Analytic Approach
In this chapter we derive the companion variational inequality approach to the reflected BSDEs of Chap. 12 . First we introduce systems of partial integro-differential variational inequalities associated with these BSDEs and we state suitable definitions
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Analytic Approach
In this chapter we derive the companion variational inequality approach to the reflected BSDEs of Chap. 12. In Sect. 13.1 we introduce systems of partial integro-differential variational inequalities associated with these BSDEs and we state suitable definitions of viscosity solutions for related problems. Existence, uniqueness and stability issues are dealt with in Sect. 13.2. In Sect. 13.2.1, we show that the value processes (first components) in the solutions of the BSDEs can be characterized in terms of the value functions for related optimal stopping or Dynkin game problems. We then establish in Sect. 13.2.2 a discontinuous viscosity solutions comparison principle that, in particular, implies uniqueness of viscosity solutions for the related obstacle problems. This comparison principle is also used in Sect. 13.2.3 for proving the convergence of stable, monotone and consistent deterministic approximation schemes.
13.1 Viscosity Solutions of Systems of PIDEs with Obstacles The assumptions of Sects. 12.3, 12.4, including the specification of the Markovian change of probability measure (12.56)–(12.57), are quite involved. In the sequel we give up all these specific assumptions, retaining only the standing Markov setup assumptions (M.0)–(M.2), and postulating in addition that the Markovian FBSDE with the data G, C, ϑ has a Markovian solution t Z t = Ft , Pt , B t , χ t , ν t , X t , Y t , Y . Remark 13.1.1 As illustrated in the previous chapter, this assumption covers various issues such as Lipschitz properties of the forward SDE coefficients b, σ , δ with respect to x, martingale representation properties, some kind of consistency between the drivers B t , χ t , ν t as t ≡ (t, x, i) varies in E, or almost sure continuity of the random function ϑ t . S. Crépey, Financial Modeling, Springer Finance, DOI 10.1007/978-3-642-37113-4_13, © Springer-Verlag Berlin Heidelberg 2013
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Analytic Approach
Our next goal is to establish the connection between a solution Z t under the above-mentioned assumptions, collectively denoted by (M) henceforth, and two systems of obstacle problems denoted by (V1) and (V2) below. We will consider this from the point of view of viscosity solutions to these problems. We refer the reader to [16, 17, 22, 27, 28] for alternative approaches in terms of weak Sobolev solutions. For the sake of existence of solutions to the related obstacle problems, we also postulate in this chapter the following assumption (E), in three parts: (E.1) all the coefficients of G are continuous with respect to (t, x); (E.2) the functions δ, f and η are locally Lipschitz with respect to (t, x), uniformly in y; (E.3) ϑ t is defined in terms of a domain O ⊆ Rd × I as in Example 12.4.6. Let D = [0, T ] × O, where O denotes the closure1 of O in Rd × I , and let Int E = [0, T ) × Rd × I, Int D = [0, T ) × O,
∂E = {T } × Rd × I ∂D := E \ Int D.
(13.1)
The “thick boundary” ∂D is motivated by the presence of the jumps in X. Given locally bounded test-functions ϕ and φ on E with ϕ o
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