By Mircea Grigoriu
Advent -- necessities of chance conception -- Random capabilities -- Stochastic Integrals -- Itô's formulation and functions -- Probabilistic types -- Stochastic usual Differential and distinction Equations -- Stochastic Algebraic Equations -- Stochastic Partial Differential Equations
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39) for the restriction of y → x = g −1 (y) to Av . 43 Let X ∼ N (0, 1) and Y = cos(X ). The distribution of Y is P(Y ≤ y) = k∈Z Φ(2kπ + cos−1 (y)) − Φ(2kπ − cos−1 (y)) for |y| ≤ 1 since {Y ≤ y} if X belongs to ∪k∈Z 2kπ − cos−1 (y), 2kπ + cos−1 (y) . 44 Let X ∼ N (0, ρ) be an Rd -valued standard Gaussian variable with ρii = 1, and set Yi = Fi−1 ◦ Φ(X i ), where Fi are continuous distributions with densities f i , i = 1, . . , d. The density of Y = (Y1 , . . , Yd ) ∈ Rd is 1 f y (y1 , . . 40) 36 2 Essentials of Probability Theory where xi = Φ −1 ◦ Fi (yi ), i = 1, .
Properties of probability measures that involve increasing/decreasing, convergent, and arbitrary sequences of events are discussed. 19 The sequence {An , n = 1, 2, . } is said to be increasing if An ⊆ An+1 for all n. If An ⊇ An+1 for all n, the sequence is decreasing. The sequence is ∞ ∞ ∞ convergent if lim supn→∞ An = ∩∞ n=1 ∪k=n Ak and lim inf n→∞ An = ∪n=1 ∩k=n Ak coincide, and we use the notation limn→∞ An = lim supn→∞ An = lim inf n→∞ An for the limit of {An }. Note that lim supn→∞ An , lim inf n→∞ An , and limn→∞ An are events in (Ω, F , P).
59 Let X = (X 0 , X 1 , . ) be a real-valued sequence defined on a probability space (Ω, F , P) and set Fn = σ (X 0 , X 1 , . . , X n ). The sequences Yn = g(X n ) and Yn = max0≤i≤n {X i } are Fn -adapted, where g : R → R is a Borel measurable function. 40 Let X = (X 0 , X 1 , X 2 , . ) be a sequence of real-valued random variables defined on a probability space (Ω, F , P) endowed with a filtration (Fn )n≥0 . The sequence X 0 , X 1 , X 2 , . . is said to be an Fn -martingale if (1) E[|X n |] < ∞, n ≥ 0, (2) X is Fn -adapted, and (3) E[X n | Fm ] = X m for 0 ≤ m ≤ n.