Questions and Answers of Probability And Stochastic Modeling

Show that the constant \(M\) in (21.18) can be chosen in the following way:\[M^{2} \geqslant 2 L^{2}+2 \sum_{j=1}^{n} \sup _{t \leqslant T}\left|b_{j}(t, 0)\right|^{2}+2 \sum_{j=1}^{n} \sum_{k=1}^{d}
The linear growth of the coefficients is essential for Corollary 21.31.a) Consider the case where \(d=n=1, b(x)=-e^{x}\) and \(\sigma(x)=0\). Find the solution of this deterministic ODE and compare
Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(b(x), \sigma(x)\) autonomous and globally Lipschitz continuous coefficients. We have seen in Corollary 21.24 that the
Let \((A, \mathfrak{D}(A))\) be the generator of a diffusion process in the sense of Definition 23.1 and denote by \(a, b\) the diffusion and drift coefficients. Show that \(a \in
Show that under the assumptions of Proposition 23.5 we can interchange integration and differentiation: \(\frac{\partial^{2}}{\partial x_{j} \partial x_{k}} \int p(t, x, y) u(y) d y=\int
Complete the proof of Proposition 23.6 (Kolmogorov's forward equation).Data From 23.6 Proposition 23.6 Proposition (forward equation. Kolmogorov 1931). Let (X+) to denote a diffusion Ex. 23.5
Let \(\left(X_{t}\right)_{t \geqslant 0}\) be a diffusion process with the infinitesimal generator \(L=L(x, D)=\left.A\right|_{\mathcal{C}_{c}^{\infty}}\) as in (23.1). Write
Let \(\left(N_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a continuous, real-valued local martingale and \(u \in \mathcal{C}^{2}(\mathbb{R})\). Show the following Itô formula \(d
Show that the covariance matrix \(C=\left(t_{j} \wedge t_{k}\right)_{j, k=1, \ldots, n}\) appearing in Theorem 2.6 is positive definite.Data From Theorem 2.6 2.6 Theorem. A one-dimensional Brownian
Verify that the matrix \(M\) in the proof of Theorem 2.6 and Corollary 2.7 is a lower triangular matrix with entries 1 on and below the diagonal. Show that the inverse matrix \(M^{-1}\) is a lower
Find out whether the processes \(X(t):=B\left(e^{t}\right)\) and \(X(t):=e^{-t / 2} B\left(e^{t}\right), t \geqslant 0\), have the no-memory property, i.e. o(X(t) ta) o(X(t + a) - X(a): t> 0) for a
Prove the time inversion property from Paragraph 2.15. W
Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use Paragraph 2.17 to show that \(\lim _{t \rightarrow \infty} B_{t} / t=0\) a.s. and in mean square sense.Data From Paragraph
Give a direct proof of the formula (3.5) using the joint probability distribution \(\left(W\left(t_{0}\right), W(t), W\left(t_{1}\right)\right)\) of the Brownian motion \(W(t)\).Data From Formula 3.5
Let for some \(T>0\). Show that for all (Bt)tzo be a BM, e c(R), (0) = 0 and f, g L
Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}, \Phi(y):=\mathbb{P}\left(B_{1} \leqslant y\right)\), and set \(X_{t}:=B_{t}+\alpha t\) for some \(\alpha \in \mathbb{R}\). Use
Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) and denote by \(\left(X_{t}^{x}\right)_{t \geqslant 0}\) the strong solution of the SDE\[d X_{t}=abla c\left(X_{t}\right) d t+d
Let \(F: \mathbb{R} ightarrow[0,1]\) be a distribution function.a) Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and a random variable \(X\) such that
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) and assume that \(X\) is a \(d\)-dimensional random variable which is independent of \(\mathscr{F}_{\infty}^{B}\).a) Show that
Let \(\left(X_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a martingale and denote by \(\mathscr{F}_{t}^{*}\) be the completion of \(\mathscr{F}_{t}\) (completion means to add all subsets of
Let \(\left(X_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a submartingale with continuous paths and \(\mathscr{F}_{t+}=\bigcap_{u>t} \mathscr{F}_{u}\). Show that \(\left(X_{t},
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find a polynomial \(\pi(t, x)\) in \(x\) and \(t\), which is of order 4 in the variable \(x\), such that \(\pi\left(t, B_{t}ight)\)
This exercise contains a recipe how to obtain "polynomial" martingales with leading term \(B_{t}^{n}\), where \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) is a \(\mathrm{BM}^{1}\).a) We know
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\). Find all \(c \in \mathbb{R}\) such that \(\mathbb{E} e^{c\left|B_{t}ight|}\) and \(\mathbb{E} e^{c\left|B_{t}ight|^{2}}\) are finite.
