stochastic calculus past exam

could you answer all the questions?

MATH6710

1

Three hours

UNIVERSITY OF MANCHESTER

STOCHASTIC CALCULUS

27 January 2017

14:00 – 17:00

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]

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 .

(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

B̃t = Bt +

∫ t
0

eBs I(Bs≤2) ds

for t ∈ [0, T ] where T > 0 is a given and fixed constant.

(5.1) Determine a probability measure P̃ under which B̃ is a standard
Brownian motion. [7 marks]

(5.2) Compute Ẽ
(
(
∫ t

0
B̃2s dBs+

∫ t
0
B̃2s e

BsI(Bs≤2) ds)2
)

for t ≥ 0 . [4 marks]
(5.3) Compute E(B̃τ ) when τ = inf { t ≥ 0 : |Bt| = 1 } . [7 marks]
(5.4) Compute Ẽ

(
e2(Bσ+

∫ σ
0 e

BsI(Bs≤2) ds)−2σ) and Ẽ(e−2σ) when
σ = inf { t ≥ 0 : Bt = 1−

∫ t
0
eBsI(Bs≤2) ds } . [7 marks]

(Recall that Ẽ denotes expectation under P̃ , and E denotes expectation under P .)

6. Let B = (Bt)t≥0 be a standard Brownian motion started at zero. Consider the stochastic
differential equation

dXt = (1+3Xt) dt + (5+2Xt) dBt

for a stochastic process X = (Xt)t≥0 where it is assumed that X0 = 1 .

(6.1) Show that there exists a unique strong solution X to this equation. [4 marks]

(6.2) Verify by Itô’s formula that this solution is given by

Xt = Yt

(
1− 9

∫ t
0
1

Ys
ds + 5

∫ t
0
1

Ys
dBs

)

for t ≥ 0 , where the process Y = (Yt)t≥0 solves

dYt = 3Yt dt + 2Yt dBt

with Y0 = 1 . [7 marks]

(6.3) Compute E
(〈X,Y 〉t

)
for t ≥ 0 . [7 marks]

(6.4) Show that the following identity in law holds:

Xt − Yt
(
1 + 5

∫ t
0
1
Ys
dBs

)
law
= −9

∫ t
0

Ys ds

for each t ≥ 0 given and fixed. [7 marks]

END OF EXAMINATION PAPER

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