Order Stochastic Calculus Past Assignment

Order Stochastic Calculus Past Assignment
Answer FOUR of the SIX questions. If more than FOUR questions are attempted, then credit will be given for the best FOUR answers.
Electronic calculators are permitted, provided they cannot store text.
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MATH67101
Answer FOUR of the six questions
1. Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let (FBt )t≥0 denote the natural filtration generated by B .
(1.1) State the definition of B . [5 marks]
(1.2) Determine whether (√
t/(1 t)B1 t )
t≥0 defines a standard Brownian motion. Explain your answer. [5 marks]
(1.3) Show that τ = inf { t > 0 : Bt = 1/t } is a stopping time with respect to (FBt )t≥0 . [5 marks]
(1.4) Show that ( B4t − 6tB2t e2Bt−2t 3t2
) t≥0 is a martingale with
respect to (FBt )t≥0 . [5 marks] (1.5) Set Mt = B
4 t − 6tB2t e2Bt−2t 3t2 for t ≥ 0 . Compute E(Mσ)
and E(σ2) when σ = inf { t ≥ 0 : |Bt| = √
3 } . [5 marks]
Order Stochastic Calculus Past Assignment
2. Let X = (Xt)t≥0 be a continuous semimartingale with values in R , let St = sup 0≤s≤t Xs for t ≥ 0 , and let F : R ×R2 → R be a C1,2,1 function.
(2.1) Apply Itô’s formula to F (t,Xt, St) for t ≥ 0 . Determine a continuous local martingale (Mt)t≥0 starting at 0 and a continuous bounded variation process (At)t≥0 such that F (t,Xt, St) = Mt At for t ≥ 0 . [5 marks]
Let B = (Bt)t≥0 be a standard Brownian motion started at zero, let X = (Xt)t≥0 be a non- negative stochastic process solving
dXt = 2
Xt dt dBt (X0 = 0)
and let F (t, x) = t2x3 for t ≥ 0 and x ∈ R .
(2.2) Explain why Itô’s formula can be applied to F (t,Xt) for t ≥ 0 . [3 marks] (2.3) Apply Itô’s formula to F (t,Xt) for t ≥ 0 . Determine a continuous
local martingale (Mt)t≥0 starting at 0 and a continuous bounded variation process (At)t≥0 such that F (t,Xt) = Mt At for t ≥ 0 . [5 marks]
(2.4) Show that (Mt)t≥0 in (2.3) is a martingale and compute 〈M, M〉t for t ≥ 0 . [6 marks]
(2.5) Compute E(τ) when τ = inf { t ∈ [0, 1] : Xt = √
1−t } . [6 marks]
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MATH67101
3. Let B = (Bt)t≥0 be a standard Brownian motion started at zero, let It = ∫ t
0 Bs ds and
St = sup 0≤s≤t Bs for t ≥ 0 , and let F : R ×R2×R → R be a C1,1,2,1 function.
(3.1) Explain why Itô’s formula can be applied to F (t, It, Bt, St) for t ≥ 0 . [5 marks] (3.2) Apply Itô’s formula to F (t, It, Bt, St) for t ≥ 0 . Determine a continuous
local martingale (Mt)t≥0 starting at 0 and a continuous bounded variation process (At)t≥0 such that F (t, It, Bt, St) = Mt At for t ≥ 0 . [6 marks]
(3.3) Show that if Ft(t, i, x, s) xFi(t, i, x, s) 1 2 Fxx(t, i, x, s) = 0 for all
(t, i, x, s) ∈ R ×R2×R with x < s and Fs(t, i, x, s) = 0 when x = s , then F (t, It, Bt, St) is a continuous local martingale for t ≥ 0 . [6 marks]
(3.4) Show that 3(St−Bt)4 B3t −18t(St−Bt)2−3It 9t2 is a martingale for t ≥ 0 . [8 marks]
4. Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let M = (Mt)t≥0 be a stochastic process defined by
Mt =
∫ √log(1 t) 0
√ 2s es
2/2 dBs
for t ≥ 0 .
Order Stochastic Calculus Past Assignment
(4.1) Show that M is a standard Brownian motion. [6 marks]
(4.2) Compute E ( M2t
∫ t 0 (Ms−1)2 ds
) for t ≥ 0 . [6 marks]
(4.3) Compute E ( M2t
∫ t 0 (Ms−1) dMs
) for t ≥ 0 . [6 marks]
(4.4) Consider the process Z = (Zt)t≥0 defined by
Zt =
√ 2(1 t)
√ log(1 t) B√
log(1 t)
for t ≥ 0 . Show by Itô’s formula that Z solves the following stochastic differential equation:
dZt = 1 2 log(1 t)
4(1 t) log(1 t) Zt dt dMt
with Z0 = 0 . [7 marks]
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MATH67101
5. Let B = (Bt)0≤t≤T be a standard Brownian motion started at zero under a probability measure P , and let B̃ = (B̃t)0≤t≤T be a stochastic process defined by