% -*- fill-column: 200; coding: utf-8 -*-
%
% Cheatsheet for Stochastic Calculus and Differential Equations
% Author: Martin Blais
%
% Copyright (C) 2009  Martin Blais
% This work is licensed under the Creative Commons
% Attribution - Non-Commercial - Share-Alike license.
% See http://creativecommons.org/licenses/by-nc-sa/2.5/ for details.
%

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\title{Stochastic Calculus Cheatsheet}
\author{}
\date{}

% Math expressions.
\newcommand{\filt}{\mathcal{F}}
\newcommand{\expect}{\bm{E}}
\newcommand{\expectpm}[1]{\bm{E}^{\mathbb{#1}}}
\newcommand{\fieldR}{\mathbb{R}}
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\newcommand{\dP}{d\,\mathbb{P}}
\newcommand{\dQ}{d\,\mathbb{Q}}
\newcommand{\lP}{\mathbb{P}}
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\newcommand{\indic}[1]{\mathbb{1}_{\{#1\}}}
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\newcommand{\esqu}[1]{e^{-\frac{#1^2}{2}}}
\newcommand{\ndist}[1]{\dfrac{1}{\sqtwopi} \int_{-\infty}^{#1} \esqu{\phi} d\phi}
\newcommand{\logSE}{\log\nicefrac{S}{E}}
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\begin{document}
\thispagestyle{empty}

\begin{center}
  {\LARGE Stochastic Calculus Cheatsheet}
\end{center}





%-------------------------------------------------------------------------------------------------------------
\section*{Standard Brownian Motion / Wiener process}

\begin{tabular}[t]{cc}
\begin{minipage}{0.48\textwidth}
{\normalsize

  \begin{eqnarray*}
    & E[dX] = 0 \qquad E[dX^2] = dt \\
    & \lim_{dt\rightarrow 0} dX^2 = dt \\
    & \textrm{Discrete approx: } dX = \phi\sqrt{dt} \quad\textrm{where }\phi \sim N(0, 1) \\
    & dX \; \textrm{is} \; O(dt^{1/2}) \qquad dt dX \; \textrm{is} \; O(dt^{3/2})
  \end{eqnarray*}

\subsection*{Itô Product Rule}
If $dX_t = \alpha dt + \beta dW_t$ and $dY_t = \gamma dt + \lambda dW_t$,
\begin{align*}
  d(X_t Y_t) & = X_t dY_t + Y_t dX_t + dX dY \\
  & = X_t dY_t + Y_t dX_t + \frac{1}{2}\beta\lambda dt
\end{align*}



}
\end{minipage}
\begin{minipage}{0.48\textwidth}
{\footnotesize

  Characterization:
  \begin{enumerate} \tightitems
  \item $X(0) = 0$
  \item Continuous everywhere, differentiable nowhere
  \item[3.] $X(t) - X(s) \sim N(0, |t-s|)$
  \item[4.] $X(t+s) - X(t)$ is independent of $X(t)$
  \end{enumerate}
  Levy's characterization:
  \begin{enumerate} \tightitems
  \item[3.] $X_t$ is a martingale w.r.t. the filtration $\filt_t$
  \item[4.] $|X|^2 - t$ is a martingale w.r.t. the filtration $\filt_t$
  \end{enumerate}

}
\end{minipage}
\end{tabular}





\section*{Stochastic Differential Equations (General Form)}
\begin{eqnarray*}
  dS & = & f(t, S)\, dt + g(t, S)\, dX_i \\
  dS_i & = & f_i(t, S_0, \dots, S_n)\, dt + g_i(t, S_0, \dots, S_n)\, dX_i
  \qquad \textrm{where}
  \quad \textrm{$f$ is the drift,}
  \ \textrm{$g$ is the diffusion}
\end{eqnarray*}



\section*{Itô's Lemma and Basic Stochastic Integration}

\subsection*{For $F(X_t)$}
\begin{tabular}[t]{cc}
\begin{minipage}{0.4\textwidth}

  \begin{displaymath}
    dF = \dfrac{dF}{dX} dX_t + \dfrac{1}{2} \dfrac{d^2F}{dX^2} dt
  \end{displaymath}

