Decoupled Mild Solutions of Path-Dependent PDEs and Integro PDEs Represented by BSDEs Driven by Cadlag Martingales

Abstract

We focus on a class of path-dependent problems which include path-dependent (possibly Integro) PDEs, and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of decoupled mild solution for which, under general assumptions, we study existence and uniqueness and its representation via the afore mentioned BSDEs. This concept generalizes a similar notion introduced by the authors in previous papers in the framework of classical PDEs and IPDEs. For every initial condition (s, η), where s is an initial time and η an initial path, the solution of such BSDE produces a couple of processes (Y s,η , Z s,η ). In the classical (Markovian or not) literature the function u(s, η) := Y s,η s constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify u as (in our language) the unique decoupled mild solution, but also to solve quite generally the so called identification problem, i.e. to also characterize the (Z s,η ) s,η processes in term of a deterministic function v associated to the (above decoupled mild) solution u.MSC 2010 Classification. 60H30; 60H10; 35D99; 35S05; 60J35; 60J75.

Gâteaux type path-dependent PDEs and BSDEs with Gaussian forward processes

“…An important step forward concerning path-dependent PDEs associated with BSDEs involving a solution of a path-dependent SDEs including the possibility of jumps and coefficients which were not necessarily continuous was done in [2]. The concept of solution was there the decoupled mild solution which is based on semigroup type techniques.…”

“…What about the case when the Brownian motion B is replaced with a (non-Markovian, nonsemimartingale) process such as fractional Brownian motion? The idea is to extend the consideration of [2] to this framework. The basic reference paper for this work is [26], that considered for the first time a BSDE which forward process was the solution of a Volterra SDE.…”

“…In the case when k(t, •) ≡ 1 [0,t] and b ≡ 0 then this driving martingale m T,s,η is È s,η -a.s. equal to X and is the conditioned Brownian motion B s,η appearing in (1.1). This case was already considered in a more general framework, in [2].…”

“…Then we introduce the operator A on a certain domain D(A) which to each Φ = Φ • m associates ( Ã( Φ)) • m. We also introduce in Definition 4.15, the bilinear operator Γ which to any Φ, Ψ ∈ D(A) maps A(ΦΨ) − ΦA(Ψ) − ΨA(Φ). This operator was already introduced in another context in [2] and extends the carré du champ operator appearing in the Markov processes literature, see [10] for instance.…”

Weak differentiability of Wiener functionals and occupation times

“…Towards a deeper understanding of the non-Markovian phenomenon, the search for a sound definition of path-dependent PDEs has attracted great interest by having functional stochastic calculus as the starting point. In this direction, see e.g Peng and Wang [42], Ekren, Keller, Touzi and Zhang [18], Ekren, Touzi and Zhang [19,20], Ekren and Zhang [21], Cosso and Russo [15], Barrasso and Russo [3], Bion-Nadal [6], Flandoli and Zanco [24], Cosso, Federico, Gozzi, Resestolato and Touzi [12] and Buckdahn, Keller, Ma and Zhang [7]. Applications to Finance and stochastic control are considered by e.g Jazaerli and Saporito [27], Possamaï, Tan and Zhou [41], Cont and Lu [11], Pham and Zhang [40].…”