Let \(p(t, x)=(2 \pi t)^{-d / 2} \exp \left(-|x|^{2} /(2 t)ight), x \in \mathbb{R}^{d}, t>0\), be the transition density of a \(d\)-dimensional Brownian motion.a) Show that \(p(t, x)\) is a solution
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(X_{t}=\exp \left(a B_{t}+b tight), t \geqslant 0\), is a martingale if, and only if, \(a^{2} / 2+b=0\).
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion. Which of the following processes are martingales?a) \(U_{t}=e^{c B_{t}}, c \in \mathbb{R}\);b)
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion and \(f \in \mathcal{C}^{1}(\mathbb{R})\). Show that \(M_{t}:=f(t) B_{t}-\int_{0}^{t} f^{\prime}(s)
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\). Show that \(X_{t}=\frac{1}{d}\left|B_{t}ight|^{2}-t, t \geqslant 0\), is a martingale.
Let \(\left(X_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(d\)-dimensional stochastic process and \(A, A_{n}, C \in \mathscr{B}\left(\mathbb{R}^{d}ight), n \geqslant 1\). Thena) \(A \subset C\)
Let \(U \subset \mathbb{R}^{d}\) be an open set and assume that \(\left(X_{t}ight)_{t \geqslant 0}\) is a stochastic process with continuous paths. Show that \(\tau_{U}=\tau_{U}^{\circ}\).
Show that the function \(d(x, A):=\inf _{y \in A}|x-y|, A \subset \mathbb{R}^{d}\), is continuous.
Let \(\tau\) be a stopping time. Check that \(\mathscr{F}_{\tau}\) and \(\mathscr{F}_{\tau+}\) are \(\sigma\)-algebras.
Let \(\tau\) be a stopping time for the filtration \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\). Show thata) \(F \in \mathscr{F}_{\tau+} \Longleftrightarrow \forall t \geqslant 0: F \cap\{\tau
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\). Find \(\mathbb{E} \tau_{R}\) where \(\tau_{R}=\inf \left\{t \geqslant 0:\left|B_{t}ight|=Right\}\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(\sigma, \tau\) be two stopping times such that \(\mathbb{E} \tau, \mathbb{E} \sigma
Let \(B=\left(B_{t}ight)_{t \geqslant 0}\) be a canonical \(\operatorname{BM}^{1}\) on Wiener space \((\Omega, \mathscr{A}, \mathbb{P})=\left(\mathcal{C}_{(\mathrm{o})},
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\).a) Show that \(X_{t}:=\left|B_{t}ight|, t \geqslant 0\), is also a Markov process for the filtration
Compare the (last part of the) first ("pedestrian") proof of Theorem 6.5 with the characterization of a BM by Lemma 5.4. Can you think of a way to use this lemma instead of the direct
Assume that \(\left(X_{t}ight)_{t \geqslant 0}\) is a uniformly bounded stochastic process with exclusively continuous sample paths, \(\left(\mathscr{F}_{t}ight)_{t \geqslant 0}\) is some filtration,
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a Brownian motion on the space \(\left(\mathcal{C}, \mathscr{B}(\mathcal{C}), \mathbb{P}^{0}ight)\). The (canonical) shift operator is the map
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and set \(M_{t}:=\sup _{s \leqslant t} B_{s}\) and \(I_{t}=\int_{0}^{t} B_{s} d s\).a) Show that the two-dimensional
Let \((B(t))_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) and let \(Z\) be a bounded \(\mathscr{F}_{\infty}^{B}\) measurable random variable. Then \(x \mapsto \mathbb{E}^{x} Z\) is in
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(\mathbb{P}\left(\sup _{s \leqslant t}\left|B_{s}ight| \geqslant xight) \leqslant 2 \mathbb{P}\left(\left|B_{t}ight|
