Mathematisch-
Naturwissenschaftliche Fakultät
Fachbereich Mathematik
Prof. Dr. Andreas Prohl Dr. Ananta Kumar Majee
Stochastische Differentialgleichungen
Summer Semester 2017 Tübingen, 28.06.2017
Homework 10
Problem 1.LetXbe anRn-valued Feller process andL be its generator.
i) Show thatL is closed.
ii) Show thatL satisfies the positive maximum principle, i.e., for allf ∈ D(L), there holds (Sf)(x0)≤0 for f(x0) = sup
x∈Rn
f(x)≥0.
Hint: fori)Letf ∈ D(L). Deduce and use that dtdStf =LStf (t >0), where d
dtStf = lim
h↓0
St+hf − Stf
h .
Problem 2.Let
Ps,t; 0≤s≤t <∞ be a family of mappings fromRn× B(Rn)to[0,1]. We call them normal transition family if for each0≤s≤t <∞, there hold:
(i) the mapsx7→Ps,t[x,A]are measurable for eachA ∈ B(Rn) (ii) Ps,t[x,·]is a probability measure onB(Rn)for eachx∈Rn (iii) the Chapman-Kolmogorov equations are valid.
Let moreover ν ∈ P(Rn) be given. Prove that there exist a probability space (Ω,F,Pν), a filtration {Ft;t≥0}and a Markov process
Xt;t≥0 on that space such that a) Pν
Xt∈A
Xs=x
=Ps,t[x,A]almost surely for each0≤s≤t <∞,x∈RnandA ∈ B(Rn).
b) L(X0) =ν.
Hint:Use Kolmogorov’s continuation theorem.
Problem 3.Let(b, σσσ)be Lipschitz, and with asymptotic linear growth. Show that the strong solution of the related SDE (with proper initial condition) is a Markov process.
Hint:Use that the solution is a Brownian flow.
Date of submission: 05.07.2017.
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