All Matches

Solution Library

Expert Answer

Textbooks

Search Textbook questions, tutors and Books

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Tutors
Study Help
Hire a Tutor
AI Study Help New

Search
Sign In
Register

study help
business
probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f, g \in \mathcal{L}_{T}^{2}\). Show thata) \(\mathbb{E}\left(f \cdot B_{t} \cdot g \cdot B_{t} \mid
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be \(\mathrm{BM}^{1}\) and \(\left(f_{n}ight)_{n \geqslant 1}\) a sequence in \(\mathcal{L}_{T}^{2}\) such that \(f_{n} ightarrow f\) in
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f \in L^{2}\left(\lambda_{T} \otimes \mathbb{P}ight.\) ) for some \(T>0\). Assume that the limit \(\lim _{\epsilon ightarrow 0}
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1},\left(Y_{t}ight)_{t \geqslant 0}\) and adapted process such that \(Y_{t} Y_{s}=Y_{t}\) for all \(s \leqslant t\) and
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \((0, \infty) i s \mapsto f(s, \omega):=\mathbb{1}_{(0, \infty)}\left(B_{s}ight)\) is continuous in \(L^{2}(\mathbb{P})\).
Let \(\left(X_{n}ight)_{n \geqslant 0}\) be a sequence of r. v. in \(L^{2}(\mathbb{P})\). Show that \(L^{2}-\lim _{n ightarrow \infty} X_{n}=X\) implies \(L^{1}-\lim _{n ightarrow \infty}
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(\int_{0}^{T} B_{s}^{2} d B_{s}=\frac{1}{3} B_{T}^{3}-\int_{0}^{T} B_{s} d s, T>0\).
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion, \(T
Show that \[\mathscr{P}=\left\{\Gamma: \Gamma \subset[0, T] \times \Omega, \Gamma \cap([0, t] \times \Omega) \in \mathscr{B}[0, t] \otimes \mathscr{F}_{t} \quad \text { for all } t \leqslant
Show that every right or left continuous adapted process is \(\mathscr{P}\) measurable.
Let \(T
Let \(\tau\) be a stopping time. Show that \(\mathbb{1}_{[0, \tau)}\) is \(\mathscr{P}\) measurable.
If \(\sigma_{n}\) is localizing for a local martingale, so is \(\tau_{n}:=\sigma_{n} \wedge n\).
The factor \(\mathbb{1}_{\left\{\sigma_{n}>0ight\}}\) is used in the definition of a local martingale, in order to avoid integrability requirements for \(M_{0}\). More precisely, if
Let \(\left(I_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a continuous adapted process with values in \([0, \infty)\) and a.s. increasing sample paths. Set for \(u \geqslant
Extend Theorem 17.1 to local martingales with continuous paths.Data From Theorem 17.1 17.1 Theorem (Doob-Meyer decomposition. Meyer 1962, 1963). For all square integ. Ex. 17.1 rable martingales Me M
For \(M, N \in \mathcal{M}_{T, \text { loc }}^{c}\) we define the covariation by \(\langle M, Nangle_{t}:=\frac{1}{4}\left(\langle M+Nangle_{t}-\langle M-Nangle_{t}ight)\). Show thata) \(t
Use Lemma 17.14 to show the following assertions:a) \(\lim _{n} \mathbb{E}\left(\sup _{t \leqslant T}\left|X_{t}^{n}-X_{t}ight|^{2}ight)=0 \Longrightarrow X^{n} \xrightarrow[n ightarrow \infty]{\text
Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\). Show that the following maps are continuous, if we use ucpconvergence in both range and domain:a) \(\mathcal{L}_{T, \text { loc }}^{2}(M) i f \mapsto
Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\) and \(f(t, \omega)\) be an adapted càdlàg (right continuous, finite left limits) process and \(t \in[0, T]\).a) Show that \(f \in
