**Exercice 5.1 (strong convergence)**

In Theorem 5.2.1, we have proved that for a SDE with globally Lipschitz coefficients $b, \sigma$, we have $$\mathbb{E}(\sup_{0\leq t \leq T}|X^{(h)}_t-X_t|^p)\leq O(h^{\frac p2})$$ for any $p>0$. The convergence order is 1/2. Show, that if $\sigma$ is constant and $b$ is ${\cal C}^2$ in space and ${\cal C}^1$ in time with bounded derivatives, the convergence order is 1.

**Exercice 5.2 (Milshtein scheme)**

Denote by $(X_t)_{t\geq 0}$ the solution of the stochastic differential equation $$X_t=x+\int_0^t \sigma(X_s){\rm d}W_s+\int_0^t b(X_s){\rm d}s$$ where $\sigma,b:{\mathbb R}\rightarrow{\mathbb R}$ are bounded ${\cal C}^2$-functions with bounded derivatives.

- Show the short time $L_2$-approximation $$\mathbb{E}\left(\left(X_t-[x+b(x) t+\sigma(x)W_t]\right)^2\right)= \frac{(\sigma\sigma'(x))^2}{2}t^2+o(t^2).$$
The above estimate is instrumental to show the strong convergence of the Euler scheme at order $1/2$: essentially, the global quadratic error (of size $N^{-1}$) is the summation of the above
*local estimates*(of order $N^{-2}$ with $t=h$) over $N$ times, to finally get the error bound for $p=2$. - Similarly, show $$\mathbb{E}\left(\left(X_t-[x+b(x)t+\sigma(x)W_t+\frac{1}{2}\sigma\sigma'(x)(W_t^2-t)]\right)^2\right)={\mathcal O}(t^3). $$ This estimate leads to a high-order scheme, called the Milshtein scheme, which is written \begin{equation} \begin{cases} X^{(h,M)}_0=&x,\\ X^{(h,M)}_{t_{i+1}}=&X^{(h,M)}_{t_{i}}+b(X^{(h,M)}_{t_{i}}) h +\sigma(X^{(h,M)}_{t_{i}}) (W_{t_{i+1}}-W_{t_{i}})\\ &+\frac{ 1 }{ 2}\sigma\sigma'(X^{(h,M)}_{t_{i}}) [(W_{t_{i+1}}-W_{t_{i}})^2-h]. \end{cases} \end{equation}
- Using the above estimate, prove that \begin{equation} \sup_{0\leq i \leq N}\mathbb{E}(|X^{(h,M)}_{t_{i}}-X_{ih}|^2)=O( h^{2}), \end{equation} i.e. a convergence rate at order $1$ for the strong convergence.

**Exercice 5.3 (convergence rate of weak convergence)**

We consider the model of geometric Brownian motion: $$X_t=x+\int_0^t \sigma X_s {\rm d}W_s+\int_0^t \mu X_s {\rm d}s$$ with $x>0$.

- Compute $\mathbb{E}(X_T^2)$.
- Let $X^h$ be the related Euler scheme with time step $h$. Set $y_i=\mathbb{E}((X^{(h)}_{t_i})^2)$. Find a relation between $y_{i+1}$ and $y_i$.
- Deduce that $\mathbb{E}((X^{(h)}_{T})^2)=\mathbb{E}(X_T^2)+O(h)$.

**Exercice 5.4 (solving SDE using change of variables)**

Consider an SDE of the form \begin{equation}X_t = x_0+\int_0^t b(X_s){\rm d}s + \int_0^t\sigma(X_s){\rm d}W_s. \end{equation} We study two transformations that lead to simpler equations and may help in the resolution of the initial SDE.

