Diffusion processes

Feller's diffusion approximation to the branching process. Let $D = \left\{ D(t): t \geq 0\right\}$ be a diffusion process with instantaneous mean $a(t, x) = \alpha x$ and instantaneous variance $b(t, x) = \beta x$ where $\alpha$ and $\beta$ are positive constants. Let $D(0) = d$. Show that the moment generating function of $D(t)$ is

Find the mean and variance of $D(t)$, and show that $\mathbb {P}\left(D(t) = 0\right) \rightarrow e^{-2 d \alpha / \beta }$ as $t \rightarrow \infty$.

Let $D$ be an Ornstein-Uhlenbeck process with $D(0) = 0$, and place reflecting barriers at $-c$ and $d$ where $c, d > 0$. Find the limiting distribution of $D$ as $t \rightarrow \infty$.

Let $D$ be a Wiener process with drift $m$, and suppose that $D(0) = 0$. Place absorbing barriers at the points $x = -a$ and $x = b$ where $a$ and $b$ are positive real numbers. Show that the probability $p_{a}$ that the process is absorbed at $-a$ is given by

Let $X = \left\{ X(t): t \geq 0\right\}$ be a simple birth-death process with parameters $\lambda_{n} = n \lambda$ and $\mu_{n} = n \mu$. Suggest a diffusion approximation to $X$.

Let $D$ be a diffusion with instantaneous mean and variance $a(t, x)$ and $b(t, x)$, and let $M(t, \theta ) = \mathbb {E}\left[e^{\theta D(t)}\right]$, the moment generating function of $D(t)$. Use the forward diffusion equation to derive Bartlett's equation:

where we interpret

if $g(t, x) = \sum_{n = 0}^{\infty } \gamma_{n}(t) x^{n}$.

Write down Bartlett's equation in the case of the Wiener process $D$ having drift $m$ and instantaneous variance $1$, and solve it subject to the boundary condition $D(0) = 0$.

Write down Bartlett's equation in the case of an Ornstein-Uhlenbeck process $D$ having instantaneous mean $a(t, x) = -x$ and variance $b(t, x) = 1$, and solve it subject to the boundary condition $D(0) = 0$.

Let $W_{1}(t), W_{2}(t), W_{3}(t)$ be independent Wiener processes. The positive $R = R(t)$ such that $R^{2} = W_{1}^{2}+W_{2}^{2}+W_{3}^{2}$ is the three-dimensional Bessel process. Show that $R$ is a Markov process. Is this result true in a general number $n$ of dimensions?

If $W$ is a Wiener process and the function $g: \mathbb {R} \rightarrow \mathbb {R}$ is continuous and strictly monotone, show that $g(W)$ is a continuous Markov process.

Exponential martingale, geometric Brownian motion. Let $W$ be a standard Wiener process and define $S(t) = e^{a t+b W(t)}$. Show that:

Find the transition density for the Markov process of Exercise (13.3.9a).

Feller's diffusion approximation to the branching process, Example (13.3.12) revisited.

Let $X = \left\{ X(t): t \geq 0\right\}$ be the diffusion model of Example (13.3.12) with $X(0) = x > 0$, and with instantaneous mean and variance $a x$ and $b x$ where $a < 0$ and $b > 0$. Let $Y = \int_{0}^{T} X(u) d u$ be the integrated size of the accumulated population up to the extinction time $T = \inf \left\{ t \geq 0: X(t) = 0\right\}$. Show that the moment generating function $m(x, \psi ) = \mathbb {E}_{x}\left[e^{-\psi Y}\right]$ satisfies

Deduce that $Y$ has a 'first-passage' distribution of the type of Exercise (13.4.2), and specify which.

Let $D$ be a standard Wiener process with drift $m$ starting from $D(0) = d > 0$, and suppose that there is a reflecting barrier at the origin. Show that the density function $f^{\mathrm{r}}(t, y)$ of $D(t)$ satisfies $f^{\mathrm{r}}(t, y) \rightarrow 0$ as $t \rightarrow \infty$ if $m \geq 0$, whereas $f^{\mathrm{r}}(t, y) \rightarrow 2\left|m\right| e^{-2\left|m\right| y}$ for $y > 0$, as $t \rightarrow \infty$ if $m < 0$.

Let $W$ be a standard Wiener process started at $W(0) = x \in (0, a)$, with absorbing barriers at 0 and $a$. Let $C$ be the process $W$ conditioned on the event that $W$ is absorbed at $a$. Show that the instantaneous mean and variance of $C$, when $C(t) = c \in (0, a)$, are $1 / c$ and $1$, respectively.

Let $T$ be the first passage time by $C$ to the point $a$. Show that $\mathbb {E}_{x}\left[T\right] = \frac{1}{3}\left(a^{2}-x^{2}\right)$ for $x \in (0, a)$.

Show that Lemma 8.10 remains true for any $d$ - dimensional Feller process $X_{t}$ with continuous paths and generator $L = \sum_{j, k = 1}^{d} a_{j k}(x) \partial_{j} \partial_{k}+\sum_{j = 1}^{d} b_{j}(x) \partial_{j}$ such that $a_{11}(x) \geqslant a_{0} > 0$ and $\left|b_{1}(x)\right| \leqslant b_{0} < \infty$.

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 $L = L\left(x, D_{x}\right) = \sum_{i j} a_{i j}(x) \partial_{i} \partial_{j}+\sum_{j} b_{j}(x) \partial_{j}+c(x)$ 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 $M_{t}^{u} = u\left(X_{t}\right) - u\left(X_{0}\right) - \int_{0}^{t} L u\left(X_{r}\right) d r$.