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3) 0 Proof. s. 14). 34 3. 3 Extensions of the Itˆ o integral The Itˆo integral f dB can be defined for a larger class of integrands f than V. 4 can be relaxed to the following: (ii)’ There exists an increasing family of σ-algebras Ht ; t ≥ 0 such that a) Bt is a martingale with respect to Ht and b) ft is Ht -adapted. Note that a) implies that Ft ⊂ Ht . The essence of this extension is that we can allow ft to depend on more than Ft as long as Bt remains a martingale with respect to the “history” of fs ; s ≤ 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. 5. 9) that Mt = Bt2 − t is an Ft -martingale. 6. Prove that Nt = Bt3 − 3tBt is a martingale. 7. A famous result of Itˆo (1951) gives the following formula for n times iterated Itˆ o integrals: n! 8) where hn is the Hermite polynomial of degree n, defined by hn (x) = (−1)n e x2 2 dn − x22 ; e dxn n = 0, 1, 2, .

3). However, as indicated earlier, the Itˆo interpretation of an integral of the form t “ f (s, ω)dBs (ω)” (∗) 0 is just one of several reasonable choices. For example, the Stratonovich integral is another possibility, leading (in general) to a different result. 2)? a. ω B (n) (t, ω) → B(t, ω) as n → ∞ (n) uniformly (in t) in bounded intervals. For each ω let Xt (ω) be the solution of the corresponding (deterministic) differential equation (n) dXt dB = b(t, Xt ) + σ(t, Xt ) t dt dt . a. (n) ω we have that Xt (ω) → Xt (ω) as n → ∞, uniformly (in t) in bounded intervals.

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