Let \(M \in \mathcal{M}_{T, \text { loc }}^{c}\) and \(f_{n}\) be a sequence in \(\mathcal{L}_{T}^{0}(M)\). If \(f_{n} ightarrow f\) in ucp-sense, then \(f \in \mathcal{L}_{T}^{0}(M)\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be \(B M^{1}\). Use Itô's formula to obtain representations of\[X_{t}=\int_{0}^{t} \exp \left(B_{s}ight) d B_{s} \quad \text { and } \quad Y_{t}=\int_{0}^{t}
Let \(\Pi=\left\{0=t_{0}
Give a direct proof for the identity \[\begin{aligned}\mathbb{E} & {\left[\left(\sum_{l=1}^{N}
Prove the following time-dependent version of Itô's formula: let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f:[0, \infty) \times \mathbb{R} ightarrow \mathbb{R}\) be a
Prove Theorem 5.6 using Itô's formula.Data From Theorem 5.6 5.6 Theorem. Let (Bt, Ft)to be a d-dimensional Brownian motion and assume that the Ex. 5.10 function fee.2 ((0, co) x Rd, R) ne([0, o) x
Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Use Itô's formula to verify that the following processes are martingales:\[X_{t}=e^{t / 2} \cos B_{t} \quad \text
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\) be a \(\mathrm{BM}^{2}\) and set \(r_{t}:=\left|B_{t}ight|=\sqrt{b_{t}^{2}+\beta_{t}^{2}}\).a) Show that the stochastic integrals
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\) be a \(\mathrm{BM}^{2}\) and \(f(x+i y)=u(x, y)+i v(x, y), x, y \in \mathbb{R}\), an analytic function. If \(u_{x}^{2}+u_{y}^{2}=1\), then
Show that the \(d\)-dimensional Itô formula remains valid if we replace the real-valued function \(f: \mathbb{R}^{d} ightarrow \mathbb{R}\) by a complex function \(f=u+i v: \mathbb{R}^{d} ightarrow
Let \(\chi \in \mathcal{C}_{c}^{\infty}(-1,1)\) with \(0 \leqslant \chi \leqslant 1, \chi(0)=1\) and \(\int \chi(x) d x=1\). Set \(\chi_{n}(x):=n \chi(n x)\).a) Show that supp \(\chi_{n} \subset[-1 /
Let \(\sigma_{\Pi}, b_{\Pi} \in \mathcal{S}_{T}\) which approximate \(\sigma, b \in \mathcal{L}_{T}^{2}\) as \(|\Pi| ightarrow 0\), and consider the Itô processes \(X_{t}^{\Pi}=X_{0}+\int_{0}^{t}
Let \(X_{t}=M_{t}+A_{t}\) and \(X_{t}=M_{t}^{\prime}+A_{t}^{\prime}\) where \(M, M^{\prime}\) are continuous local martingales and \(A, A^{\prime}\) are continuous processes which have locally a.s.
State and prove the \(d\)-dimensional version of Lemma 19.1.Data From Lemma 19.1 19.1 Lemma. Let (Bt, Ft)o be a BM, f L (ATP) for all T> 0, and assume that \f(s, w) < C for some C > 0 and all s > 0
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion, i.e. \(\left(b_{t}ight)_{t \geqslant 0},\left(\beta_{t}ight)_{t \geqslant 0}\) are independent
Let \((B(t))_{t \geqslant 0}\) be a \(B M^{1}\) and \(T
Let \((\mathcal{L},\langle\cdot, \cdotangle)\) be a Hilbert space and \(\mathcal{H}\) some linear subspace. Show that \(\mathcal{H}\) is a dense subset of \(\mathcal{L}\) if, and only if, the
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}, n \geqslant 1,0 \leqslant t_{1}
Assume that \(\left(B_{t}, \mathscr{G}_{t}ight)_{t \geqslant 0}\) and \(\left(M_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) are independent martingales. Show that \(\left(M_{t}ight)_{t \leqslant T}\)
Show the following addition to Lemma 19.16:\[\tau(t-)=\inf \{s \geqslant 0: a(s) \geqslant t\} \quad \text { and } \quad a(t-)=\inf \{s \geqslant 0: \tau(s) \geqslant t\}\]and \(\tau(s) \geqslant t
Let \(f:[0, \infty) ightarrow \mathbb{R}\) be absolutely continuous. Show that \(\int f^{a-1} d f=f^{a} / a\) for all \(a>0\).