\end{minipage}
\begin{minipage}{0.6\textwidth}

  \begin{displaymath}
    F(X_t) = F(X_0) + \int_0^t \dfrac{dF}{dX} dX_\tau +
    \dfrac{1}{2} \int_0^t \dfrac{d^2F}{dX^2} d\tau
  \end{displaymath}

\end{minipage}
\end{tabular}

%====================

\subsection*{For $F(X_t, t)$}
\begin{tabular}[t]{cc}
\begin{minipage}{0.4\textwidth}

  \begin{displaymath}
    dF = \dfrac{\partial F}{\partial X} dX_t + \left( \dfrac{\partial F}{\partial t} + \dfrac{1}{2} \dfrac{\partial^2 F}{\partial X^2} \right) dt
  \end{displaymath}

\end{minipage}
\begin{minipage}{0.6\textwidth}

  \begin{displaymath}
    F(X_t, t) = F(X_0, 0) + \int_0^t \dfrac{\partial F}{\partial X} dX_\tau +
    \int_0^t \left(\dfrac{\partial F}{\partial t}  + \dfrac{1}{2} \dfrac{\partial^2 F}{\partial X^2} \right) d\tau
  \end{displaymath}

\end{minipage}
\end{tabular}







\newcommand{\pade}[3]{\dfrac{\partial^{#2} V}{\partial #1_{#3}^{#2}}}

\section*{Functions of Stochastic Functions}

\begin{tabular}[t]{cc}
\begin{minipage}{0.58\textwidth}

  \subsection*{1-dimensional: $V(t, S)$}
  \begin{eqnarray*}
    dV & = & \pade{t}{}{} dt + \pade{S}{}{} dS  + \dfrac{1}{2} g^2 \pade{S}{2}{} dt \\
       & = & \left( \pade{t}{}{} + f \pade{S}{}{} + \dfrac{1}{2} g^2 \pade{S}{2}{} \right) dt + g \pade{S}{}{} dX
  \end{eqnarray*}

\end{minipage}
\begin{minipage}{0.38\textwidth}

  \begin{enumerate} \tightitems\small
  \item Apply Taylor expansion on $V$
  \item Apply It\^o's Lemma:
    \begin{itemize}
    \item $dX_i^2 \rightarrow dt$
    \item $dX_i dX_j \rightarrow \rho_{ij} dt$
    \end{itemize}
  \item Regroup the terms in $dt$ and $dX_i$
  \item Sto.integ.: integrate the resulting DE
  \end{enumerate}


\end{minipage}
\end{tabular}

  \subsection*{2-dimensional: $V(t, S_1, S_2)$}
  \begin{displaymath}
    %% dV =  \dfrac{\partial V}{\partial t} dt + \dfrac{\partial V}{\partial S_1} dS_1 + \dfrac{\partial V}{\partial S_2} dS_2 +
    %%  \left( \dfrac{1}{2} g_1^2 \dfrac{\partial^2 V}{\partial S_1^2}
    %% + \rho g_1 g_2 \dfrac{\partial^2 V}{\partial S_1 \partial S_2}
    %% + \dfrac{1}{2} g_2^2 \dfrac{\partial^2 V}{\partial S_2^2} \right) dt
    %%
    dV = \left( \pade{t}{}{} + f_1 \pade{S}{}{1} + f_2 \pade{S}{}{2} +
         \half g_1^2 \pade{S}{2}{1} + \rho g_1 g_2 \dfrac{\partial^2 V}{\partial S_1 \partial S_2} + \half g_2^2 \pade{S}{2}{2} \right) dt
         + g_1 \pade{S}{}{1} dX_1 + g_2 \pade{S}{}{2} dX_2
  \end{displaymath}

  \subsection*{n-dimensional: $V(t, S_1, \dots, S_n)$}
  \begin{displaymath}
    dV = \left( \pade{t}{}{} + \sum_{i=1}^{n} f_i \pade{S}{}{i} +
         \half \sum_{i=1}^{n} g_i^2 \pade{S}{2}{i} +
         \sum_{i=1, j>1}^{n} \rho_{ij} \, g_i \, g_j \dfrac{\partial^2 V}{\partial S_i \partial S_j} \right ) dt
         + \sum_{i=1}^{n} g_i \pade{S}{}{i} dX_i
  \end{displaymath}





