A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps

Abstract

We consider a stochastic functional delay differential equation, namely an equation whose evolution depends on its past history as well as on its present state, driven by a pure diffusive component plus a pure jump Poisson compensated measure. We lift the problem in the infinite dimensional space of square integrable Lebesgue functions in order to show that its solution is an L 2 −valued Markov process whose uniqueness can be shown under standard assumptions of locally Lipschitzianity and linear growth for the coefficients. Coupling the aforementioned equation with a standard backward differential equation, and deriving some ad hoc results concerning the Malliavin derivative for systems with memory, we are able to derive a non-linear Feynman-Kac representation theorem under mild assumptions of differentiability.Remark 1.1. In what follows we will only consider the 1−dimensional case, the case of a R d −valued stochastic process, perturbed by a general R m − dimensional Wiener process and a R n −dimensional Poisson random measure, with d > 1, m > 1 and n > 1, can be easily obtained from the present one.In order to take into account the delay component, we study the equation (1.2) in the Delfour-Mitter space defined as follows M 2 := L 2 ([−r, 0]; R) × R, endowed with the scalar product (X t , X(t)), (Y t , Y (t)) M2 = X t , Y t L 2 + X(t) · Y (t) ,

Drawbacks and Limitations of Black-Scholes Model for Options Pricing

“…Lin et al (2018) discussed the pricing of European options on two underlying assets with delays whose price processes satisfy geometric Brownian motions with delays. Cordoni et al (2017) discussed stochastic functional delay differential equation, whose evolution depends on its past history as well as on its present state. A different view of the valuation of financial derivatives by binomial tree methods is provided by Hyong-chol et al (2016).…”

A BSDE with Delayed Generator Approach to Pricing under Counterparty Risk and Collateralization

“…Since then, such results have been then generalized in several directions; see, for example, [26][27][28][29][30][31][32] and the references therein. We would like to mention that the path-dependent calculus has revealed itself since its inception as a powerful tool to model financial markets exhibiting delay and also path-dependent options.…”

Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations

“…The latter has been also proved for Semilinear Parabolic Equations in [25], where the definition of the generalized directional gradient is firstly introduced. The concept of mild solution together with the generalized directional gradient to handle path-dependent Kolmogorov equation with jumps and delay has been widely analyzed in the functional formulation, see, e.g., [13]. Moreover, a discrete-time approximation for solutions of a system of decoupled FBSDEs with jumps have been proved in [6] by means of Malliavin calculus tools.…”

“…The present paper is structured as follows: we start providing notations and problem setting in Section 2, according to the theoretical framework developed by [15] and [17]; in Section 3 we study the well-posedness of the path-dependent BSDE mentioned appearing in the Markovian FBSDEs stystem (1.6) following the approach in [11] by additionally considering jumps; in Section 4 we provide a Feynman-Kac formula relating the BSDE to the Kolmogorov Equation defined in (1.1) to then generalise results in [13] by considering a dependence in the generator f of the backward dynamic on a delayed L 2 term, namely Y t,φ r , for a small delay δ; in Section 5 we derive the existence of a mild solution for the Kolmogorov Equation within the setting developed in [25]; finally, in Section 6, we provide an application based on the analyzed theoretical setting, i.e. a version of the Large Investor Problem characterised by a jump-diffusion dynamic.…”