- Table 8.1 No . 2 records the solution to the initial value problem (8.6). Explain how this result follows from our considerations regarding (8.3) and (8.7).Data From Equation (8.3)Data From Problem
- Let \(\left(X_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a real-valued process whose paths \(t \mapsto X_{t}(\omega)\) are (a.s.) right continuous and have finite left-hand limits (this
- Let \(\tau:=\tau_{(-a, b)^{c}}^{\circ}\) be the first entrance time of a \(\mathrm{BM}^{1}\) into the set \((-a, b)^{c}\).a) Show that \(\tau\) has finite moments \(\mathbb{E} \tau^{n}\) of any order
- Let \(\left(X_{t}\right)_{t \geqslant 0}\) be a \(d\)-dimensional Feller process and let \(f, g \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)\). Show that the function \(x \mapsto
- Let \(\left(P_{t}\right)_{t \geqslant 0}\) be the transition semigroup of a \(\mathrm{BM}^{d}\) and denote by \(L^{p}, 1 \leqslant p
- Let \(\left(T_{t}\right)_{t \geqslant 0}\) be a Markov semigroup given by \(T_{t} u(x)=\int_{\mathbb{R}^{d}} u(y) p_{t}(x, d y)\) where \(p_{t}(x, C)\) is a kernel in the sense of Remark 7.6. Show
- Show that \(p_{t_{1}, \ldots, t_{n}}^{x}\left(C_{1} \times \cdots \times C_{n}\right)\) of Remark 7.6 define measures on \(\mathscr{B}\left(\mathbb{R}^{d \cdot n}\right)\).Data From 7.6 Remark 7.6
- Let \(\left(T_{t}\right)_{t \geqslant 0}\) be a Feller semigroup, i.e. a strongly continuous, positivity preserving sub-Markovian semigroup on
- Complete the following alternative argument for Example 7.16. Assume that \(\left(u_{n}\right)_{n \geqslant 1} \subset \mathcal{C}_{\infty}^{2}(\mathbb{R})\) such that \(\left(\frac{1}{2}
- Let \((A, \mathfrak{D}(A))\) be the generator of a \(\mathrm{BM}^{d}\). Adapt the arguments of Example 7.25 and show that \(\mathcal{C}_{\infty}^{2}\left(\mathbb{R}^{d}\right) \varsubsetneqq
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and consider the two-dimensional process \(X_{t}:=\left(t, B_{t}\right), t \geqslant 0\).a) Show that \(\left(X_{t}\right)_{t
- A Poisson process is a real-valued stochastic process \(\left(N_{t}\right)_{t \geqslant 0}\) such that \(N_{0}=0\), \(N_{t}-N_{s} \sim N_{t-s}\) and for \(t_{0}=00\). In particular,
- Let \(C\) denote Cantor's discontinuum which is obtained if we remove recursively the open middle third of any remaining interval:\[[0,1] \leadsto\left[0, \frac{1}{3}\right] \cup\left[\frac{2}{3},
- Denote by \(\mathcal{H}^{1}\) the Cameron-Martin space (13.2) and by \(\mathcal{H}_{\circ}^{1}\) the set defined in (13.3).a) Show that \(\mathcal{H}^{1}\) is a Hilbert space with the canonical norm
- a) Show that b) Show that Let B, H, T be real random variables on the same probability space. We write "", and "H", for "independent", and "conditionally independent given H", respectively.
- Show that the limit (15.19) does not depend on the approximating sequence.Data From (15.19) L(P)-lim fn Bt 11-00 (15.19)
- Use Theorem 15.15.c) to show that the stochastic integrals for the right and left continuous simple processes \(f(t, \omega):=\sum_{j=1}^{n} \phi_{j-1}(\omega) \mathbb{1}_{\left[s_{j-1},
- Let \(\left(B_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a one-dimensional Brownian motioin and \(\tau\) a stopping time. Show that \(f(s, \omega):=\mathbb{1}_{[0, T \wedge \tau(\omega))}(s), 0
- Show that the process \(f^{2} \bullet\langle Mangle_{t}:=\int_{0}^{t}|f(s)|^{2} d\langle Mangle_{s}\) appearing in Theorem 17.9.b) is adapted.Data From 17.9 Theorem 17.9 Theorem. Let (Mt, ter be in
- The following exercise contains an alternative proof of Itô's formula (18.1) for a one-dimensional Brownian motion \(\left(B_{t}\right)_{t \geqslant 0}\).a) Let \(f \in
- a) Use the (two-dimensional, deterministic) chain rule \(d(F \circ G)=F^{\prime} \circ G d G\) to deduce the formula for integration by parts for Stieltjes integrals:\[\int_{0}^{t} f(s) d g(s)=f(t)
- Show that \(\beta_{t}=\int_{0}^{t} \operatorname{sgn}\left(B_{s}\right) d B_{s}\) is a \(\mathrm{BM}^{1}\).Use Lévy's characterization of a \(\mathrm{BM}^{1}\), Theorem 9.13 or 19.5.Data From
- State and prove a \(d\)-dimensional version of the Burkholder-Davis-Gundy inequalities (19.21) if \(p \in[2, \infty)\).Use the fact that all norms in \(\mathbb{R}^{d}\) are equivalentData From
- Let and assume that Show that f.g L(R.)