\clearpage

\section*{Transition Density Functions}

% Simple versions
%% \begin{tabular}[t]{cc}
%% \begin{minipage}{0.5\textwidth}
%%
%%   \subsection*{Forward}
%%   \begin{displaymath}
%%     \dfrac{\partial p}{\partial t'} = c^2 \dfrac{\partial^2 p}{\partial y'^2}
%%   \end{displaymath}
%%
%% \end{minipage}
%% \begin{minipage}{0.5\textwidth}
%%
%%   \subsection*{Backward}
%%   \begin{displaymath}
%%     \dfrac{\partial p}{\partial t} - c^2 \dfrac{\partial^2 p}{\partial y^2} = 0
%%   \end{displaymath}
%%
%% \end{minipage}
%% \end{tabular}
%%
%% \subsection*{Solutions}
%% \begin{displaymath}
%%   p(y, t; y', t') = \dfrac{1}{2c\sqrt{\pi (t'-t)}} \exp\left( -\dfrac{(y'-y)^2}{4c^2(t'-t)}  \right)
%% \end{displaymath}



\begin{tabular}[t]{cc}
\begin{minipage}{0.5\textwidth}

  \subsection*{Forward Kolmogorov}
  \begin{displaymath}
    \dfrac{\partial p}{\partial t'} = \frac{1}{2} \dfrac{\partial^2 }{\partial y'^2} \left( B(y',t')^2 p \right)
    - \dfrac{\partial }{\partial y'} \left( A(y',t') p \right)
  \end{displaymath}

\end{minipage}
\begin{minipage}{0.5\textwidth}

  \subsection*{Solution}
  \begin{displaymath}
    p(S, t; S', t') = \dfrac{1}{\sigma S' \sqrt{2\pi (t'-t)}} e^{
      - \dfrac{\left( \log\frac{S}{S'} + \left(\mu - \frac{1}{2}\sigma^2\right) (t'-t) \right)^2}
      {2\sigma^2(t'-t)}
      }
  \end{displaymath}

\end{minipage}
\end{tabular}



\section*{Common Processes/Dynamics}

\begin{tabular}[t]{cc}
\begin{minipage}{0.5\textwidth}

\subsection*{Brownian Motion with Drift}
  \begin{displaymath}
    dS = \mu \,dt + \sigma \,dX
  \end{displaymath}

\end{minipage}
\begin{minipage}{0.5\textwidth}

\subsection*{Geometric Brownian Motion (Lognormal)}
  \begin{displaymath}
    dS = \mu S \,dt + \sigma S \,dX
  \end{displaymath}
  \begin{displaymath}
    \dfrac{dS}{S} = \mu \,dt + \sigma \,dX
  \end{displaymath}

\end{minipage}
\end{tabular}

%====================

\begin{tabular}[t]{cc}
\begin{minipage}{0.5\textwidth}

\subsection*{Vasiček (1977)}
  \begin{displaymath}
    dS = \gamma(\bar{r} - r) \,dt + \sigma \,dX
  \end{displaymath}

\end{minipage}
\begin{minipage}{0.5\textwidth}

\subsection*{Cox, Ingersoll, Ross}
  \begin{displaymath}
    dS = (\upsilon - \sigma S) \,dt + \sigma S^{\frac{1}{2}} \,dX
  \end{displaymath}

\end{minipage}
\end{tabular}


FIXME TODO add others, Ho Lee and company...




\vspace{2in}
%-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\section*{All you need to know about Sto.Calc}

(FIXME integrate these words of wisdom from Antoine.)

\begin{itemize}
\item If $X_t \rightarrow N(\mu, \sigma)$ then $E(x^{X_t}) = e^{\mu + \frac{\sigma^2}{2}}$.
\item Itô: $d( f(X_t) )$
\item Itô: $d( X_t Y_t ) = X_t dY_t + Y_t dX_t + \frac{1}{2}\beta\lambda dt$ where $dX_t = \alpha dt + \beta dW_t$ and $dY_t = \gamma dt + \lambda dW_t$
\item $E[ \int X_t dW_t ] = 0$
\item $Var[ \int X_t dW_t ] = \int X_t^2 dt$
\item Girsanov's theorem.
\item Generating correlated $X$ and $Y$.
\end{itemize}







