Download Peacocks and Associated Martingales, with Explicit by Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc PDF
By Francis Hirsch, Christophe Profeta, Bernard Roynette, Marc Yor
We name peacock an integrable technique that is expanding within the convex order; this type of idea performs an incredible function in Mathematical Finance. A deep theorem because of Kellerer states approach is a peacock if and provided that it has a similar one-dimensional marginals as a martingale. this kind of martingale is then stated to be linked to this peacock.
In this monograph, we show a variety of examples of peacocks and linked martingales with the aid of various tools: development of sheets, time reversal, time inversion, self-decomposability, SDE, Skorokhod embeddings… they're constructed in 8 chapters, with a few hundred of exercises.
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Extra resources for Peacocks and Associated Martingales, with Explicit Constructions
Example text
Then (ϕ (Xt ),t ≥ 0) is a peacock. Proof. We first prove Point 1). Upon replacing ϕ by ϕ − ϕ (0), we may, without loss of generality, assume that ϕ (0) = 0. We must prove that, for ψ ∈ C, E[ψ (ϕ (tX))] is an increasing function of t. But, since E[ϕ (tX)] does not depend on t, upon replacing ψ (x) by ψ (x) − ψ (0) − ψ (0)x, we may suppose that ψ (0) = 0 and that ψ is positive. Then, for 0 ≤ s ≤ t, we have by convexity, since ϕ (sX) ∈ [0, ϕ (tX)]: ψ (ϕ (sX)) ≤ λ ψ (0) + (1 − λ )ψ (ϕ (tX)) ≤ ψ (ϕ (tX)) (λ ∈ [0, 1]).
Then (Xt ,t ≥ 0) is a peacock. Indeed, for ψ ∈ C: E[ψ (Xt )] = E[E[ψ (Xt )|Xs ]] ≥ E[ψ (E[Xt |Xs ])] (from Jensen’s inequality) = E[ψ (Xs )]. 4). 4 (Fair 2-processes and martingales). 8) and which is not a martingale. 8); more generally, it satisfies, for every 0 ≤ t1 ≤ t2 ... Xtk−1 ] = Xtk−1 . e. 5 (A fair k-process for every k is a martingale). Using the terminology introduced above, prove that (Xt ,t ≥ 0) is a fair k-process for every k ∈ N if and only if it is a martingale. 6 (Decreasing processes in the convex order and inverse martingales).
Then: (Xt := tX,t ≥ 0) is a peacock. 1. ’s. αn be n positive and increasing functions (from R+ to R+ ). Then: n ∑ αi (t)Xi ,t ≥ 0 is a peacock. i=1 Proof. 12. 20. 1. 21 (If E[X|Y ] = 0, then (Y +tX, t ≥ 0) is a peacock). ’s such that E[X|Y ] = 0. Prove that (Y + tX,t ≥ 0) is a peacock. (Hint: let ψ be a convex function and 0 ≤ s < t. Prove that: ψ (Y + sX) ≤ 1 − s s ψ (Y ) + ψ (Y + tX) t t and ψ (Y ) ≤ E[ψ (Y + tX)|Y ]; then take the expectations. Alternative proof: let ψ ∈ C. 22 (Weak self-similarity and peacocks).