- Let \(f \in L^{2}\left(\mathbb{R}_{m}^{+}\right), g \in L^{2}\left(\mathbb{R}_{n}^{+}\right)\)and assume that the functions are symmetric: \(f=\widehat{f}\), \(g=\widehat{g}\). We define the
- Let \(X=\left(X_{t}\right)_{t \geqslant 0}\) be the process from Example 21.2. Show that \(X\) is a Gaussian process with independent increments and find \(C(s, t)=\mathbb{E} X_{s} X_{t}, s, t
- Let \(B_{t}=\left(b_{t}, \beta_{t}\right)\) be a \(\mathrm{BM}^{2}\). Solve the \(\mathrm{SDE}\)\[X_{t}=x+b \int_{0}^{t} X_{s} d s+\sigma_{1} \int_{0}^{t} X_{s} d b_{s}+\sigma_{2} \int_{0}^{t} X_{s}
- Show that in Example 21.7\[X_{t}^{\circ}=\exp \left(-\int_{0}^{t}\left(\beta(s)-\delta^{2}(s) / 2\right) d s\right) \exp \left(-\int_{0}^{t} \delta(s) d B_{s}\right)\]and verify the expression given
- Show that Remark 22.5.a) remains valid if we replace \(f(x)\) by \(f(s, x)\). 22.5 Remark. a) The Stratonovich integral satisfies the classical chain rule. Indeed, if fe e(R) and dX = (t) dB + b(t)
- Show that Example 22.7 remains valid if we assume that \(\sigma(0)=0\) and \(\sigma(x)>0\) for \(x eq 0\). 22.7 Example. Let (Bt)to be a BM and assume that = c(R) is either strictly positive or
- Let \(B=\left(B_{t}\right)_{t \in[0,1]}\) be a \(\mathrm{BM}^{1}\) and set \(W=\left(W_{t}\right)_{t \in[0,1]}\) where \(W_{t}:=B_{t}-t B_{1}\).a) Show that \(W\) is a mean zero Gaussian process and
- Let \(\left(\mathcal{C}_{(\mathrm{o})}, \mathscr{B}\left(\mathcal{C}_{(\mathrm{o})}\right), \mu\right)\) be the canonical Wiener space (we assume \(d=1\) ). We will now consider the space
- We continue with the set-up introduced in Problem 7.Denote by \(\left(\mathcal{C}_{(0)}, \mathscr{B}\left(\mathcal{C}_{(0)}\right), \mu\right)\) the canonical Wiener space and set
- Let \(B=\left(B_{t}\right)_{t \geqslant 0}\) be a canonical \(\mathrm{BM}^{1}\) on Wiener space \((\Omega, \mathscr{A}, \mathbb{P})=\left(\mathcal{C}_{(0)}, \mathscr{B}\left(\mathcal{C}_{(0)}\right),
- What is Stats?
