By George Biddell Airy

ISBN-10: 1418179310

ISBN-13: 9781418179311

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**Extra resources for A treatise on trigonometry**

**Example text**

30)). t t 0 But what happens if St hits the boundaries 1 or s¯ of the interval [1, s¯]? Say, at time t0 > 0, that we have for the first time after t = 0 that once again we have St0 = 1. Consider the Brownian motion W = (Wt )t ≥0 during the infinitesimal interval [t0, t0 + dt]. ) two possibilities for the increment of W: either dWt0 := Wt0 +dt − Wt0 = dt 1/2 or dWt0 := Wt0 +dt − Wt0 = −dt 1/2 . Let us start with the former case. 32), d S˜t0 := g(St0 +dt ) − g(St0 ) = g (St0 ) dSt0 + 21 g (St0 ) d S t0 g (1) 2 2 = St0 (µ dt + σ dt 1/2 ) + S σ dt 2 t0 g (1) 2 = σ dt 1/2 + µ + σ dt.

T . 8. 7. 3)) via the identification ϕˆ−1 −1 1 ϕˆt−1 = Hˆ t , 0 ϕˆt−1 = 1 ˜ −ϕˆt−1 St−1 ˜ t−1, + (Hˆ • S) t = 1, . . 34) t = 1, . . 35) ˜ T. as well as ϕˆT1 = 0, ϕˆT0 = (Hˆ • S) Proof. Again the proof reduces to the economically obvious fact that trading in the frictionless market S˜ yields better terms of trade than in the market S under transaction costs λ. 5. 35). 9. In the above analysis, the notion of the Legendre transform played a central role. 36) in the present financial application.

52 3 Growth-optimal portfolio in the Black–Scholes model A self-financing trading strategy starting with zero endowment is a pair of rightcontinuous, adapted, finite variation processes (ϕt0, ϕt1 )0≤t ≤T such that 0 = ϕ1 = 0; (i) ϕ0− 0− (ii) ϕt0 = ϕt0,↑ − ϕt0, ↓ and ϕt1 = ϕt1,↑ − ϕt1,↓, where ϕt0,↑ , ϕt0,↓ , ϕt1, ↑, and ϕt1,↓ are the decompositions of ϕ0 and ϕ1 into the difference of increasing processes, 0,↑ 0, ↓ 1,↑ 1,↓ starting at ϕ0− = ϕ0− = ϕ0− = ϕ0− = 0, and satisfying dϕt0,↑ ≤ (1 − λ)St dϕt1,↓, dϕt0, ↓ ≥ St dϕt1,↑, 0 ≤ t ≤ T.

### A treatise on trigonometry by George Biddell Airy

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