Use the strong Markov property to show the stationary and independent increments property in Lemma 19.32.c) directly, i.e. for all \(0=u_{0}Data From Lemma 19.32 19.32 Lemma. Let (Bt)to be a BM, Lo
Show that the Hermite Polynomials are linearly independent.
Let \((X, Y)\) be a mean zero Gaussian random variable. Show that \(\mathbb{E}\left(X^{2} Yight)=0\). Use this observation to show that \(I^{2}(f)-\|f\|_{L^{2}}^{2}\) is and
Pick any orthonormal basis \(\left(e_{k}ight)_{k \geqslant 1}\) of \(H\). Let \(\alpha \in \mathbb{N}_{0}^{(\infty)}\), i.e. \(\alpha=\left(\alpha_{1}, \ldots, \alpha_{N}, 0,0, \ldotsight)\) where
Let \(F, G\) be smooth, bounded cylindrical functions. Show thata) \(D_{t}(F G)=G D_{t} F+F D_{t} G\).b) \(D_{t}^{m}(F G)=\sum_{k=0}^{m}\left(\begin{array}{c}m \\ k\end{array}ight) D_{t}^{k} F \cdot
(Special case of Problem 9) Use Itô's formula to show that for any two symmetric f, g L (R)
Let \(F=\exp \left[I_{1}(g)-\frac{1}{2} \int_{0}^{\infty} g^{2}(s) d sight]\) for some \(g \in L^{2}\left(\mathbb{R}_{+}ight)\). Find all derivatives \(D^{k} F\) as well as the Wiener-Itô chaos
Find the Wiener-Itô chaos decomposition for the following random variables ( \(T>0\) is fixed):a) \(\exp \left(B_{T}-T / 2ight)\),b) \(B_{T}^{5}\),c) \(\int_{0}^{1}\left(r^{3} B_{t}^{3}+2 t
Find the Wiener-Itô chaos decomposition for the following random variables ( \(T>0\) is fixed):a) \(\sinh \left(B_{T}ight)\)b) \(\cosh \left(B_{T}ight)\).
Let \(\mathcal{D} \subset L^{2}(\mu)\) and write \(\Sigma:=\operatorname{span}(\mathcal{D})\). Show that \(\Sigma\) is dense if, and only if \(\mathcal{D}\) is total, i.e. \[\bar{\Sigma}=L^{2}(\mu)
Let \((B(t))_{t \in[0,1]}\) be a \({B M^{1}}^{1}\) and set \(t_{k}=k / 2^{n}\) for \(k=0,1, \ldots, 2^{n}\) and \(\Delta t=2^{-n}\). Consider the difference equation\[\Delta
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). The solution of the \(\mathrm{SDE}\)\[d X_{t}=-\beta X_{t} d t+\sigma d B_{t}, \quad X_{0}=\xi\]with \(\beta>0\) and \(\sigma \in
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find an SDE which has \(X_{t}=t B_{t}, t \geqslant 0\), as unique solution.
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find for each of the following processes an SDE which is solved by it:(a) \(U_{t}=\frac{B_{t}}{1+t}\);(b) \(V_{t}=\sin B_{t}\);(c)
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Solve the following SDEsa) \(d X_{t}=b d t+\sigma X_{t} d B_{t}, X_{0}=x_{0}\) and \(b, \sigma \in \mathbb{R}\);b) \(d
Show for all \(x, y \in \mathbb{R}^{n}\) the inequality\[|x-y|(1+|x|)^{-1}(1+|y|)^{-1} \leqslant\left.|| xight|^{-2} x-|y|^{-2} y \mid .\]Use a direct calculation after squaring both sides.
Letbe two globally Lipschitz continuous functions. If \(f\) is bounded, then \(h(x):=f(x) g(x)\) is locally Lipschitz continuous and at most linearly growing. fg: R-R
Let \(A: \mathcal{C}_{c}^{\infty}\left(\mathbb{R}^{d}ight) ightarrow \mathcal{C}_{\infty}\left(\mathbb{R}^{d}ight)\) be a closed operator such that \(\|A u\|_{\infty} \leqslant C\|u\|_{(2)}\); here
Let \(L=L\left(x, D_{x}ight)=\sum_{i j} a_{i j}(x) \partial_{i} \partial_{j}+\sum_{j} b_{j}(x) \partial_{j}+c(x)\) be a second order differential operator. Determine its formal adjoint
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(X_{t}:=\left(B_{t}, \int_{0}^{t} B_{s} d sight)\) is a diffusion process and determine its generator.
Consider the \(\operatorname{ODE} \partial_{t} \xi(t, x)=x+b(\xi(t, x)), \xi(0, x)=x\), where \(b \in \mathcal{C}_{b}^{2}\left(\mathbb{R}^{d}, \mathbb{R}^{d}ight)\) such that
