On diffusions

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 $a \in \mathcal{C}\left(\mathbb {R}^{d}, \mathbb {R}^{d \times d}\right)$ and $b \in \mathcal{C}\left(\mathbb {R}^{d}, \mathbb {R}^{d}\right)$.

Show that under the assumptions of Proposition 23.5 we can interchange integration and differentiation: $\frac{\partial^{2}}{\partial x_{j} \partial x_{k}} \int p(t, x, y) u(y) d y = \int \frac{\partial^{2}}{\partial x_{j} \partial x_{k}} p(t, x, y) u(y) d y$ and that the resulting function is in $\mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$.

Let $A: \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right) \rightarrow \mathcal{C}_{\infty }\left(\mathbb {R}^{d}\right)$ be a closed operator such that $\left\| A u\right\|_{\infty } \leqslant C\left\| u\right\|_{(2)}$; here $\left\| u\right\|_{(2)} = \left\| u\right\|_{\infty }+\sum_{j}\left\| \partial_{j} u\right\|_{\infty }+\sum_{j k}\left\| \partial_{j} \partial_{k} u\right\|_{\infty }$. Show that $\mathcal{C}_{\infty }^{2}\left(\mathbb {R}^{d}\right) = \overline{\mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right)}{ }^{\left\| \cdot \right\|_{(2)}} \subset \mathfrak {D}(A)$.

Let $L = L\left(x, D_{x}\right) = \sum_{i j} a_{i j}(x) \partial_{i} \partial_{j}+\sum_{j} b_{j}(x) \partial_{j}+c(x)$ be a second order differential operator. Determine its formal adjoint $L^{*}\left(y, D_{y}\right)$, i.e. the linear operator satisfying $\left\langle L u, \phi \right\rangle_{L^{2}} = \left\langle u, L^{*} \phi \right\rangle_{L^{2}}$ for all $u, \phi \in \mathcal{C}_{c}^{\infty }\left(\mathbb {R}^{d}\right)$. What is the formal adjoint of the operator $L(x, D)+\frac{\partial }{\partial t}$ ?

Complete the proof of Proposition 23.6 (Kolmogorov's forward equation).

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Show that $X_{t}: = \left(B_{t}, \int_{0}^{t} B_{s} d s\right)$ is a diffusion process and determine its generator.

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 $M_{t}^{u} = u\left(X_{t}\right) - u\left(X_{0}\right) - \int_{0}^{t} L u\left(X_{r}\right) d r$.

Consider the $\operatorname {ODE} \partial_{t} \xi (t, x) = x+b(\xi (t, x)), \xi (0, x) = x$, where $b \in \mathcal{C}_{b}^{2}\left(\mathbb {R}^{d}, \mathbb {R}^{d}\right)$ such that $\left|\partial_{x}^{\alpha } b(x)\right|,\left|\alpha \right| = 2$, is Lipschitz continuous. Show that all $\partial_{x}^{\alpha } \xi (t, x),\left|\alpha \right| \leqslant 2$, are polynomially bounded.

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

where $c \in \mathcal{C}_{c}^{2}\left(\mathbb {R}^{d}, \mathbb {R}\right)$. The following problem makes a connection between $X$ and Brownian motion killed with rate $V(\cdot )$.

Let $\left(M^{(1)}, \ldots , M^{(d)}\right)$ be a local martingale with $M_{0} = 0$ and denote its quadratic variation by $\left\langle M\right\rangle = \left(\left\langle M^{(j)}, M^{(k)}\right\rangle \right)_{j, k}$. Show that $\left\langle M\right\rangle = 0$ implies that $M = 0$.

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 $d u\left(N_{t}\right) = u^{\prime }\left(N_{t}\right) d N_{t}+\frac{1}{2} u^{\prime \prime }\left(N_{t}\right) d\left\langle N\right\rangle_{t}$.

Let $\left(N_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a continuous local martingale. Show that $\tau_{n}: = \left\{ s \geqslant 0:\left|N_{t}\right| \geqslant n\right\}$ is a localizing sequence which makes $N_{t \wedge \tau_{n}}$ bounded.

Prove the converse of Corollary 23.20.