Exercise 4.1 (linear transformation of Brownian motion)
- Let $W$ be a standard $d$-dimensional Brownian motion and let $U$ be an orthogonal matrix (i.e. $U^*=U^{-1}$). Prove that $UW$ defines a new standard $d$-dimensional Brownian motion.
- Application: let $W_1$ and $W_2$ be two independent Brownian motions. For any $\rho\in [-1,1]$, justify that $\rho W_1+\sqrt{1-\rho^2}W_2$ and $-\sqrt{1-\rho^2}W_1+\rho W_2$ are two independent Brownian motions.
Exercise 4.2 (approximation of the integral of a stochastic process)
For a standard Brownian motion, we study the convergence rate of the approximation
$$\Delta I_n:=\int_0^1 W_s {\rm d}s-\frac{ 1 }{n }\sum_{i=0}^{n-1} W_{\frac in}$$ as $n\to+\infty$.
- We start by a rough estimate. Prove that $${\mathbb E}(|\Delta I_n|)\leq \sum_{i=0}^{n-1} {\mathbb E}\left(\int_{\frac in}^{\frac{i+1}n} |W_s-W_{\frac in}|{\rm d}s\right)=O(n^{-1/2}).$$
- Using Lemma A.1.4, prove that $\Delta I_n$ is Gaussian distributed. Compute its parameters and conclude that $${\mathbb E}(|\Delta I_n|)=O(n^{-1}).$$
- A more generic proof of the above estimate consists of writing $$\Delta I_n:=\sum_{i=0}^{n-1} \int_{\frac in}^{\frac{i+1}n} (\frac{ i+1 }{n }-s){\rm d}W_s$$ where we have applied the Itô formula to $s\mapsto ( \frac{ i+1 }{n }-s)(W_s-W_{\frac in})$ on each interval $[\frac in,\frac{i+1}n]$. Using the Itô isometry, derive ${\mathbb E}(|\Delta I_n|^2)= O(n^{-2})$ and therefore the announced estimate.
- Proceeding as in (3), extend the previous estimate to $$\int_0^1 X_s{\rm d}s-\frac{ 1 }{n }\sum_{i=0}^{n-1} X_{\frac in}$$ where $X$ is a scalar Itô process with bounded coefficients (Definition 4.2.4).
Exercise 4.3 (approximation of stochastic integral)
We consider the convergence rate of the approximation $$\Delta J_n:=\int_0^1 Z_s {\rm d}W_s-\sum_{i=0}^{n-1} Z_{\frac in} (W_{\frac {i+1}n}-W_{\frac in})$$ where $Z_s:=f(s,W_s)$ for some function $f$, such that ${\mathbb E}\int_0^1 |Z_s|^2 {\rm d}s+\sup_{i\leq n-1}\mathbb{E}|Z_{\frac in}|^2<+\infty$. We illustrate that the convergence order is, under mild conditions, equal to $1/2$ but it can be smaller for irregular $f$.
- Show that ${\mathbb E}|\Delta J_n|^2= {\mathbb E} \left( \sum_{i=0}^{n-1} \int_{\frac in}^{\frac{i+1}n} |Z_s- Z_{\frac in}|^2 {\rm d}s \right)$.
- When $Z_s=W_s$, show that ${\mathbb E}|\Delta J_n|^2 \sim {\rm Cst}\ n^{-1}$ for some positive constant.
- Assuming that $f$ is bounded, smooth with bounded derivatives, prove that ${\mathbb E}|\Delta J_n|^2=O(n^{-1})$.
- Assume that $Z$ is a square-integrable martingale. Show that ${\mathbb E}|Z_s- Z_{\frac in}|^2\leq {\mathbb E}|Z_{\frac {i+1}n}|^2-{\mathbb E}|Z_{\frac in}|^2$, and thus ${\mathbb E}|\Delta J_n|^2\leq ({\mathbb E}|Z_{1}|^2-{\mathbb E}|Z_{0}|^2)n^{-1}$.