\clearpage
%-------------------------------------------------------------------------------------------------------------
\section*{Martingales}

\begin{tabular}[t]{cc}
\begin{minipage}{0.5\textwidth}
{\normalsize

  \begin{center}
    \includegraphics[height=1in]{ranvar.pdf}
  \end{center}
}
\end{minipage}
\begin{minipage}{0.5\textwidth}
{\normalsize

  \begin{center}
    \includegraphics[height=1in]{chgprob.pdf}
  \end{center}
}
\end{minipage}
\end{tabular}




\begin{tabular}[t]{cc}
\begin{minipage}{0.5\textwidth}
{\normalsize

\subsection*{Probability Spaces}

% \textbf{Random variable}: a \emph{function} which assigns to each individual event $\omega \in \Omega$ a
% numerical value.
%
% \textbf{Stochastic Process}: $S(t)$ can be viewed as a sequence of random variables indexed by time $t$.
%
% \textbf{Adapted/Measurable Process}: $S_t$ is adapted to the filtration $\filt_t$, or measurable with respect
% to $\filt_t$, if the value of $S$ at time $t$ is known given the information set $\filt_t$.

``Let $(\Omega, \mathcal{F}, \probmeas)$ be a probability space\dots''
\begin{itemize}\tightitems
\item $\Omega$: sample space
\item $\filt$: filtration (information set), \\
  (Note that $\filt_{t_1} \subseteq \filt_{t_2} \subseteq \filt_{T} \equiv \filt$)
\item $\probmeas$: probability measure
\end{itemize}

}
\end{minipage}
\begin{minipage}{0.5\textwidth}
{\normalsize

\subsection*{Unconditional Expectation}

Expected value under a prob. measure (Lebesgue integral):
\begin{align*}
  \expect[h(X)] & = \int_\Omega h(x) p(x) dx = \int_\Omega h(x) d(\mathbb{P(x)}) = \int_\Omega h(x) \dP \\
  \expect[\indic{X\in A}] & = \int_\Omega \indic{X\in A} \dP = \int_A \dP = \mathbb{P}(A)
\end{align*}

}
\end{minipage}
\end{tabular}

%-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\vspace{4mm}

\begin{tabular}[t]{cc}
\begin{minipage}[t]{0.48\textwidth}
{\small

\subsection*{Martingales (Definition)}
\begin{align*}
  & \expect[M_t] < \infty \\
  & \expect[M_{t+1} | \filt_t] = M_t \quad \forall 0 \leq s \leq t \\
  & \expect[M_{t+1} | \filt_t] \leq M_t \quad \textrm{(supermartingale)} \\
  & \expect[M_{t+1} | \filt_t] \geq M_t \quad \textrm{(submartingale)}
\end{align*}

{\footnotesize
Wiener $\in$ Martingale (driftless) $\subset$ Markov (memoryless) $\subset$ non-Markov
}


\subsection*{Equivalent Measures}
Absolute continuity: if $ P(A) = 0 \rightarrow Q(A) = 0 \qquad \forall A$.

$\mathbb{Q}$ is ``absolutely continuous'' w.r.t. $\mathbb{P}$, and
$\mathbb{Q} << \mathbb{P}$.

\begin{quote}
  \begin{center}
    ``It is allright to tinker with the probabilities as long as we do not tinker
    with the (im)possibilities.''
  \end{center}
\end{quote}

Equivalent measures: if $\mathbb{Q} << \mathbb{P}$ and $\mathbb{P} << \mathbb{Q}$.

