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An Introduction to Variational Inequals and Their Applns by D. Kinderlehrer, et al.,

By D. Kinderlehrer, et al.,

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9 shows). For our purposes the Itˆo integral will be most convenient, so we will base our discussion on that from now on. Exercises Unless otherwise stated Bt denotes Brownian motion in R, B0 = 0. 1. 6) that t t sdBs = tBt − 0 Bs ds . 2. 3. 3. − Bs ds . e. {Ht }t≥0 is the filtration of the process {Xt }t≥0 ). t. t. its own filtration {Ht }t≥0 . t Ht , then E[Xt ] = E[X0 ] for all t ≥ 0 . t. its own filtration. 4. t. {Ft }: (i) Xt = Bt + 4t (ii) Xt = Bt2 (iii) Xt = t2 Bt − 2 t sBs ds 0 (iv) Xt = B1 (t)B2 (t), where (B1 (t), B2 (t)) is 2-dimensional Brownian motion.

Prove that Bt : = Bt0 +t − Bt0 ; t≥0 is a Brownian motion. 13. Let Bt be 2-dimensional Brownian motion and put Dρ = {x ∈ R2 ; |x| < ρ} Compute for ρ > 0 . P 0 [Bt ∈ Dρ ] . 14. Let Bt be n-dimensional Brownian motion and let K ⊂ Rn have zero n-dimensional Lebesgue measure. Prove that the expected total length of time that Bt spends in K is zero. (This implies that the Green measure associated with Bt is absolutely continuous with respect to Lebesgue measure. See Chapter 9). 15. e. U U T = I. Prove that Bt : = U Bt is also a Brownian motion.

Prove that Wt cannot have continuous paths. (Hint: Consider (N ) (N ) E[(Wt − Ws )2 ], where (N ) Wt = (−N ) ∨ (N ∧ Wt ), N = 1, 2, 3, . ) . 12. 9 we let ◦dBt denote Stratonovich differentials. 13. A stochastic process Xt (·): Ω → R is continuous in mean square if E[Xt2 ] < ∞ for all t and lim E[(Xs − Xt )2 ] = 0 s→t for all t ≥ 0 . a) Prove that Brownian motion Bt is continuous in mean square. e. there exists C < ∞ such that |f (x) − f (y)| ≤ C|x − y| Prove that Yt : = f (Bt ) is continuous in mean square.

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