prof. dr. t. meyer-brandis m. bauer, d. ritter winter term 2018/19 mathematical finance ii · 2018....
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Prof. Dr. T. Meyer-Brandis M. Bauer, D. Ritter Winter term 2018/19
Mathematical Finance IIProblem Sheet 11
Exercise∗ 1 (9 points): Let f : [0,∞)×R→ R be bounded and continuous. Assume that f istwice continuously differentiable in the interior (0,∞) × R of its domain and fulfills there theheat equation
∂f
∂t(t, x) =
1
2
∂2f
∂x2(t, x).
Let B be standard Brownian Motion and Mt := maxBs : 0 ≤ s ≤ t its running maximum.
i) Show that for any t > 0 and x ∈ R, the process (f(t− s, x−Bs))0≤s≤t is a martingale.
ii) Show that E[Mt ∨m] = g(t,m) for t > 0 and m ≥ 0 with
g(t,m) := 2mΦ(t−12m)−m+ 2
√tϕ(t−
12m),
where ϕ(z) = 1√2πe−
z2
2 denotes the density of the standard normal distribution and Φ(x) =∫ x−∞ ϕ(z)dz its cumulative distribution function.
Hint: You may use here that P(Mt ≥ m) = P(|Bt| ≥ m) = 2P(Bt ≥ m) for all t ≥ 0 andm ≥ 0.
iii) Show that (g(t− s, x−Bs))0≤s≤t is a martingale.
iv) Let T > 0 be a finite time horizon. Show that for 0 ≤ t < T we have P-a.s.
E[MT |Ft] = Bt + g(T − t,Mt −Bt),
where (Ft)t∈[0,T ] is the natural filtration of (Bt)t∈[0,T ].
Hint: You may use the Markov property of the Brownian motion.
Exercise 2: Let B = (Bt)t∈R+ be a real-valued Brownian motion and T > 0. For each X ∈L2([0, T ]), find the process Φ ∈ Λ2([0, T ]) such that
X = E[X]+
∫ T
0
Φs dBs,
holds.
i) X = B3T
ii) X = eBT
iii) X =∫ T0Bt dt
iv) X = sinBT
Exercise 3: Let B = (Bt)t∈[0,1] be a real-valued (Ft)t∈[0,1]-Brownian motion on a filtered prob-ability space (Ω,F , (Ft)t∈[0,1],P). Consider the process
(Xa,bt
)t∈[0,1) which is given by
Xa,bt = a(1− t) + bt+ (1− t)
∫ t
0
1
1− sdBs for all 0 ≤ t < 1, (∗∗)
where a, b ∈ R.
i) Is the process Xa,bt well-defined? Explain your answer.
ii) Show that Xa,bt solves on [0, 1) the stochastic differential equation
dXt = b−Xt
1−t dt+ dBt
X0 = a.
iii) Show that Xa,bt converges to b in L2(Ω) as t→ 1.
iv) Show thatX0,0t has the same distribution as the Brownian BridgeBt−tB1 for all 0 ≤ t < 1.
Exercise 4: Consider the SDE given bydXt = f(t,Xt)dt+ σ(t)XtdBt
X0 = x,
where f : R+ × R → R and σ : R+ → R are continuous functions, σ ∈ Λ2 and B is a1-dimensional Brownian motion. Moreover, define
Zt := exp
(−∫ t
0
σ(s)dBs +1
2
∫ t
0
σ2(s)ds
).
i) Show thatd(ZtXt) = Ztf(t,Xt)dt,
and thus by setting Yt(ω) = Zt(ω)Xt(ω) for each ω ∈ Ω, we get
dYt(ω)
dt= Zt(ω)f
(t, Z−1t (ω)Yt(ω)
), Y0 = x.
Note that for each ω ∈ Ω this is a deterministic differential equation.
ii) Use i) to solve the SDE
dXt =1
Xt
dt+ αXtdBt, X0 = x > 0,
where α ∈ R.Hint: Use the separation of variables.