Let \(\left(M^{(1)}, \ldots, M^{(d)}ight)\) be a local martingale with \(M_{0}=0\) and denote its quadratic variation by \(\langle Mangle=\left(\left\langle M^{(j)}, M^{(k)}ightangleight)_{j, k}\).
Let \(\left(N_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a continuous local martingale. Show that \(\tau_{n}:=\left\{s \geqslant 0:\left|N_{t}ight| \geqslant night\}\) is a localizing sequence
Prove the converse of Corollary 23.20.Data From Corollary 23.20 23.20 Corollary (Lvy 1948). Let (Xt, Ft)to be a stochastic process with values in Rd and continuous sample paths. If for all j, k =
Show that there exists a stochastic process \(\left(X_{t}ight)_{t \geqslant 0}\) such that the random variables \(X_{t}\) are independent \(\mathrm{N}(0, t)\) random variables. This process also
Let \(S\) be any set and \(\mathscr{E}\) be a family of subsets of \(S\). Show that\[\sigma(\mathscr{E})=\bigcup\{\sigma(\mathscr{C}): \mathscr{C} \subset \mathscr{E}, \mathscr{C} \text { is
Let \(\emptyset eq E \in \mathscr{B}\left(\mathbb{R}^{d}ight)\) contain at least two elements, and \(\emptyset eq I \subset[0, \infty)\) be any index set. We denote by \(E^{I}\) the product space
Let \(\mu\) be a probability measure on \(\left(\mathbb{R}^{d}, \mathscr{B}\left(\mathbb{R}^{d}ight)ight), v \in \mathbb{R}^{d}\) and consider the \(d\) dimensional stochastic process
Let \(X, Y, X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}, n \geqslant 1\), be random variables. a) If, for all n > 1, Xn Yn and if (Xn, Yn) (X, Y), then XIL Y. b) Let X Y such that X, Y ~ B1/2 = (80
Let \(X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}^{d}, n \geqslant 1\), be two sequences of random variables such that \(X_{n} \xrightarrow{d} X\) and \(X_{n}-Y_{n} \xrightarrow{\mathbb{P}} 0\). Then
Let \(X_{n}, Y_{n}: \Omega ightarrow \mathbb{R}, n \geqslant 1\), be two sequences of random variables.a) If \(X_{n} \xrightarrow{d} X\) and \(Y_{n} \xrightarrow{\mathbb{P}} c\), then \(X_{n} Y_{n}
Let Xn,X,Y:ΩightarrowR,n⩾1Xn,X,Y:ΩightarrowR,n⩾1, be random variables. Ifra \left or missing \ri lim E(f(x)g(Y)) = E(f(x)g(Y)) for all = C(R), g = B(R), n-co
Let \(\delta_{j}, j \geqslant 1\), be iid Bernoulli random variables with \(\mathbb{P}\left(\delta_{j}= \pm 1ight)=1 / 2\). We set\[S_{0}:=0, \quad S_{n}:=\delta_{1}+\cdots+\delta_{n} \quad \text {
Consider the condition for all \(s
Let \(G\) and \(G^{\prime}\) be two independent real N \((0,1)\)-random variables. Show that It: !! G+ G' ~N(0, 1) and . 2
Let \(\left(X_{t}ight)_{t \geqslant 0}\) be a stochastic process on the measure space \((\Omega, \mathscr{A}, \mathbb{P})\) satisfying (B1)(B3). Assume further that there is some measurable set
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space and let \(\mathbb{P}^{*}(Q):=\inf \{\mathbb{P}(A): A \supset Q, A \in \mathscr{A}\}\) be the outer measure. If \(\Omega_{0} \subset
Let \((\Omega, \mathscr{A}, \mathbb{P})\) be a probability space. The completion \(\left(\Omega, \mathscr{A}^{\prime}, \mathbb{P}^{\prime}ight)\) is the smallest extension such that
Show that there exist a random vector \((U, V)\) such that \(U\) and \(V\) are one-dimensional Gaussian random variables but \((U, V)\) is not Gaussian.Try \(f(u, v)=g(u) g(v)(1-\sin u \sin v)\)
Let \(G \sim \mathrm{N}(m, C)\) be a \(d\)-dimensional Gaussian random vector and \(A \in \mathbb{R}^{d \times d}\). Show that \(A G \sim \mathrm{N}\left(A m, A C A^{\top}ight)\).
