Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale
Анотація
Let for each n ∈ ℕ X n be an ℝ d ‐valued locally square integrable martingale w.r.t. a filtration ( ℱ n ( t ), t ∈ ℝ + ) (probability spaces may be different for different n ). It is assumed that the discontinuities of X n are in a sense asymptotically small as n → ∞ and the relation holds for all t > s > 0, row vectors z , and bounded uniformly continuous functions f . Under these two principal assumptions and a number of technical ones, it is proved that the X n ′s are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes ( X n (0), 〈 X n 〉) converge in distribution to some , then a sequence ( X n ) converges in distribution to a continuous local martingale X with initial value and quadratic characteristic H , whose finite‐dimensional distributions are explicitly expressed via those of .
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