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.

**Read or Download Peacocks and Associated Martingales, with Explicit Constructions PDF**

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**Additional resources for Peacocks and Associated Martingales, with Explicit Constructions **

**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.