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and denote by \(\tau_{b}=\inf \left\{s \geqslant 0: B_{s}=bight\}\) the first time when \(B_{t}\) reaches \(b \in \mathbb{R}\). Show
Let \(\tau=\tau_{(a, b)^{c}}^{\circ}\) be the first exit time of a Brownian motion from the interval \((a, b)\).a) Find \(\mathbb{E}^{x} e^{-\lambda \tau}\) for all \(x \in(a, b)\) and
Let \(W=\left(W_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{2}\) such that \(W_{0}=(a, b)\) with \(a, b>0\). What is the probability that \(W_{t}\) hits first the positive part of the \(x\)-axis
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and set \(M_{t}:=\sup _{s \leqslant t} B_{s}\). Find the distribution of \(\left(M_{t}, B_{t}ight)\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and let \(\tau_{0}\) be the first hitting time of 0 . Find the "density" of \(\mathbb{P}^{x}\left(B_{t} \in d z, \tau_{0}>tight)\),
Let \(K \subset \mathbb{R}^{d}\) be a compact set. Show that there is a decreasing sequence of continuous functions \(\phi_{n}(x)\) such that \(\mathbb{1}_{K}=\inf _{n} \phi_{n}\).Let \(U \supset K\)
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find the distribution of \(\widetilde{\xi}_{t}:=\inf \left\{s \geqslant t: B_{s}=0ight\}\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and set \(M_{t}=\sup _{s \leqslant t} B_{s}\). Denote by \(\xi_{t}\) the largest zero of \(B_{s}\) before time \(t\) and by
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(0
Show that \(\mathcal{C}_{\infty}:=\left\{u: \mathbb{R}^{d} ightarrow \mathbb{R}ight.\) : continuous and \(\left.\lim _{|x| ightarrow \infty} u(x)=0ight\}\) equipped with the uniform topology is a
Let \(A, B \in \mathbb{R}^{d \times d}\) and set \(P_{t}:=\exp (t A):=\sum_{j=0}^{\infty}(t A)^{j} / j\) !.a) Show that \(P_{t}\) is a strongly continuous semigroup. Is it contractive?b) Show that
Let \(\left(P_{t}ight)_{t \geqslant 0}\) and \(\left(T_{t}ight)_{t \geqslant 0}\) be two Feller semigroups with generators \((A, \mathfrak{D}(A))\), resp. \((B, \mathfrak{D}(B))\).a) If
Let \(U_{\alpha}\) be the \(\alpha\)-potential operator of a \(\mathrm{BM}^{d}\). Give a probabilistic interpretation of \(U_{\alpha} \mathbb{1}_{C}\) and \(\lim _{\alpha ightarrow 0} U_{\alpha}
Let \(U_{0}\) be the potential operator of a \(\mathrm{BM}^{d}\) in dimension \(d=1\) or \(d=2\). Show that every \(u \in \mathfrak{D}\left(U_{0}ight)\) such that \(u \geqslant 0\) is trivial, i.e.
Let \(\left(U_{\alpha}ight)_{\alpha>0}\) be the \(\alpha\)-potential operator of a \(\mathrm{BM}^{d}\). Use the resolvent equation to prove the following formulae for \(f \in \mathcal{B}_{b}\) and
Let \(\left(f_{n}ight)_{n \geqslant 1} \subset \mathcal{C}_{\infty}\left(\mathbb{R}^{d}ight)\) be a sequence of functions such that \(0 \leqslant f_{n} \leqslant f_{n+1}\) and \(f:=\sup _{n} f_{n}
Let \(t \mapsto X_{t}\) be a right continuous stochastic process. Show that for closed sets \(F\)\[\mathbb{P}\left(X_{t} \in F \quad \forall t \in \mathbb{R}^{+}ight)=\mathbb{P}\left(X_{q} \in F
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}, f: \mathbb{R}^{d} ightarrow \mathbb{R}\) be a continuous function such that \(\int_{0}^{t} f\left(B_{s}ight) d s=0\) for all \(t>0\).