- Set $Z_s:={\cal N}'(W_s/\sqrt{1-s})/\sqrt{1-s}$. Establish that $n^{1/2}{\mathbb E}|\Delta J_n|^2$ is bounded away from 0, for $n$ large enough.
Exercise 4.4 (exact simulation of Ornstein-Uhlenbeck process)
Two processes $(X_t)_{t \ge 0}$ and $(Y_t)_{t \ge 0}$ have the same distribution if for any $n \in \mathbb{N}$ and any $0 \leq t_1 < \cdots < t_n$, the vectors $(X_{t_1},...,X_{t_n})$ and $(Y_{t_1}, ..., Y_{t_n})$ have the same distribution. Let us consider the Ornstein-Uhlenbeck process $(X_t)_{t\geq 0}$, solution of $$X_t= x_0-a\int_0^t X_s{\rm d}s + \sigma W_t,$$ where $x_0\in \mathbb{R}$, $\sigma \geq 0$, and $(W_t)_{t \ge 0}$ is a standard Brownian motion.
- By applying the Itô formula to $e^{at}X_t$, give an explicit representation for $X_t$ in terms of stochastic integrals.
- Deduce the explicit distribution of $(X_{t_1},\cdots,X_{t_n})$.
- Find two functions $\alpha(t)$ and $\beta(t)$ such that $(X_t)_{t\geq 0}$ has the same distribution as $(Y_t)_{t\geq 0}$ with $Y_t=\alpha(t)(x_0+W_{\beta(t)})$. Design a scheme for the exact simulation of the Ornstein-Uhlenbeck process.
Exercise 4.5 (Transformations of SDE and PDE)
For any $t \in [0,T)$ and $x \in \mathbb{R}$, we denote by $(X^{t,x}_s, s \in [t,T])$ the solution to $$X_s = x+\int_t^s b(X_r){\rm d}r + \int_t^s \sigma(X_r){\rm d}W_r, \qquad t \le s \le T$$ where the coefficients $b,\sigma:\mathbb{R} \to \mathbb{R}$ are smooth with bounded derivatives, and $\sigma(x)\ge c>0$.
For a given Borel set $A\subset \mathbb{R}$, we define $u(t,x):={\mathbb P}(X^{t,x}_T \in A)$. We assume in the following $u(t,x)>0$ for any $(t,x)\in [0,T)\times \mathbb{R}$, and that appropriate smoothness assumptions are satisfied (namely, $u \in {\cal C}^{1,2}([0,T)\times \mathbb{R})$).
- Let $x_0 \in \mathbb{R} $ and $f$ be a bounded continuous function. Using the PDE satisfied by $u$ on $[0,T) \times \mathbb{R}$, show that\[{\mathbb E}[f(X_t)|X_T \in A] = \frac{{\mathbb E}[f(X_t) u(t,X_t)]}{u(0,x_0)}, \qquad \forall t < T,\] where $X_t=X^{0,x_0}_t$ to simplify.
- We assume that for any $s \le t \le T$ the equation \begin{align*}\overline{X}_r &= x+\int_s^r\Bigl(b(\overline{X}_w) + \sigma^2(\overline{X}_w) \frac{\partial_x u}{u}(w,\overline{X}_w) \Bigr){\rm d} w \\&+ \int_s^r \sigma(\overline{X}_w){\rm d}W_w, \qquad s \le r \le t \end{align*}has a unique solution, denoted by $(\overline{X}^{s,x}_r, s \le r \le t)$. We set $v_t(s,x):={\mathbb E}[ f(\overline{X}^{s,x}_t)]$.
- What is the PDE solved by $(s,x) \mapsto v_t(s,x)$ on $[0,t) \times\mathbb{R}$?
- Applying the Itô formula to $u(s,X_s)$ and $v_t(s,X_s)$, $0 \le s \le t$, and then to $u(s,X_s)v_t(s,X_s)$, show\[{\mathbb E}[f(X_t)u(t,X_t)] = v_t(0,x_0)u(0,x_0), \qquad \forall t < T.\]
- Conclude that for any $t < T$, the distribution of $X_t$ given $\{X_T \in A\}$ is the distribution of $\overline{X}^{0,x_0}_t$.
