Theory of Martingales

Limit theorems on large deviations for semimartingales

“…3.2.1 in [2]). Furthermore, U n being an increasing finite-valued predictable process, is locally bounded (Lemma 1.6.1 in [2]). It follows then that (3.17) implies both the local integrability of U n and the fact that U n is the compensator of U n .…”

“…Setting, basic concept, method 1. Let X n = (X n t ) t≥0 , n ≥ 1 be a sequence of stochastic processes with path in D. For each n, X n is a semimartingale on a stochastic basis (Ω, F , F n = (F n t ) t≥0 , P ) and is a weak solution to the Ito equation [1] X n t = x + where x ∈ R, (W n t ) t≥0 is a Winer process, p n (dt, du) is an integer valued random measure on (R + × E, B(R + ) ⊗ E), q n (dt, du) is the compensator of p n (dt, du) with q n (dt, du) = ndtq(du), (1.2) q(du) is a measure on (E, E) (for all the definitions and facts from the martingale theory we refer the reader to [2] and [3]. We assume that functionals a(t, X), b(t, X) and f (t, X, u) (t ∈ R + , X ∈ D, u ∈ E) are P(D)-and P(D)-measurable and satisfy the following conditions:…”

“…By Theorem 1.2 and the definition of C-exponential tightness it suffices to prove that (1.10) implies (1.6) for φ ∈ C. First we recall the definition of the Skorokhod-Lindvall metric in D (see, e.g. [2]). …”

“…The process Z n (2λ) = (Z n t (2λ)) t≥0 is a positive local martingale and hence, it is a positive supermartingale ( [2]), Problem 1.4.4). So EZ n τ (2λ) ≤ 1, and (4.5) holds, provided |G(α; s, X)| is uniformly bounded in s for almost all s ≤ τ for any α > 0.…”

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

“…The following lemma recalls some well-known martingale inequalities of which the first one is the Burkholder-Gundy inequality. Their proofs can be seen, for example, in Liptser and Shiryayev (1989).…”