Peacocks and Associated Martingales, with Explicit by Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc

By Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor

We name peacock an integrable method that is expanding within the convex order; the sort of concept performs an enormous position in Mathematical Finance. A deep theorem as a result of Kellerer states strategy is a peacock if and provided that it has an identical one-dimensional marginals as a martingale. one of these martingale is then acknowledged to be linked to this peacock.

In this monograph, we express a number of examples of peacocks and linked martingales with assistance from diverse tools: building of sheets, time reversal, time inversion, self-decomposability, SDE, Skorokhod embeddings… they're constructed in 8 chapters, with a couple of hundred of exercises.

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Example text

1. ’s. X is said to be dominated by Y for the convex order if, for every convex function ψ : R → R such that E[|ψ (X)|] < ∞ and E[|ψ (Y )|] < ∞, one has: E[ψ (X)] ≤ E[ψ (Y )]. 1) We denote this order by: (c) X ≤ Y. 2) The class C C denotes the class of convex C 2 -functions ψ : R −→ R such that ψ has a compact support. We note that if ψ ∈ C: • |ψ | is a bounded function, • there exist k1 and k2 ≥ 0 such that: |ψ (x)| ≤ k1 + k2 |x|. 3) The class C+ We denote by C + the class of convex functions ψ ∈ C such that ψ is positive and increasing.

Ii) Prove that (Vt ,t ≥ 0) is a (Gt ,t ≥ 0) martingale, hence a peacock. 14]): 14 1 Some Examples of Peacocks • Let (Ju , u ≥ 0) be a (Gu , u ≥ 0) predictable process; then there exist two positive (Fu , u ≥ 0) predictable processes (Ju+ , u ≥ 0) and (Ju− , u ≥ 0) such that: ∀u ≥ 0, Ju = Ju− 1[0,Λ ] (u) + Ju+ 1]Λ ,+∞[ (u). ) We now particularize the framework of this Question. s. We set N t := sup Ns and t→+∞ s≤t Λ := sup{t ≥ 0; N t = Nt }. 32], At = log(N t ). We shall now prove directly that (Vt ,t ≥ 0) is a peacock (without using the fact that (Vt ,t ≥ 0) is a (Gt ,t ≥ 0)martingale).

C) Proof. i) We prove that 1) ⇒ 3). Assume that X ≤ Y . 1) first with ψ (x) = x, then with ψ (x) = −x, we deduce that E[X] = E[Y ]. 3)) and similarly for Y . ii) If ψ ∈ C, then there exist a and b such that x −→ a + bx + ψ (x) belongs to C + . This shows 3) ⇒ 2). iii) Since any convex function ψ is the envelope from below of the affine functions which are smaller than ψ , one sees that any convex function is an increasing limit of a sequence of functions in C. Then, 2) ⇒ 1) follows from the monotone convergence theorem.

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