- In the case $b=0$, $\sigma(x)=1$ and $A=(y-R,y+R)$, show that $\frac{\partial_x u(t,x)}{u(t,x)} \rightarrow -\frac{x-y}{T-t}$ for any $(t,x)$ as $R \to 0$. Interpret the solution to the following equation in terms of a Brownian bridge: \[\overline{X}_t =x_0 -\int_0^t \frac{\overline{X}_s-y}{T-s}{\rm d}s+ W_t.\]
Exercise 4.6 (Exit time from a domain)
Consider the solution $(X^x_t)_t$ of a Stochastic Differential Equation in $\mathbb{R}^d$, starting from $x$ at time 0, with time-independent coefficients $(b,\sigma)$ satisfying the usual Lipschitz conditions of Theorem 4.3.1. Let $D$ be a non-empty open connected set of $\mathbb{R}^d$ and set \[\tau_D^x = \inf\{t \ge 0 : X^x_t \notin D \} \] for the first exit time from $D$.
- Assume first that $\sup_{x \in D}{\mathbb E}[\tau^x_D] \le c$, for some constant $c>0$.
- Show that ${\mathbb E}[(\tau^x_D)^k] \le k! c^k$ for any $k \in \mathbb{N}$.
Hint: use the identity $\frac1k T^k =\int_0^T (T-t)^{k-1} {\mathrm d}t$ and the Markov property of $X^x$. - Deduce that $\sup_{x \in D} {\mathbb E}[e^{\lambda \tau^x_D}] < \infty $ for any $\lambda < c^{-1}$. What are the consequences of this result on the simulation of the path of $X$ up to $\tau^x_D$?
- Set $\gamma(t) = \sup_{x \in D} {\mathbb P}(\tau^x_D > t)$. Show $\gamma(t+s) \le \gamma(t)\gamma(s)$ for any $t,s \ge 0$.
- The previous question shows that the function $t \mapsto \ln\gamma(t) \in [-\infty,0]$ is sub-additive: by the Fekete lemma, the limit\[\lim_{t \to \infty} \frac1t \ln\gamma(t) = \inf_{t >0} \frac1t \ln\gamma(t) =: -\alpha_{D}\] exists in $[-\infty,0]$. Show that $\alpha_{D} \ge c^{-1}$.
- Assume now that $D$ is bounded. The infinitesimal generator of $X$ is denoted by $\cal L$.
- Suppose there exists $f \in {\cal C}^2(\mathbb{R}^d,\mathbb{R})$ such that $f(x) \ge 0$ and ${\cal L} f(x) \le -1$ for $x \in D$. Show that $\sup_{x \in D} {\mathbb E}[\tau^x_D] \le c =: \sup_{y \in D} f(y)$.
- In the case $\inf_{x\in D}[\sigma \sigma^*]_{i,i}(x)>0$ for some $i$, exhibit such a function $f$.
Exercise 4.7 (Bismut-Elworthy-Li formula)
Let $X^x$ be the Ornstein-Uhlenbeck process solution of \[X_t= x-a\int_0^t X_s{\mathrm d}s + \sigma W_t, \] with $x\in \mathbb{R}$ and $\sigma>0$. We aim at computing $\partial_x
- Using the sensitivity formula of Theorem 4.5.3, show that $$\partial_x{\mathbb E}(f(X^{x}_T))={\mathbb E}\Big(f(X^{x}_T)\int_0^{T} \frac{ e^{-at} } {\sigma T}{\mathrm d}W_t\Big).$$
- Show that the sensitivity formula is still valid by replacing $\int_0^{T} \frac{ e^{-at} } {\sigma T}{\mathrm d}W_t$ by its conditional expectation given $X^x_T$. Compute this conditional expectation explicitly.
- Prove that the new formula coincides with that given by the likelihood ratio method using the Gaussian distribution of $X^x_T$ (Proposition 2.2.9).
- Which representation among that (1) or (3) has the smallest variance?
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