Stochastische Prozesse

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Stochastische Prozesse

^{WS 15/16}

Vorlesung: Prof. Dr. Thorsten Schmidt Exercise: Dr. Tolulope Fadina

http://www.stochastik.uni-freiburg.de/lehre/2015WiSe/inhalte/2015WiSeStochProz

Exercise 6

Submission: 24-11-2015

Problem 1 (4 Points). (a) LetX be a submartingale,n∈Z⁺ and letλ >0. Show that λP

1≤i≤nmax |X_i| ≥3λ

≤4E[|X₀|] + 3E[|X_n|].

(b) Let{X_n}^∞_n=1 be a sequence of integrable random variables on a probability space(Ω,F,P) which converges weakly in L¹(P) to an integrable random variableX. Show that for each σ-eld G ⊂ F, the sequenceE[Xn|G]converges to E[X|G]weakly in L¹(P)

Problem 2 (4 Points). (a) Let X = (Xt)0≤t<∞ be a local martingale and τ is a stopping time. Show that Yt=Xt∧τ is also a local martingale.

(b) X = (X_n)n∈N be i.i.d withP(X₁ = 1) =pand P(X₁=−1) =q= 1−p. Furthermore, Sn=

n

X

i=1

Xi

and

τ = inf{n≥1 :S_n≥b} (1)

whereb∈N. For {· · · }=∅ in (1) setτ =∞ and on{τ =∞}

S_τ = lim

n→∞S_n, if the limit exists. Show that

P(τ <∞) = p

q b

for p < q.

Problem 3 (4 Points). Let X = (Xt)0≤t<∞ be a right-continuous martingale with respect to F_t. X is said to be square integrable if E[X_t²]<∞and X₀= 0 a.s., and we write X∈ M₂. Let X be a process in M₂ or in M_loc, and we assume its quadratic variationhXiis integrable.

i.e., E[hXi_∞]<∞.Show that (a) X is a martingale

(b) X and submartingale X² are both uniformly integrable, in particular X∞= lim

t→∞X_t exists almost surely and

E[X_∞² ] =E[hXi_∞]

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Hint: Conditions(a)−(d)are equivalent: (a)Xis uniformly integrable family of random variables, (b) X converges in L¹ ast → ∞, (c) X converges almost surely to an integrable variable X∞, such thatX_tis a martingale (respectively submartingale), (d) there exists an integrable random variable Y such that Xt =E[Y|F_t]P-a.s. for everyt≥0. Note: conditions(a)−(c) also holds for non-negative right-continuous submartingale X.

If X∈ M_loc andτ is a stopping time ofF_t, thenE[X_τ²]≤E[hXi_τ], where X_∞² = lim_t→∞X_t².

Problem 4 (4 Points). (a) Show that for any optional timeτ and predictable process X, the random variable X_τ1_{{τ <∞}} isF_τ−-measurable.

(b) Let A ∈ V. Show that there exist a unique pair (B, C) of adapted increasing processes such thatA=B−C andV ar(A) =B+C.

Hint: If A is predictable,B, C and V ar(A) are also predictable.