Let Show that the random variable \(\beta\left(\sigma_{n}ight)\) has a probability density. (b, ) be a BM and = infit > 1/n b(t) = 0}
Let \(\left(\Pi_{n}ight)_{n \geqslant 1}\) be a sequence of refining (i.e. \(\left.\Pi_{n} \subset \Pi_{n+1}ight)\) partitions of \([0,1]\) such that \(\lim _{n ightarrow
Let \(f=(g, h):[0, \infty) ightarrow \mathbb{R}^{2}\) and \(p>0\). Show that \(\operatorname{VAR}_{p}(f ;[0, t])
Let \(f \in \mathcal{C}[0,1]\). For every partition \(\Pi=\left\{t_{0}=0
Let \(f\) be continuous. Show that it does not affect the finiteness of \(\operatorname{VAR}_{p}(f ;[0, t])\) and the numerical value of \(\operatorname{var}_{p}(f ;[0, t])\) if we restrict ourselves
(Quadratic variation) Let \((B(t))_{t \geqslant 0}\) be a one-dimensional Brownian motion. Consider the random variables
Show that \(\mathrm{BM}^{1}\) is almost surely not \(1 / 2\)-Hölder continuous:a) For all \(Z \sim \mathrm{N}(0,1)\) and \(x>0\) we have\[\frac{1}{\sqrt{2 \pi}} \frac{x e^{-x^{2} / 2}}{x^{2}+1}x)c
Let \(X \sim \mathrm{N}(0,1)\). Show that for every \(\lambda_{0} \in\left(0, \frac{1}{2}ight)\) there is a constant \(C=C\left(\lambda_{0}ight)\) such that \(\sup _{\lambda \leqslant \lambda_{0}}
Let \(X\) be a real-valued random variable on \((\Omega, \mathscr{A}, \mathbb{P})\) and \(\mathscr{F} \subset \mathscr{A}\) be \(\sigma\)-algebra. Show thatShow that \(\mathbb{E}\left(e^{i \xi X}
Prove that in \(\mathbb{R}^{n}\) all \(\ell^{p}\)-norms \((1 \leqslant p \leqslant \infty)\) are equivalent:\[\max _{1 \leqslant j \leqslant n}\left|x_{j}ight|
Show that for \(\alpha \in(0,1)\) the function \(Z \mapsto \mathbb{E}\left(|Z|^{\alpha}ight)\) is subadditive and complete the argument in the proof of Theorem 10.1 for this case.Data From Theorem
The proof of Theorem 10.3 actually shows, that almost all Brownian paths are nowhere Lipschitz continuous. Modify the argument of this proof to show that almost all Brownian paths are nowhere Hölder
Use Theorem 10.6 to show that the strong \(p\)-variation \(\operatorname{VAR}_{p}(B ;[0,1])\) of \(\mathrm{BM}^{1}\) is for \(p>2\) finite.Data From Theorem 10.6 10.6 Theorem (Lvy 1937). Let (B)o
Verify that s-dimensional Hausdorff measure is an outer measure.
Show that the \(d\)-dimensional Hausdorff measure of a bounded open set \(U \subset \mathbb{R}^{d}\) is finite and positive: \(0
Let \(E \subset \mathbb{R}^{d}\). Show that Hausdorff dimension \(\operatorname{dim} E\) coincides with the numbers (E)-co}, sup [a30: sup {a 0 (E) > 0} inf (a 0:(E)
Let \(\left(E_{j}ight)_{j \geqslant 1}\) be subsets of \(\mathbb{R}^{d}\) and \(E:=\bigcup_{j \geqslant 1} E_{j}\). Show that \(\operatorname{dim} E=\sup _{j \geqslant 1} \operatorname{dim} E_{j}\).
Let \(E \subset \mathbb{R}^{d}\). Show that \(\operatorname{dim}\left(E \times \mathbb{R}^{n}ight)=\operatorname{dim} E+n\). Does a similar formula hold for \(E \times F\) where \(E \subset
Let \(f: \mathbb{R}^{d} ightarrow \mathbb{R}^{n}\) be a bi-Lipschitz map, i.e. both \(f\) and \(f^{-1}\) are Lipschitz continuous. Show that \(\operatorname{dim} f(E)=\operatorname{dim} E\). Is this
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\). Show that for \(\lambda
Let \(F \subset \mathbb{R}\) be a non-void perfect set, i.e. a closed set such that each point in \(F\) is an accumulation point of \(F\). Show that a perfect set is uncountable.
