Stochastic Differential Equations: An Introduction with by Bernt K. Oksendal

By Bernt K. Oksendal

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By Bernt K. Oksendal

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Additional info for Stochastic Differential Equations: An Introduction with Applications, Edition 5, Corrected Printing

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Hida (1980), Adler (1981), Rozanov (1982), Hida, Kuo, Potthoff and Streit (1993) or Holden, Øksendal, Ubøe and Zhang (1996). 3) where Xj = X(tj ), Wk = Wtk , ∆tk = tk+1 − tk . We abandon the Wk -notation and replace Wk ∆tk by ∆Vk = Vtk+1 − Vtk , where {Vt }t≥0 is some suitable stochastic process. The assumptions (i), (ii) and (iii) on Wt suggest that Vt should have stationary independent increments with mean 0. It turns out that the only such process with continuous paths is the Brownian motion Bt .

G. Hoffman (1962, p. ) So by bounded convergence T (h(t, ω) − gn (t, ω))2 dt → 0 E as n → ∞ , S as asserted. Step 3. Let f ∈ V. Then there exists a sequence {hn } ⊂ V such that hn is bounded for each n and T (f − hn )2 dt → 0 as n → ∞ . E S Proof. Put  if f (t, ω) < −n  −n hn (t, ω) = f (t, ω) if −n ≤ f (t, ω) ≤ n  n if f (t, ω) > n . Then the conclusion follows by dominated convergence. That completes the approximation procedure. We are now ready to complete the definition of the Itˆo integral T f (t, ω)dBt (ω) for f ∈ V .

7)  t1 In     t1 In  C=   ..    . t1 In Hence 13 t1 In t2 In .. 8) and E x [(Bt − x)2 ] = nt, E x [(Bt − x)(Bs − x)] = n min(s, t) . 10) since E x [(Bt − Bs )2 ] = E x [(Bt − x)2 − 2(Bt − x)(Bs − x) + (Bs − x)2 ] = n(t − 2s + s) = n(t − s), when t ≥ s . e. Bt1 , Bt2 − Bt1 , · · · , Btk − Btk−1 are independent for all 0 ≤ t1 < t2 · · · < tk . 11) To prove this we use the fact that normal random variables are independent iff they are uncorrelated. (See Appendix A). 12) which follows from the form of C: E x [Bti Btj − Bti−1 Btj − Bti Btj−1 + Bti−1 Btj−1 ] = n(ti − ti−1 − ti + ti−1 ) = 0 .

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