Let (left(B_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{1}) and (b(x), sigma(x)) autonomous and globally Lipschitz continuous coefficients.

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 solution of the stochastic differential equation \(d X_{t}=b\left(X_{t}\right) d t+\sigma\left(X_{t}\right) d B_{t}\) is a Markov process. Recall that \(\mathcal{C}_{b}(\mathbb{R})\) and \(\mathcal{C}_{\infty}(\mathbb{R})\) are the bounded continuous functions and the continuous functions vanishing at infinity.

a) Find the transition semigroup \(T_{t}\) of this Markov process and show that \(T_{t}\) maps \(\mathcal{C}_{b}(\mathbb{R})\) into itself.

b) Use Corollary 21.31 to show that \(T_{t}\) maps \(\mathcal{C}_{\infty}(\mathbb{R})\) into itself.

c) Find the infinitesimal generator \(A\) of \(T_{t}\) on \(\mathcal{C}_{c}^{2}(\mathbb{R})\). Show that \(A\) has an extension onto \(\mathcal{C}_{b}^{2}(\mathbb{R})\).
Use Itô's formula.

d) Prove the following generalization of Dynkin's formula: if \(b, \sigma\) are bounded and if \(\tau\) is a stopping time with \(\mathbb{E}^{x} \tau)\right]-u(0, x)=\mathbb{E}^{x}\left[\int_{0}^{\tau}\left(\partial_{t} u\left(s, X_{s}\right)+A_{x} u\left(s, X_{s}\right)\right) d s\right] .\]

Data From  21.31 Corollary

Transcribed Image Text:

21.31 Corollary. Let (Bt, Ft)to be a d-dimensional Brownian motion and assume that the coefficients b: [0, co) x R"R" and : [0,00) x R" Rnxd of the SDE (21.1) satisfy the Lipschitz condition (21.17) for all x, y = R" with the global Lipschitz constant L. Denote by (X) to the solution of the SDE (21.1) with initial condition X = x = R". Then lim |X(w)] = co_almost surely for every t> 0. x-00 Proof. For x = R" \ {0} we set x = x/1x12 R", and define for every t> 0 (21.41) [(1+1 +IX) if x 0, Y = 0 if x = 0. Observe that |x| 0 if, and only if, |x|co. If x, y 0 we find for all p > 2 |Y - YP YP YP \\X}{| - | || < |Y}\| || \X-XP