Show that Corollary 11.26 also follows from Lemma 11.23.Calculate \(\frac{\partial^{2}}{\partial u \partial s}\left(1-\frac{2}{\pi} \arccos \sqrt{\frac{s}{u}}ight)\).Data From Corollary 11.26Data
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use the Borel-Cantelli lemma to show that the running maximum \(M_{n}\) := \(\sup _{0 \leqslant t \leqslant n} B_{t}\) cannot grow
Show that the proof of Khinchine's LIL, Theorem 12.1, can be modified to give \[\varlimsup_{t ightarrow \infty} \frac{\sup _{s \leqslant t}|B(s)|}{\sqrt{2 t \log \log t}} \leqslant 1\]Use in Step
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use Theorem 12.5 to show that \(\kappa(t)=(1+\epsilon) \sqrt{2 t \log |\log t|}\) is an upper function for \(t ightarrow 0\).Data
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Deduce from Theorem 12.5 the following test for upper functions in large time. Assume that \(\kappa \in \mathcal{C}[1, \infty)\) is
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion, \(a, b>0\) and \(\tau:=\inf \left\{t \geqslant 0:\left|B_{t}ight|=b \sqrt{a+t}ight\}\). Show thata) \(\mathbb{P}(\tau
Let \(w \in \mathcal{C}_{(\mathrm{o})}[0,1]\) and assume that for every fixed \(t \in[0,1]\) the number \(w(t)\) is a limit point of the family \(\left\{Z_{s}(t): s>eight\} \subset \mathbb{R}\). Show
Let \(\mathcal{K}\) be the set from Theorem 13.1. Show that for \(w \in \mathcal{K}\) the estimate \(|w(t)| \leqslant \sqrt{t}\), \(t \in[0,1]\) holds.Data From Theorem 13.1 13.1 Theorem (Strassen
Let \(u \in \mathcal{H}^{1}\) and denote by \(\Pi_{n}, n \geqslant 1\), a sequence of partitions of \([0,1]\) such that \(\lim _{n ightarrow \infty}\left|\Pi_{n}ight|=0\). Show that the
Let \(\phi \in \mathrm{BV}[0,1]\) and consider the following Riemann-Stieltjes integral\[G^{\phi}(w)=\phi(1) w(1)-\int_{0}^{1} w(s) d \phi(s), \quad w \in \mathcal{C}_{(0)}[0,1]\]Show that
Let \(w \in \mathcal{C}_{(0)}[0,1]\). Find the densities of the following bi-variate random variables:a) \(\left(\int_{1 / 2}^{t} s^{2} d w(s), w(1 / 2)ight)\) for \(1 / 2 \leqslant t \leqslant
Show that \(F:=\left\{w \in \mathcal{C}_{(0)}[0,1]: \sup _{q^{-1} \leqslant c \leqslant 1} \sup _{0 \leqslant r \leqslant 1}|w(c r)-w(r)| \geqslant 1ight\}\) is for every \(q>1\) a closed subset of
Check the (in)equalities from Step \(3^{\circ}\) in the proof of Lemma 13.20.Data From Leema 13.20 13.20 Lemma. (w)>() for almost all w = 0. Proof. 1 Since for every w = D() we have 2rw = (r), r 0 a
In the proof of Theorem 14.2 we assume that \(\left(B_{t}ight)_{t \geqslant 0}\) and \((U, W)\) are independent. Show that \(\mathscr{F}_{t}:=\sigma\left(B_{s}, s \leqslant t ; U, Wight)\) is an
Let \(\left(M_{n}, \mathscr{F}_{n}ight)_{n \geqslant 0}\) and \(\left(N_{n}, \mathscr{F}_{n}ight)_{n \geqslant 0}\) be \(L^{2}\) martingales; then \(\left(M_{n} N_{n}-\langle M, Nangle_{n},
Let \(\left(M_{n}, \mathscr{F}_{n}ight)_{n \geqslant 0}\) and \(\left(N_{n}, \mathscr{F}_{n}ight)_{n \geqslant 0}\) be \(L^{2}\) martingales. Show that \(\left|\langle M, Nangle_{n}ight| \leqslant\)
Show that \(\|M\|_{\mathcal{M}_{T}^{2}}:=\left(\mathbb{E}\left[\sup _{s \leqslant T}\left|M_{s}ight|^{2}ight]ight)^{\frac{1}{2}}\) is a norm in the family \(\mathcal{M}_{T}^{2}\) of equivalence

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