Stochastic differential equations

Show that $\left\| \cdot \right\|_{t}$ defined by (24.7) gives rise to a complete normed linear space.

Suppose $g(t)$ is non-negative and bounded on each finite subinterval of $[0, \infty )$. Suppose there exist constants $A$ and $B$ such that

for each $t \geq 0$. Prove that $g(t) \leq A e^{B t}$ for all $t \geq 0$. This result is known as Gronwall's lemma.

Hint: Write

use (24.21) to substitute for $g(r)$, and iterate.

The starting point in (24.1) can be random. Suppose $Y$ is a random variable that is measurable with respect to $\mathcal{F}_{0}, Y$ is square integrable, and $\sigma$ and $b$ are bounded and Lipschitz. Prove pathwise existence and uniqueness for the equation

The functions $\sigma$ and $b$ in (24.1) can depend on time as well as space. Suppose $\sigma :[0, \infty ) \times \mathbb {R} \rightarrow \mathbb {R}$, $b:[0, \infty ) \times \mathbb {R} \rightarrow \mathbb {R}$ are bounded and uniformly Lipschitz in the second variable: there exists $c$ independent of $s$ such that $\left|\sigma (s, x)-\sigma (s, y)\right| \leq c\left|x-y\right|$ and similarly for $b$. Prove pathwise existence and uniqueness for the equation

Here is a multidimensional analog of (24.1). Suppose the functions $\sigma_{i j}: \mathbb {R}^{d} \rightarrow \mathbb {R}$, $1 \leq i, j \leq d$, are bounded and Lipschitz, and $b_{i}: \mathbb {R}^{d} \rightarrow \mathbb {R}$, $i=1, \ldots , d$, are bounded and Lipschitz, $W^{j}$ are independent one-dimensional Brownian motions, $x_{0}=\left(x_{0}^{(1)}, \ldots , x_{0}^{(d)}\right)$, and $X_{t}=\left(X_{t}^{(1)}, \ldots , X_{t}^{(d)}\right)$ satisfies

for $i=1, \ldots , d$. Prove pathwise existence and uniqueness for this system of equations.

Suppose $f$ and $g$ map $[0, \infty ) \rightarrow \mathbb {R}$ with $\int_{0}^{\infty } f(t)^{2} d t<\infty$ and $\int_{0}^{\infty } g(t)^{2} d t<\infty$. Show that $\int_{0}^{\infty } f(t) d W_{t}$ is a mean zero Gaussian random variable, the same with $f$ replaced by $g$, and

Show that if $W_{t}$ is a $d$-dimensional Brownian motion, then $\left|W_{t}\right|^{2}$ is the square of a Bessel process of order $d$.

Prove (1), (3), and (4) of Proposition 24.7.

Let $X$ be the solution to $d X_{t}=\sigma \left(X_{t}\right) d W_{t}+b\left(X_{t}\right) d t$, where $W$ is a one-dimensional Brownian motion, $\sigma$ and $b$ are Lipschitz continuous real-valued functions, and $\left|\sigma (x)\right| \leq c(1+\left|x\right|)$ and $\left|b(x)\right| \leq c(1+\left|x\right|)$. Let $t_{0}>0$. Prove that if $p \geq 2$, then

Let $W$ be a one-dimensional Brownian motion and let $X_{t}^{x}$ be the solution to

Suppose $\sigma$ and $b$ are $C^{\infty }$ functions and that $\sigma$ and $b$ and all their derivatives are bounded. Show that for each $t$ the map $x \rightarrow X_{t}^{x}$ is continuous in $x$ with probability one. Show that the map is differentiable in $x$.

Suppose $A(t)$ and $B(t)$ are deterministic functions of $t$. Find an explicit solution to the one-dimensional SDE