Let \(X, \Gamma \sim \mathrm{N}(0, t)\). Show that - X + I and X ~ N(0, 2t).
Let \(G_{n} \sim \mathrm{N}\left(0, t_{n}ight), n \geqslant 1\), be sequence of Gaussian random variables. Show that \(G_{n} \xrightarrow{\mathrm{d}} G\) if, and only if, the sequence
Let \(A, B \in \mathbb{R}^{n \times n}\) be symmetric matrices. If \(\langle A x, xangle=\langle B x, xangle\) for all \(x \in \mathbb{R}^{n}\), then \(A=B\).
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Decide which of the following processes are Brownian motions:a) \(X_{t}:=2 B_{t / 4}\);b) \(Y_{t}:=B_{2 t}-B_{t}\);c)
Let \((B(t))_{t \geqslant 0}\) be a \(B M^{1}\).a) Find the density of the random vector \((B(s), B(t))\) where \(0
Find the correlation function \(C(s, t):=\mathbb{E}\left(X_{S} X_{t}ight), s, t \geqslant 0\), of the stochastic process \(X_{t}:=B_{t}^{2}-t, t \geqslant 0\), where \(\left(B_{t}ight)_{t \geqslant
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}, \alpha>0\) and consider the stochastic process \(X_{t}:=e^{-\alpha t / 2} B_{e^{\alpha t}}, t \geqslant 0\).a) Determine
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Show that \(\mathscr{F}_{\infty}^{B}=\bigcup_{J \text { countable }}^{J \subset[0, \infty)} \sigma(B(t): t \in J)\).
Let \(X(t), Y(t)\) be any two stochastic processes. Show that (X(u1),..., X(up)) (Y(u),..., Y(up)) for all u 1 implies (X(S1),..., X(Sn)) (Y(t),..., Y(tm)) for all s < < Sm, t < ... < tn, m, n 1.
Let (Ft)t⩾0(Ft)t⩾0 and (Gt)t⩾0(Gt)t⩾0 be any two filtrations and define F∞=σ(⋃t⩾0Ft)F∞=σ(⋃t⩾0Ft) and G∞=σ(⋃t⩾0Gt)G∞=σ(⋃t⩾0Gt). Show that Ft Gt (Vt > 0) if, and
Use Lemma 2.8 to show that (all finite dimensional distributions of) a \(\mathrm{BM}^{d}\) is invariant under rotations. 2.8 Lemma. Let (X+)to be a d-dimensional stochastic process. X satisfies
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion.a) Show that \(W_{t}:=\frac{1}{\sqrt{2}}\left(b_{t}+\beta_{t}ight)\) is a \(\mathrm{BM}^{1}\).b) Are
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion. For which values of \(\lambda, \mu \in \mathbb{R}\) is the process \(X_{t}:=\lambda b_{t}+\mu \beta_{t}
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion. Decide whether for \(s>0\) the process \(X_{t}:=\left(b_{t}, \beta_{s-t}-\beta_{t}ight), 0 \leqslant t
Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\), be a two-dimensional Brownian motion and \(\alpha \in[0,2 \pi)\). Show that \(W_{t}=\left(b_{t} \cdot \cos \alpha+\beta_{t} \cdot \sin
Let \(X\) be a \(Q\)-BM. Determine \(\operatorname{Cov}\left(X_{t}^{j}, X_{s}^{k}ight)\), the characteristic function of \(X(t)\) and the transition probability (density) of \(X(t)\).
Show that (B1) is equivalent to Bt-Bs for all 0 < s < t.
Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and set \(\tau_{a}=\inf \left\{s \geqslant 0: B_{s}=aight\}\) where \(a \in \mathbb{R}\). Show that \(\tau_{a} \sim \tau_{-a}\) and
A \(\mathbb{R}\)-valued stochastic process \(\left(X_{t}ight)_{t \geqslant 0}\) such that \(\mathbb{E}\left(X_{t}^{2}ight)

Showing 6200 - 6300 of 6914

First
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70

Services

Sitemap
Fun
Definitions
Become Tutor
Study Help Categories
Recent Questions
Expert Questions

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Campus Ambassador

Secure Payment

Download Our App

© 2024 SolutionInn. All Rights Reserved