HW4 MA5161. Due February 24

Exercise. Let (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P}) be a filtered probability space that satisfies the usual conditions. We denote

\mathcal{F}_{\infty}=\sigma \left( \mathcal{F}_t , t \ge 0 \right)

and for t\ge 0, \mathbb{P}_{/\mathcal{F}_t} is the restriction of \mathbb{P} to \mathcal{F}_t. Let \mathbb{Q} be a probability measure on \mathcal{F}_{\infty} such that for every t \ge 0,

\mathbb{Q}_{/\mathcal{F}_t} \ll \mathbb{P}_{/\mathcal{F}_t}.

  • Show that there exists a right continuous and left limited martingale (D_t)_{t \ge 0} such that for every t \ge 0,D_t=\frac{d\mathbb{Q}_{/\mathcal{F}_t}}{d\mathbb{P}_{/\mathcal{F}_t}},\text{ }\mathbb{P}-a.s.
  • Show that the following properties are equivalent:
    1) \mathbb{Q}_{/\mathcal{F}_\infty} \ll \mathbb{P}_{/\mathcal{F}_\infty};
    2) The martingale (D_t)_{t \ge 0} is uniformly integrable;
    3) (D_t)_{t \ge 0} converges in L^1;
    4) (D_t)_{t \ge 0} almost surely converges to an integrable and \mathcal{F}_\infty measurable random variable D such that D_t =\mathbb{E}(D\mid \mathcal{F}_t), \quad t \ge 0.

 

Posted in Uncategorized | Leave a comment

HW4 MA5311. Due February 24

Exercise 1. Let M be a smooth manifold and  \phi: C^\infty (M,R) \to C^\infty (M,R) be a linear operator such that for every smooth functions f,g: M \to R, V(fg)=fVg+gVf. Show that there exists a vector field U on M such that for every smooth function g, Vg(x)=df_x (U(x)).

Exercise 2. Let B_n be the open unit ball in R^n. Let y in B_n. Show that there exists a smooth vector field on R^n, such that e^V(0)=y and V(x)=0 if x is not in B_n.

Posted in Uncategorized | Leave a comment

HW3 MA5311. Due February 15

Exercise. Let X \subset \mathbf{R}^k be a subset homeomorphic to the closed ball B_n \subset \mathbf{R}^n.  Show that if f: X \to X is continuous, then there exists x \in X such that f(x)=x.

Exercise. Let X be a one-dimensional compact manifold with boundary. Show that X is diffeomorphic to a finite union of segments and circles (You may use the appendix in Milnor’s lecture notes).

Posted in Uncategorized | Leave a comment

HW3 MA5161. Due February 15

Exercise. (First hitting time of a closed set by a continuous stochastic process)
Let (X_t)_{t \ge 0} be a continuous process adapted to a filtration (\mathcal{F}_t)_{t \ge 0}. Let

T=\inf \{ t \ge 0, X_t \in F \},

where F is a closed subset of \mathbb{R}. Show that T is a stopping time of the filtration (\mathcal{F}_t)_{t\ge 0}.

Exercise. (Closed martingale)
Let (\mathcal{F}_t)_{t \ge 0} be a filtration defined on a probability space (\Omega, \mathcal{F},\mathbb{P}) and let X be an integrable and \mathcal{F}-measurable random variable. Show that the process \left( \mathbb{E}(X\mid \mathcal{F}_t) \right)_{t \ge 0} is a martingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0}.

Exercise. Let (\mathcal{F}_t)_{t \ge 0} be a filtration defined on a probability space (\Omega, \mathcal{F},\mathbb{P}) and let (M_t)_{t \ge 0} be a submartingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0}. Show that the function t \rightarrow \mathbb{E} (M_t) is non-decreasing.

Exercise. Let (\mathcal{F}_t)_{t \ge 0} be a filtration defined on a probability space (\Omega, \mathcal{F},\mathbb{P}) and let (M_t)_{t \ge 0} be a martingale with respect to the filtration (\mathcal{F}_t)_{t \ge 0}. Let now \psi : \mathbb{R} \rightarrow \mathbb{R} be a convex function such that for t \ge 0, \mathbb{E} \left( \mid \psi(M_t) \mid \right) < + \infty. Show that the process (\psi(M_t))_{t \ge 0} is a submartingale.

Posted in Uncategorized | Leave a comment

HW2 MA5161. Due February 3

Exercise 1. Let m:\mathbb{R}_{\ge 0} \rightarrow \mathbb{R} and let R: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \rightarrow \mathbb{R} be a symmetric and positive function. Show that there exists a probability space \left( \Omega , \mathcal{F}, \mathbb{P} \right) and a Gaussian process (X_t)_{t \ge 0} defined on it, whose mean function is m and whose covariance function is R.

Exercise 2. Let (X_t)_{t \ge 0} be a continuous process adapted to a filtration \mathcal{F}_t. Show that (X_t)_{t \ge 0} is progressively measurable.

Posted in Uncategorized | Leave a comment

HW2 MA5311: Due February 3

Exercise 1. Let g : \mathbf{R}^2 \to \mathbf{R}^4, (u,v) \to (\cos u , \sin u, \cos v, \sin v). Show that g( \mathbf{R}^2) is a 2-dimensional smooth manifold homeomorphic to the torus \mathbf{S}^1 \times \mathbf{S}^1.

Exercise 2. Let h_+: \mathbf{S}^2-N \to \mathbf{C} be the stereographic projection from the north pole N, and h_- be the stereographic projection from the south pole S.

  1. Show that for z \neq 0, h_+ h_-^{-1} =\frac{1}{\bar z}.
  2. Show that if P is a non constant polynomial, the map f=h_+^{-1} P h_+, f(N)=N is smooth.
  3. More generally, if Q: \mathbf{C} \to \mathbf{C} is smooth, find a condition on Q so that f=h_+^{-1} Q h_+, f(N)=N is smooth.
  4. By following Milnor’s argument in the proof of the fundamental theorem of algebra, find sufficient conditions so that a smooth map Q: \mathbf{C} \to \mathbf{C} is onto.

 

Exercise 3.  By using Sard’s theorem, prove that the set of regular values of a smooth map f : M \to N is dense in N.

Posted in Uncategorized | Leave a comment

MA5161. HW 1 due Wednesday 1/25

Exercise 1. Show that the  \sigma-algebra \mathcal{T}(\mathbb{R}_{\ge 0},\mathbb{R}^d) is also generated by the following families:

  • \{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d ), f(t_1) \in B_1,...,f(t_n) \in B_n \} where t_1,...,t_n \in \mathbb{R}_{\ge 0} and where B_1,...,B_n are Borel sets in \mathbb{R}^d.
  • \{ f \in \mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), (f(t_1),...,f(t_n)) \in B \} where t_1,...,t_n \in \mathbb{R}_{\ge 0} and where B is a Borel set in (\mathbb{R}^{d})^{\otimes n}.

 

Exercise 2.  Show that the following sets are in \mathcal{B} ([0,1],\mathbb{R}):

  • \{ f \in \mathcal{C}([0,1], \mathbb{R}), \sup_{t\in [0,1]} f(t) <1 \}
  • \{ f \in \mathcal{C}([0,1], \mathbb{R}), \exists t\in [0,1] f(t) =0 \}
Posted in Uncategorized | Leave a comment