- Let \(\left(\mathscr{F}_{t}\right)_{t \geqslant 0}\) be an admissible filtration for the Brownian motion \(\left(B_{t}\right)_{t \geqslant 0}\). Mimic the proof of Lemma 2.14 and show that for
- Assume that the processes in Lemma 5.7 or in Lemma 5.8 are only "almost surely right continuous", resp. "almost surely continuous". Identify all steps in the proofs where this becomes relevant and
- Let \(\left(B_{t}, \mathscr{F}_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) such that \(\mathscr{F}_{0}\) contains all measurable null sets and let \(\tau\) be a stopping time. Show that
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) with transition semigroup \(\left(P_{t}\right)_{t \geqslant 0}\). Show, using arguments similar to those in the proof of
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{d}\) and set \(u(t, x):=P_{t} u(x)\). Adapt the proof of Proposition 7.3.g) and show that for \(u \in
- a) Let \(d(x, A):=\inf _{a \in A}|x-a|\) be the distance between the point \(x \in \mathbb{R}^{d}\) and the set \(A \in \mathscr{B}\left(\mathbb{R}^{d}\right)\). Show that \(x \mapsto d(x, A)\) is
- Show that Dynkin's formula (7.30) in Proposition 7.31 follows from Theorem 7.30.Data From Theorem 7.30 Ex. 7.18 7.30 Theorem. Let (Xt, F)to be a Feller process on Rd with transition semigroup
- Assume that \(L: \mathcal{C}_{c}^{\infty}\left(\mathbb{R}^{d}\right) \rightarrow \mathcal{C}\left(\mathbb{R}^{d}\right)\) is a local operator (cf. Definition 7.36) which is almost positive, i.e. for
- Let \(f\) be the initial value in the problem considered in Lemma 8.1 and write \(u(t, x):=\) \(P_{t} f(x)=\mathbb{E} f\left(B_{t}+x\right)\) where \(\left(B_{t}\right)_{t \geqslant 0}\) is a BM \({
- Assume that in Lemma 8.1 the initial datum satisfies \(f \in \mathcal{C}_{\infty}\left(\mathbb{R}^{d}\right)\) but not necessarily \(f \in \mathfrak{D}(A)\). Consider (8.3) with \(P_{\epsilon} f\)
- Complete the approximation argument for Lévy's arc-sine law from \(\S 8.9\) :a) Show, by a direct calculation, that \(v_{n, \lambda}(x)\) converges as \(n \rightarrow \infty\). Conclude from (8.16)
- Show Theorem 8.5 with semigroup methods.Observe that \(A \int_{0}^{t} P_{s} g d s=P_{t} g-g=\frac{d}{d t} \int_{0}^{t} P_{s} g d s-g\).Data From Theorem 8.5 8.5 Theorem. Let (Bt)to be a BMd and g =
- Find the solution to the Dirichlet problem in dimension \(d=1: u^{\prime \prime}(x)=0\) for all \(x \in(0,1), u(0)=a, u(1)=b\) and \(u\) is continuous in [0,1]. Compare your findings with Wald's
- Use Lemma 7.33 and give an alternative derivation of the result of Lemma 8.10.Data From Lemma 7.33Data From Leema 8.10 7.33 Lemma. Let (X,,F)o be a Feller process with a right continuous filtration,
- Show that Lemma 8.10 remains true for any \(d\)-dimensional Feller process \(X_{t}\) with continuous paths and generator \(L=\sum_{j, k=1}^{d} a_{j k}(x) \partial_{j} \partial_{k}+\sum_{j=1}^{d}
- Let \(g:[0, \infty) \times \mathbb{R}^{d} \rightarrow \mathbb{R}\) be a bounded continuous function such that \(g(t, \cdot)\) is \(\kappa\)-Hölder continuous with a Hölder constant which does not
- Use the LIL (Corollary 12.2) to give an alternative proof for the fact that a onedimensional Brownian motion oscillates in every time interval \([0, \epsilon]\) infinitely often around its starting
- Let \(d \geqslant 2\). A flat cone in \(\mathbb{R}^{d}\) is a cone in \(\mathbb{R}^{d-1}\). Adapt the argument of Example 8.18.e) to show the following useful regularity criterion for a
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a one-dimensional Brownian motion. Show that \[\operatorname{dim} B^{-1}(A) \leqslant \frac{1}{2}+\frac{1}{2} \operatorname{dim} A \quad \text { a.s. for
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and denote by \(g_{t}\) the last zero before time \(t>0\).a) Use Theorem 11.25 to give a further proof of Lévy's arc-sine law
- Let \(\left(B_{t}\right)_{t \geqslant 0}, d_{t}\) and \(g_{t}\) be as in Corollary 11.26. Define\[L_{t}^{-}:=t-g_{t} \quad \text { and } \quad L_{t}:=d_{t}-g_{t}\]Find the laws of \(\left(L_{t}^{-},
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Apply Doob's maximal inequality (A.13) to the exponential martingale \(M_{t}^{\xi}:=\exp \left(\xi B_{t}-\frac{1}{2} \xi^{2}
- Let \(X=\left(X_{t}\right)_{t \geqslant 0}\) be a one-dimensional process satisfying (B0), (B1), (B2). Assume, in addition, that \(X\) is continuous in probability, i.e. \(\lim _{t \rightarrow 0}
