Download Theory of Stochastic Processes: With Applications to by Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya PDF

By Dmytro Gusak, Alexander Kukush, Alexey Kulik, Yuliya Mishura, Andrey Pilipenko

ISBN-10: 0387878610

ISBN-13: 9780387878614

ISBN-10: 0387878629

ISBN-13: 9780387878621

This e-book is a set of routines overlaying the entire major issues within the sleek concept of stochastic strategies and its purposes, together with finance, actuarial arithmetic, queuing idea, and danger theory.

The goal of this ebook is to supply the reader with the theoretical and useful fabric important for deeper realizing of the most subject matters within the thought of stochastic strategies and its comparable fields.

The ebook is split into chapters in line with a few of the themes. each one bankruptcy comprises difficulties, tricks, ideas, in addition to a self-contained theoretical half which supplies the entire worthwhile fabric for fixing the issues. References to the literature also are given.

The routines have quite a few degrees of complexity and range from easy ones, worthwhile for college kids learning simple notions and procedure, to very complicated ones that exhibit a few vital theoretical proof and constructions.

This publication is among the greatest collections of difficulties within the conception of stochastic strategies and its functions. the issues during this ebook should be precious for undergraduate and graduate scholars, in addition to for experts within the concept of stochastic approaches.

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Extra resources for Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory

Sample text

Let {W (t),t ≥ 0} be the Wiener process. Prove that there exists the limit in probability 2 n k+1 k P-lim ∑ W , −W n→∞ n n k=0 and find this limit. Prove that n P-lim ∑ W n→∞ k=0 k+1 −W n k n = ∞. 20. Prove that almost all trajectories of the Wiener process have unbounded variation on [0, 1]. 21. Let {W (t), t ∈ R+ } be the Wiener process. Prove that, with probability one, λ 1 {t ≥ 0| W (t) = 0} = 0. 3 Trajectories. Modifications. 22. Prove that, with probability one, the Wiener process attains its maximum value on [0, 1] only once.

S. Every trajectory of the process Y is continuous as a Lebesgue integral with varying upper bound, therefore Y (·, ω ) ≡ 0 for almost all ω . Then X(·, ω ) ≡ 12 for the same ω . It is impossible. 13. a) γ < 12 ; b) γ < H. 32 3 Trajectories. Modifications. 27. , the points t ∈ R+ for which P(τ = t) > 0). The required condition is as follows. For any t ∈ Kτ there exists ε > 0 such that P(τ ∈ (t,t + ε )) = 0. 31. 30, θk = τ1/k , k ∈ N are random variables. 6). Thus τ = ∑∞ k=1 θk · 1IZk ∈Γ · ∏m>k 1IZm ∈Γ is a random variable too.

12. 7. (e) If ξ ∼ N(a, σ 2 ) then Eξ 4 = a4 + 6a2 σ 2 + 3σ 4 . Prove this formula and use it. 16. 1. 4 Continuity. Differentiability. Integrability 39 (2) Let ϕ ∈ C0∞ (R) be a compactly supported nonnegative infinitely differentiable ∞ ∞ ϕ (x)dx = 1. Define ϕn (x) = nϕ (nx), Xn (t) := −∞ X(t − s)ϕn (s) function with −∞ ∞ ds = −∞ X(s)ϕn (t − s)ds. 2) in the mean square. s. 3) E Xn (s) − X (s) ds ∞ and E|Xn (s) − X (s)| ≤ E|X (s)| + −∞ E|X (s1 )|ϕn (s − s1 )ds1 . 2) and the Lebesgue dominated convergence theorem.

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