MATH67101 STOCHASTIC CALCULUS 2020
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MATH67101
STOCHASTIC CALCULUS
2020
SECTION A
Answer FOUR of the six questions
1. Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let (F)t≥0 natural filtration generated by B .
(1.1) State the definition of B .
(1.2) Determine whether (B1+2t − B1)/t≥0 defines a standard
Brownian motion. Explain your answer.
(1.3) Show that τ = inf {t > 0 : Bt = log(t)} is a stopping time with respect to (F)t≥0 .
(1.4) Show that Bt+e√2Bt −tt≥0 is a martingale with respect to (F)t≥0 .
(1.5) Set Mt = Bt + e√2Bt −t for t ≥ 0 . Compute E(Mσ) and E(e−σ) when σ = inf {t ≥ 0 : Bt = 1 } .
denote the
[5 marks]
[5 marks]
[5 marks] [5 marks]
[5 marks]
2. Let X = (Xt)t≥0 be a continuous semimartingale with values in R , let St = sup0≤s≤t Xs and It = Ss ds for t ≥ 0 , and let F : R3 → R be a C2,1,1 function.
(2.1) Apply Itˆo’s formula to F(Xt,St,It) 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(Xt,St,It) = 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 = 3dt + 2XtdBt (X0 = 0)
and let F(t,x) = tx2 for t ≥ 0 and x ∈ R+ .
(2.2) Explain why Itˆo’s formula can be applied to F(t,Xt) for t ≥ 0 .
(2.3) Apply Itˆo’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 .
(2.4) Show that (Mt)t≥0 in (2.3) is a martingale and compute M,Mt for t ≥ 0 .
(2.5) Compute E(τ) when τ = inf {t ∈ [0, 5] : Xt = 1 −t/5 } .
[3 marks]
[5 marks]
[6 marks] [6 marks]
3. Let B = (Bt)t≥0 be a standard Brownian motion started at zero, let St = sup0≤s≤t Bs and It = inf0≤s≤t Bs for t ≥ 0 , and let F : R+×R ×R+ ×R− → R be a C1,2,1,1 function.
(3.1) Explain why Itˆo’s formula can be applied to F(t,Bt,St,It) for t ≥ 0 .
(3.2) Apply Itˆo’s formula to F(t,Bt,St,It) 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,Bt,St,It) = Mt+At for t ≥ 0 .
(3.3) Show that if Ft(t,x,s,i)+Fxx(t,x,s,i) = 0 for all (t,x,s,i) with Fs(t,x,s,i) = 0 for x = s and Fi(t,x,s,i) = 0 for x = i , then F(t,Bt,St,It) is a continuous local martingale for t ≥ 0 .
(3.4) Show that (St − Bt)4 +(Bt − It)2 − (6t − 1)(St − Bt)2 +3t2 − 2t is
a martingale for t ≥ 0 .
[5 marks]
[6 marks]
[6 marks]
[8 marks]
4. Let B = (Bt)t≥0 be a standard Brownian motion started at zero, and let M stochastic process defined by
Mt = 0et − 1 dBs
for t ≥ 0 .
(4.1) Show that M is a standard Brownian motion.
(4.2) Compute E(1+Mt)2 M ds for t ≥ 0 .
(4.3) Compute E(1+Mt)2 M dMs for t ≥ 0 .
(4.4) Consider the process Z = (Zt)t≥0 defined by
Zt = f(t)Bet − 1
for t ≥ 0 where f : R+ → R is a C1 function. Using Itˆo’s formula examine whether f can be chosen so that Z solves
dZt = − Zt dt + dMt (Z0 = 0).
If this is possible, determine f explicitly.
= (Mt)t≥0 be a
[5 marks] [6 marks] [6 marks]
[8 marks]
5. Let B = (Bt)0≤t≤T be a standard Brownian motion started at zero under a probability measure P , and let = (t)0≤t≤T be a stochastic process defined by
t = Bt − 0t Bs eBs I(Bs ≤ −1) ds
for t ∈ [0,T] , where T > 0 is a given and fixed constant.
(5.1) Determine a probability measure under which is a standard
Brownian motion.
(5.2) Compute ( sdBs+t − sBs eBs I(Bs ≤ −1) ds)2 for t ≥ 0 . (5.3) Compute E(|τ|) when τ = inf {t ≥ 0 : Bt = − 1 or Bt = 2 } .
(5.4) Compute e6(Bσ −0σ BseBs I(Bs≤−1) ds) − 18σ and e− 18σ when σ = inf {t ≥ 0 : Bt = + Bs eBs I(Bs ≤ −1) ds} .
[7 marks] [4 marks] [7 marks]
[7 marks]
(Recall that denotes expectation under , 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+2Xt)dt + (3+Xt)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. (6.2) Verify by Itˆo’s formula that this solution is given by
Xt = Yt 1 − 20t ds + 30t dBs
for t ≥ 0 , where the process Y = (Yt)t≥0 solves
dYt = 2Yt dt + YtdBt
with Y0 = 1 .
(6.3) Compute E X,Y t for t ≥ 0 .
(6.4) Show that the following identity in law holds
Xt − Yt 1 + 30t dBs 20t Ys ds
for each t ≥ 0 given and fixed.
[4 marks]
[7 marks] [7 marks]
[7 marks]
2022-01-26