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(\tau\) some a.s. finite stopping time such that \(\left|B_{t \wedge \tau}\right|, t \geqslant 0\), is bounded. Show that
- Let \(f \in \mathcal{S}_{T}\) be a simple process and \(M \in \mathcal{M}_{T}^{2, c}\). Show that the definition of the stochastic integral \(\int_{0}^{T} f(s) d M_{S}\) (cf. Definition 15.9) does
- Use the fact that any continuous, square-integrable martingale with bounded variation paths is constant (cf. Proposition 17.2) to show the following: \(\langle f \cdot
- The quadratic covariation of two continuous \(L^{2}\) martingales \(M, N \in \mathcal{M}_{T}^{2, c}\) is defined as in the discrete case by the polarization formula (15.8).Let \(\left(B_{t}\right)_{t
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f \in \operatorname{BV}[0, T], TData From 13.5 Paragraph G (w) = $(s) dw(s) = (1)w(1) - [w(s) d(s), w C(o)[0, 1], 1)-\ - is
- Adapt the proof of Proposition 15.18 and prove that we also have \[\lim _{|\Pi| \rightarrow 0} \mathbb{E}\left[\sup _{t \leqslant T}\left|\int_{0}^{t} f(s) d B_{s}-\sum_{j=1}^{n}
- Let \(\tau\) be a stopping for a \(\mathrm{BM}^{1}\left(B_{t}\right)_{t \geqslant 0}\) and define the stochastic intervalthatApproximate \(\tau\) by stopping times with finitely many values as in
- Show that Proposition 17.2 and Corollary 17.3 remain valid for local martingales with continuous paths (cf. Definition 16.7).Data From Proposition 17.2Data From Corollary 17.3 17.2 Proposition. Let
- Show that the limits (18.5) and (18.6) actually hold in \(L^{2}(\mathbb{P})\) and uniformly for \(T\) from compact sets.To show that the limits hold uniformly, use Doob's maximal inequality. For the
- Let \(\left(N_{t}\right)_{t \geqslant 0}\) be a Poisson process with intensity \(\lambda=1\) (see Problem 10.1 for the definition). Show that for \(\mathscr{F}_{t}^{N}:=\sigma\left(N_{r}: r \leqslant
- Let XtXt be as in Problem 19.5 and set a) Use Problem 19.5 to find the probability density of ˆτbτ^b if α,b>0α,b>0.b) Find and , respectively.Data From Problem 19.5 b := inf{t 0 : X
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\), denote by \(\mathcal{H}_{T}^{2}\) the space used in Lemma 19.10 and pick \(\xi_{1}, \ldots, \xi_{n} \in \mathbb{R}\) and
- Show that, in Theorem 19.17, the quadratic variation \(\langle Mangle_{t}\) is a \(\mathscr{G}_{t}\) stopping time.Direct calculation, use Lemma 19.16.c) and A. 15Data From Theorem 19.17 19.17
- We have seen in Lemma 19.27.a) that \(\operatorname{supp}\left[d L_{t}^{0}(\omega)\right] \subset\left\{t \geqslant 0: B_{t}(\omega)=0\right\}\) for almost all \(\omega\). Show that
- The proof of Theorem 19.29 uses, implicitly, the following beautiful result due to Skorokhod [239] which is to be proved:Lemma. Let \(b:[0, \infty) \rightarrow \mathbb{R}\) be a continuous function
- Show that in Lemma 19.32.a) the following stronger assertion holds: \(\left(L_{t}\right)_{t \geqslant 0}\) is an additive functional, i.e. \(L_{t+s}=L_{t}+L_{s} \circ \theta_{t}\) holds for all \(s,
- Let \(\left(X_{t}, \mathscr{G}_{t}\right)\) be an adapted, real-valued process with right continuous paths and finite left limits. Assume that \(\mathbb{P}\left(X_{t}-X_{s} \in A \mid
- Show that the definition of the double Itô integral for off-diagonal simple functions (Definition 20.4) is independent of the representation of the simple function.Data From Definition 20.4 20.4
- Repeat the calculation from the end of Example 20.5 for a general measure \(\mu\). What happens on the diagonal?Data From Example 20.5 20.5 Example. Let us return to Example 20.1. If II = {0 = to
- Show that the definition of the iterated Itô integral for \(f \in L^{2}\left(\mathbb{R}_{+}^{2}\right)\) (Definition 20.9) is independent of the approximating sequence.Data From 20.9 Definition
- Prove (20.20), i.e. show that \(\mathbb{E}\left[B_{T}^{2 n} e^{-B_{T}^{2}}\right]=\frac{(2 n-1) ! !}{\sqrt{2 T+1}}\left(\frac{T}{2 T+1}\right)^{n}\). Here, \((-1) ! !:=1\), \((2 n-1) ! !=1 \cdot 3
- Verify the claim made in Example 21.9 using Itô’s formula.Derive from the proof of Lemma 21.8 explicitly the form of the transformation and the coefficients in Example 21.9.Integrate the condition
- Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use Lemma 21.10 to find the solution of the following SDE:\[d X_{t}=\left(\sqrt{1+X_{t}^{2}}+\frac{1}{2} X_{t}\right) d
- 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

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