driven by G-Brownian Motions

In this paper, the authors consider a reflected backward stochastic differential equation driven by a G-Brownian motion (G-BSDE for short), with the generator growing quadratically in the second unknown. The authors obtain the existence by the penalty method, and some a priori estimates which imply the uniqueness, for solutions of the G-BSDE. Moreover, focusing their discussion at the Markovian setting, the authors give a nonlinear Feynman-Kac formula for solutions of a fully nonlinear partial differential equation.

Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs

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.

Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

The PDEs and Numerical Scheme for Derivatives under Uncertainty Volatility

We use the stochastic differential equations (SDE) driven by G-Brownian motion to describe the basic assets (such as stocks) price processes with volatility uncertainty. We give the estimation method of the SDE’s parameters. Then, by the nonlinear Feynman-Kac formula, we get the partial differential equations satisfied by the derivatives. At last, we give a numerical scheme to solve the nonlinear partial differential equations.

Forward–backward SDEs with distributional coefficients

Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Feynman-Kac representation formula. The associated semi-linear second order parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated; our analysis uses mild solutions, Sobolev spaces and semigroup theory.

BSDE representation and randomized dynamic programming principle for stochastic control problems of infinite-dimensional jump-diffusions

We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.

Quadratic reflected BSDEs and related obstacle problems for PDEs

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.

The obstacle problem of integro-partial differential equations with applications to stochastic optimal control/stopping problem

Abstract This paper is devoted to existence and uniqueness of minimal mild super solutions to the obstacle problem governed by integro-partial differential equations. We first study the well-posedness and local Lipschitz regularity of Lp solutions (p ≥ 2) to reflected forward-backward stochastic differential equations (FBSDEs) with jump and lower barrier. Then we show that the solutions to reflected FBSDEs provide a probabilistic representation for the mild super solution via a nonlinear Feynman–Kac formula. Finally, we apply the results to study stochastic optimal control/stopping problems.

Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes

We consider a classical finite horizon optimal control problem for continuous-time pure jump Markov processes described by means of a rate transition measure depending on a control parameter and controlled by a feedback law. For this class of problems the value function can often be described as the unique solution to the corresponding Hamilton–Jacobi-Bellman equation. We prove a probabilistic representation for the value function, known as nonlinear Feynman–Kac formula. It relates the value function with a backward stochastic differential equation (BSDE) driven by a random measure and with a sign constraint on its martingale part. We also prove existence and uniqueness results for this class of constrained BSDEs. The connection of the control problem with the constrained BSDE uses a control randomization method recently developed by several authors. This approach also allows to prove that the value function of the original non-dominated control problem coincides with the value function of an auxiliary dominated control problem, expressed in terms of equivalent changes of probability measures.

BSDE, path-dependent PDE and nonlinear Feynman-Kac formula

In this paper, we introduce a type of path-dependent quasilinear (parabolic) partial differential equations in which the (continuous) paths on an interval [0,t] becomes the basic variables in the place of classical variables (t,x). This new type of PDE are formulated through a classical backward stochastic differential equation (BSDEs, for short) in which the terminal values and the generators are allowed to be general functions of Brownian paths. In this way we have established a new type of nonlinear Feynman-Kac formula for a general non-Markovian BSDE. Some main properties of regularities for this new PDE was obtained.

Renormalized solutions of semilinear equations involving measure data and operator corresponding to Dirichlet form

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function $u$ is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$ can be represented by a suitable nonlinear Feynman-Kac formula. This implies in particular that for a broad class of local and nonlocal semilinear equations there exists a unique renormalized solution.