The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem and the linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.     

Asymptotic behavior of the solution of heat equation driven by fractional white noise

This note deals with the asymptotic behavior of a weak solution of the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We study the solution given by the Feynman-Kac formula by the method of moments.     

On the Reference Probability Approach to the Equations of Non-Linear Filtering

The filtering of diffusions from their noisy observations is considered in this paper. The introduction of various reference probability measures and the use of a stochastic Feynman-Kac formula is shown to lead to new and already known filtering equations. In some cases, which include extensions of the Benes filtering problem, the new equations we propose possess a nice Gaussian solution, yielding an explicit finite dimensional filter     

Comparison theorems for stochastic evolution equations

This paper establishes an anticipating stochastic differential equation of parabolic type for the expectation of the solution of a stochastic differential equation conditioned on complete knowledge of the path of one of its components. Conversely, it is shown that any appropriately regular solution of this stochastic p.d.e. must be given by the conditional expectation. These results generalize the connection, known as the Feynman-Kac formula, between parabolic equations and expectations of functions of a diffusion. As an application, we derive an equation for the unnormalized smoothing law of a filtering problem with observation feedback.     

On semi-linear degenerate backward stochastic partial differential equations

 In this paper we study a class of one-dimensional, degenerate, semilinear backward stochastic partial differential equations (BSPDEs, for short) of parabolic type. By establishing some new a priori estimates for both linear and semilinear BSPDEs, we show that the regularity and uniform boundedness of the adapted solution to the semilinear BSPDE can be determined by those of the coefficients, a special feature that one usually does not expect from a stochastic differential equation. The proof follows the idea of the so-called bootstrap method, which enables us to analyze each of the derivatives of the solution under consideration. Some related results, including some comparison theorems of the adapted solutions for semilinear BSPDEs, as well as a nonlinear stochastic Feynman-Kac formula, are also given. Received: 16 January 2001 / Revised version: 11 October 2001 / Published online: 14 June 2002     

Reflected solutions of backward stochastic differential equations driven by G-Brownian motion

In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion. The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected G-BSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization. We then give some applications including a generalized Feynman-Kac formula of an obstacle problem for fully nonlinear partial differential equation and option pricing of American types under volatility uncertainty.     

A Feynman-Kac type formula for a fixed delay CIR model

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 fixed delay CIR process. Moreover, for the fixed delay CIR process, we derive a Feynman-Kac type formula, leading to a generalized exponential-affine formula, which is used to determine a bond pricing formula when the interest rate follows the delay’s equation. It turns out that, for each maturity time T, the instantaneous forward rate is an affine function (with time dependent coefficients) of the rate process and of an auxiliary process (also depending on T). The coefficients satisfy a system of deterministic differential equations.     

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps, whose data (the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

Traveling Wave of Three-Species Stochastic Lotka-Volterra Competitive System

This paper is devoted to a three-species stochastic competitive system with multiplicative noise. The existence of stochastic traveling wave solution can be obtained by constructing sup/sub-solution and using random dynamical system theory. Furthermore, under a more restrict assumption on the coefficients and by applying Feynman-Kac formula, the upper/lower bounds of asymptotic wave speed can be achieved.     

Two-point Correlation Function and Feynman-Kac Formula for the Stochastic Heat Equation

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on \(\mathbb {R}\). The bounds for p-th moments proved in Chen and Dalang (Ann. Probab. 2015) are simplified. We validate the Feynman-Kac formula for the p-point correlation function of the solutions to this equation with measure-valued initial data.     

On the Feynman-Kac formula and its applications to filtering theory

In this article the Feynman-Kac formula is obtained for a Markov process (X _t) whose transition probability function is not stationary. A converse to the Feynman-Kac formula is also obtained. This is used to prove the uniqueness of the solution to a measure-valued equation satisfied by the optimal filter in the white-noise approach to nonlinear filtering theory.Research partially supported by the Air Force Office of Scientific Research Contract No. F49620 85 C 0144 and by the Indian Statistical Institute.     

Qp-valued jump processes associated with linear and nonlinear pseudo-differential equations

We consider stochastic differential equations in the space Q_p of p-adic numbers associated with nonlinear pseudo-differential equations and systems. We construct both scalar and matrix-valued multiplicative functionals of Q_p-valued Markov processes and derive probabilistic representations of solutions to the Cauchy problem for nonlinear pseudo-differential equations and systems which extend the Feynman-Kac formula. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 5–37.     

The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.     

Backward doubly stochastic differential equations and systems of quasilinear SPDEs

Summary We introduce a new class of backward stochastic differential equations, which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDE's.The research of this author was partially supported by DRET under contract 901636/A000/DRET/DS/SRThe research of this author was supported by a grant from the French Ministère de la Recherche et de la Technologie, which is gratefully acknowledged     

A STOCHASTIC REPRESENTATION FOR THE LEVEL SET EQUATIONS

ABSTRACT

A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicial choices of controls. The sublevel sets of the unique level set solution of the geometric equation is shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.     

PRICING EUROPEAN OPTION IN A DOUBLE EXPONENTIAL JUMP-DIFFUSION MODEL WITH TWO MARKET STRUCTURE RISKS AND ITS COMPARISONS

Using Fourier inversion transform, P.D.E. and Feynman-Kac formula, the closedform solution for price on European call option is given in a double exponential jump-diffusion model with two different market structure risks that there exist CIR stochastic volatility of stock return and Vasicek or CIR stochastic interest rate in the market. In the end, the result of the model in the paper is compared with those in other models, including BS model with numerical experiment. These results show that the double exponential jump-diffusion model with CIR-market structure risks is suitable for modelling the real-market changes and very useful.     

Duality and Multiplicative Stochastic Processes on Quantum Groups

An analogue of McKean's stochastic product integral is introduced and used to define stochastic processes with independent increments on quantum groups. The explicit form of the dual pairing (q-analogue of the exponential map) is calculated for a large class of quantum groups. The constructed processes are shown to satisfy generalized Feynman-Kac type formulas, and polynomial solutions of associated evolution equations are introduced in the form of Appell systems. Explicit calculations for Gauss and Poisson processes complete the presentation.     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

Some remarks about backward itô formula and applications

By using a suitable generalization of a bar.kwa.rd Itô formula, we treat in a. unified way Girsanow and stochastic Feynman-Kac formulae.