*Lamperti transformation*. We assume that the SDE is defined on ${\mathbb R}$, and we assume that the coefficients $b:{\mathbb R}\to{\mathbb R}$ and $\sigma:{\mathbb R}\to(0,\infty)$ are of class ${\cal C}^1$ with bounded derivative, that the function $\frac1{\sigma(x)}$ is not integrable at $\pm \infty$, and that the function $b/\sigma - \sigma'/2$ is Lipschitz.- Verify that the function $f(x)=\int_{x_0}^x \frac{{\rm d}y}{\sigma(y)}$ is a bijection from ${\mathbb R}$ into ${\mathbb R}$.
- Show that the solution to the SDE can be put in the form $X_t=f^{-1}(Y_t)$, where the process $Y$ solves
\[
Y_t = \int_0^t \tilde{b} (Y_s) {\rm d}s + W_t,
\]
for some function $\tilde{b}$ to explicit.
*Comment*: The interest in Lamperti transformation is to retrieve an SDE with unit-diffusion coefficient; the approximation by Euler scheme of this SDE becomes more accurate (Exercise 5.1). *Doss-Sussmann transformation*. We consider now the case where the coefficients $b,\sigma:{\mathbb R}^d \to {\mathbb R}^d$ are Lipschitz, $\sigma$ is of class ${\cal C}^2_b$ (bounded with bounded derivatives), and $W_t$ is still a scalar Brownian motion. Denote by $F(\theta,x)$ the flow of the differential equation $x'_\theta=\sigma(x_\theta)$ in ${\mathbb R}^d$, i.e. \[ \partial_\theta F(\theta,x) = \sigma(F(\theta,x)), \qquad F(0,x)=x, \qquad \forall \: (\theta,x) \in {\mathbb R} \times {\mathbb R}^d, \] and by $\partial_x F(\theta,x)$ the Jacobian matrix of $F$. Using that $F$ is of class ${\cal C}^2$ on ${\mathbb R}\times{\mathbb R}^d$ and that $\partial_x F(\theta,x)$ is invertible for any $(\theta,x)$, show that the solution $X_t$ to the SDE is written in the form \[ X_t(\omega)=F(W_t(\omega),Z_t(\omega)) \] for any $t$, where the differentiable process $Z_t$ solves --- $\omega$ by $\omega$ --- the Ordinary Differential Equation (ODE) \[ Z_t = x_0+\int_0^t (\partial_x F(W_s,Z_s))^{-1} \Bigl(b - \frac12 (\partial_x \sigma) \sigma \Bigr)(F(W_s,Z_s)) {\rm d}s, \] with the matrix $(\partial_x F(\theta,x))^{-1}$ being the inverse of $\partial_x F(\theta,x)$.

**Exercice 5.5 (simulation of exit time)**

Write a simulation program to compute ${\mathbb E}(\mathbb{1}_{\sup_{0\leq t\leq T}S_t\leq U}(K-S_T)_+)$ where $S$ is a geometric Brownian motion. Compare the three schemes (discrete time approximation, Brownian bridge method, boundary shifting method).

**Exercice 5.6 (exit time, non convergence)**

In the theoretical results of Section 5.4, the assumption of non-degeneracy (i.e. $\sigma(x)$ invertible) plays an important role in the validity of the convergence results. Otherwise, we may face some pathological situations, as illustrated below.

- In dimension 1, consider the model with $b(y)=y$, $\sigma(y)=0$, $x_0 =1$ and $D=(-\infty,\exp(1))$, $T=1$. Prove that $\tau^{(h)}_{disc.}>1$ and $\tau=1$, so that $$\mathbb{P}(\tau^{(h)}_{disc.}> T)-\mathbb{P}(\tau> T)=1$$ does not converge to 0 as $h\to0$.
- In dimension 1, consider the model $b(x) = \cos(x), \sigma(x) = \sin(x)$, $x_0=\pi/2$ with $D =(-\pi,2\pi)$. Prove that $\tau=+\infty$ almost surely and $\tau^{(h)}_{disc.}<+\infty$ almost surely.

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

## No comments:

## Post a Comment