updated montecarlo

Mathematics/4th/Stochastic_processes/Stochastic_processes.tex

@@ -370,9 +370,9 @@
       p_{ij}^{(n+1)} & =\Prob(X_{n+1}=j\mid X_0=i)                     \\
                      & =\sum_{k\in I}\Prob(X_{n+1}=j, X_n=k\mid X_0=i) \\
       \begin{split}
-        &=\sum_{k\in I}\Prob(X_{n}=k\mid X_0=i)\cdot\\
-        &\hspace{2.5cm}\cdot\Prob(X_{n+1}=j\mid X_n=k, X_0=i)
-      \end{split}             \\
+         & =\sum_{k\in I}\Prob(X_{n}=k\mid X_0=i)\cdot          \\
+         & \hspace{2.5cm}\cdot\Prob(X_{n+1}=j\mid X_n=k, X_0=i)
+      \end{split}          \\
                      & =\sum_{k\in I}p_{ik}^{(n)}p_{kj}^{(1)}
     \end{align*}
     where the penultimate equality follows from \mcref{SP:lema1Markov} and the last equality follows from \mcref{SP:lema2Markov} and the Markov property because if $D=\{X_n=k, X_0=i\}$ we have that:
@@ -589,10 +589,10 @@
     where $A=\{X_0=i,X_1\ne i, \ldots, X_{m_1-1}\ne i,X_{m_1}=i, X_{m_1+1}\ne i,\ldots, X_{m_1+\cdots+m_{\ell-1}}=i\}$. So, by the \mnameref{SP:substitutionPrinciple} we have:
     \begin{align*}
       \begin{split}
-        p_\ell&=\Prob_i(X_{m_1+\cdots+m_\ell}=i,X_{m_1+\cdots+m_\ell-1}\ne i,\ldots,\\
-        &\hspace{4cm}X_{m_1+\cdots+m_{\ell-1}+1}\ne i\mid A)
+        p_\ell & =\Prob_i(X_{m_1+\cdots+m_\ell}=i,X_{m_1+\cdots+m_\ell-1}\ne i,\ldots, \\
+               & \hspace{4cm}X_{m_1+\cdots+m_{\ell-1}+1}\ne i\mid A)
       \end{split} \\
-       & =\Prob_i(X_{m_\ell}=i,X_{m_\ell-1}\ne i,\ldots,X_{1}\mid X_{0}=i)          \\
+       & =\Prob_i(X_{m_\ell}=i,X_{m_\ell-1}\ne i,\ldots,X_{1}\mid X_{0}=i)                                     \\
        & =\Prob(\tau_i=m_\ell)
     \end{align*}
   \end{proof}
@@ -652,14 +652,14 @@
   \begin{proof}
     Note that $\{X_n=j\}\subseteq \{\tau_j\leq n\}=\bigsqcup_{m=1}^n\{\tau_j=m\}$. Hence, $\{X_n=j\}=\bigsqcup_{m=1}^n[\{X_n=j\}\cap \{\tau_j=m\}]$. Thus:
     \begin{align*}
-      p_{ij}^{(n)} & =\Prob_i(X_n=j)                                              \\
-                   & =\sum_{m=1}^n \Prob_i(X_n=j,\tau_j=m)                        \\
-                   & =\sum_{m=1}^n \Prob_i(X_n=j\mid\tau_j=m)\Prob_i(\tau_j=m)    \\
+      p_{ij}^{(n)} & =\Prob_i(X_n=j)                                               \\
+                   & =\sum_{m=1}^n \Prob_i(X_n=j,\tau_j=m)                         \\
+                   & =\sum_{m=1}^n \Prob_i(X_n=j\mid\tau_j=m)\Prob_i(\tau_j=m)     \\
       \begin{split}
-        & =\sum_{m=1}^n \Prob_i(X_n=j\mid X_m=j,X_{m-1}\ne j,\ldots, X_1\ne j)\cdot \\
-        &\hspace{7cm}\cdot f_{ij}^{(m)}
+         & =\sum_{m=1}^n \Prob_i(X_n=j\mid X_m=j,X_{m-1}\ne j,\ldots, X_1\ne j)\cdot \\
+         & \hspace{7cm}\cdot f_{ij}^{(m)}
       \end{split} \\
-                   & =\sum_{m=1}^n \Prob_j(X_{n-m}=j)f_{ij}^{(m)}                 \\
+                   & =\sum_{m=1}^n \Prob_j(X_{n-m}=j)f_{ij}^{(m)}                  \\
                    & =\sum_{m=1}^nf_{ij}^{(m)}p_{jj}^{(n-m)}
     \end{align*}
   \end{proof}
@@ -926,9 +926,9 @@
     \begin{align*}
       p_{ij}(t+s) & =\sum_{k\in I} \Prob(X_{t+s}=j,X_s=k\mid X_0=i) \\
       \begin{split}
-        & =\sum_{k\in I} \Prob(X_{t+s}=j\mid X_s=k,X_0=i)\cdot\\
-        &\hspace{3.5cm}\cdot\Prob(X_s=k\mid X_0=i) \\
-      \end{split}         \\
+         & =\sum_{k\in I} \Prob(X_{t+s}=j\mid X_s=k,X_0=i)\cdot \\
+         & \hspace{3.5cm}\cdot\Prob(X_s=k\mid X_0=i)            \\
+      \end{split}       \\
                   & =\sum_{k\in I} p_{ik}(t)p_{kj}(s)
     \end{align*}
   \end{proof}

