Weak Solutions for SPDEs and Backward Doubly Stochastic Differential Equations

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Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations (2005, background): “Moreover, see [ 2 ], under suitable conditions, u is a weak solution of (1.2) in the Sobolev sense.”
About the Pricing Equations in Finance (2011, unknown): “, [11, 12, 8, 5, 4]), dMt = ∇u (t,Xt)dWt + u(t,Xt−)dNt .”
A class of globally solvable Markovian quadratic BSDE systems and applications (2016, background): “…notion of a weak solution - typically a starting point for any regularity analysis in the PDE world - is too weak for us; indeed, the very deﬁnition of a solu-tion to a BSDE requires Y to be a semimartingale (see, however, [BL97], [BM01], [Lej02], and [MX08] for developments in Lipschitz systems).”
Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients (2006, methodology): “Now using some ideas of Theorem 2.1 in [ 3 ], similar to the argument as in Section 4 in [25], we know that v n (s,x) = (� ∗ ∇u n )(s,x) and u n is the weak”