Mathematics/5th/Montecarlo_methods/Montecarlo_methods.tex

@@ -420,13 +420,44 @@
     \begin{itemize}
       \item $\vf{b},\vf\sigma\in\mathcal{C}^3$ with bounded derivatives and $\vf\sigma$ uniformly elliptic.
       \item $D$ is a half-space or $\Fr{D}$ is bounded of class $\mathcal{C}^3$.
-      \item $g$ is measurable with polynomial growth and vanishes at a neighbourhood of $\Fr{D}$.
+      \item $g$ is measurable with polynomial growth and vanishes at a neighborhood of $\Fr{D}$.
     \end{itemize}
     Then:
     $$
       \Exp\left(\indi{\tau>T}g(X_T) \right)-\Exp(\indi{\tilde\tau^m>T} g(\tilde{X}^m_T))=\O{\frac{1}{\sqrt{m}}}
     $$
     where $\tilde\tau^m:=\min\{ t_i: \tilde{X}^m_{t_i}\notin D\}$ and $\tilde{X}^m$ is the Euler scheme.
   \end{proposition}
+  \begin{proposition}
+    Assume that:
+    \begin{itemize}
+      \item $\vf{b},\vf\sigma\in\mathcal{C}^5$ with bounded derivatives and $\vf\sigma$ uniformly elliptic.
+      \item $D$ is a half-space or $\Fr{D}$ is bounded of class $\mathcal{C}^5$.
+      \item $g$ is measurable with polynomial growth and vanishes at a neighborhood of $\Fr{D}$.
+    \end{itemize}
+    Then:
+    $$
+      \Exp\left(\indi{\tau>T}g(X_T) \right)-\Exp(\indi{\tau^m>T} g(\tilde{X}^m_T))=\O{\frac{1}{m}}
+    $$
+  \end{proposition}
+  \begin{proposition}
+    Assume that $\mathcal{D}$ is a half-space with hyperplane orthogonal to $\vf\nu\in\RR^d$ passing by $\vf{z}\in\RR^d$, i.e.:
+    $$
+      \mathcal{D}=\{ \vf{y}\in\RR^d : \transpose{\vf\nu}(\vf{y}-\vf{z})>0\}
+    $$
+    If $\vf{x}_i,\vf{x}_{i+1}\in \mathcal{D}$, then:
+    \begin{multline*}
+      \Prob(\exists t\in [t_i,t_{i+1}], \tilde{X}_t^m\notin\mathcal{D}:\tilde{X}_{t_i}^m=\vf{x}_i, \tilde{X}_{t_{i+1}}^m=\vf{x}_{i+1})=\\=\exp{
+        -2\frac{m}{T}\frac{\transpose{\vf\nu}(\vf{x}_{i}-\vf{z})\transpose{\vf\nu}(\vf{x}_{i+1}-\vf{z})}{\norm{\transpose{\vf\sigma(\vf{x}_i)}\vf\nu}^2}
+      }
+    \end{multline*}
+  \end{proposition}
+  \begin{lemma}
+    Let ${(B_t)}_{t\geq 0}$ be a Brownian motion. Then, $\forall a\leq 0$ and $b\geq a$ we have:
+    $$
+      \Prob\left(\min_{t\in[0,h]}B_t\leq a\mid B_h=b\right)=\exp{-\frac{2}{h}a(a-b)}
+    $$
+  \end{lemma}
+  \subsection{Computation of sensitivities}
 \end{multicols}
 \end{document}