\subsection*{Radon-Nikod\'ym Theorem}
\begin{displaymath}
  \mathbb{Q}(A) = \int_A \Lambda\, \dP \qquad \textrm{where} \; \Lambda = \dfrac{\dQ}{\dP} \; \textrm{is the R.N. derivative.}
\end{displaymath}

}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
{\small
\subsection*{Conditional Expectation}
{\footnotesize (Use these to prove that a process is a Martingale; use the definition.)}

\begin{enumerate}
\item Linearity: $\expect[aX + bY|\filt] = a\expect[X|\filt] + b\expect[Y|\filt]$
\item \textbf{Tower Property}: if $\filt \subset \mathcal{G}$,
  \begin{align*}
  \expect[\expect[X|\mathcal{G}] \,|\filt] &= E[X|\filt] \\
  \expect[\expect[X|\filt]] &= E[X]
  \end{align*}
\item Taking out what is known:
  \begin{align*}
    & \expect[X|\filt] = X \\
    \textrm{(if $X$ is $\filt$-measurable but not $Y$:)}\quad
      & \expect[XY|\filt] = X\expect[Y|\filt]
  \end{align*}

\item Independence: if $X$ is independent from $\filt$,  \\
  $\expect[X|\filt] = E[X]$
\item Positivity: if $X \geq 0$ then $\expect[X|\filt] \geq 0$
\item Jensen' Inequality: if $f$ is a convex function, then $f(\expect[X|\filt]) \leq \expect[f(X)|\filt]$
\end{enumerate}


\subsection*{Exponential Martingale}
\begin{displaymath}
  M(t) = \exp(S_t + f(t)) \qquad \textrm{where}\; f(t) = -(\mu + \dfrac{1}{2}\sigma^2) t
\end{displaymath}


}
\end{minipage}
\end{tabular}




%-------------------------------------------------------------------------------------------------------------
\section*{It\^o Integrals \& Martingales}

\begin{tabular}[t]{cc}
\begin{minipage}[t]{0.5\textwidth}
{\normalsize

% \subsection*{It\^o Integral}
It\^o integrals are Martingales:
\begin{displaymath}
  \expect[ \int_0^T g(t,X_t) dX_t ] = 0
\end{displaymath}

\subsection*{Martingale Representation Theorem}

If $M$ is a Martingale, there exists $g(t,X)$ such that 
\begin{displaymath}
  M_T = M_0 + \int_0^T g(t,X) dX_t
\end{displaymath}
{\footnotesize The rightmost term is an Itô integral (and thus also a Martingale).}

\subsection*{Fubini's Theorem}
\begin{displaymath}
  \expect\left[ \int_0^T f(X_t) dt \right] = \int_0^T \expect\left[ f(X_t) \right] dt
\end{displaymath}

}
\end{minipage}
\begin{minipage}[t]{0.5\textwidth}
{\normalsize

\subsection*{Properties of It\^o Integrals}

\begin{enumerate}
\item Linearity: 
  \begin{displaymath}
    \int_0^T (\alpha f(t) + \beta g(t)) dX_t = \int_0^T \alpha f(t) dX_t + \int_0^T \beta g(t) dX_t
  \end{displaymath}

\item Isometry: 
  \begin{displaymath}
    \expect\left[ \left| \int_0^T f(t) dX_t \right|^2 \right] = \expect\left[ \int_0^T |f(t)|^2 dt \right]
  \end{displaymath}

\item Martingale: 
  \begin{displaymath}
    \expect\left[ \int_0^T f(t) dX_t \Bigg| \filt_s \right] = \int_0^s f(t) dX_t
  \end{displaymath}

\end{enumerate}





}
\end{minipage}
\end{tabular}










\clearpage
\section*{Application of Martingales to Asset Pricing}

\begin{center}
  \textit{Warning: I still need to complete and arrange this page of notes.}
\end{center}

\begin{center}
  \parbox{3in}{
    \subsection*{Fundamental Asset Pricing Formula}
    \begin{displaymath}
      \textrm{Value} = \expect^{\textrm{Meas.}}\left[ PV(\textrm{expected cash flows}) \right]
    \end{displaymath}
  }
\end{center}


\begin{tabular}[t]{cc}
\begin{minipage}{0.48\textwidth}
{\normalsize

\subsection*{Risk-free Asset}
\begin{align*}
  dB_t& = r B_t dt, \quad B(0) = B_0 \\
  B(t)& = B_0 e^{rt}
\end{align*}

\subsection*{Underlying S}
\begin{align*}
  dS_t & = \mu S_t dt + \sigma S_t dX, \quad S(0) = S_0 \\ 
  S(t) & = S_0 e^{\mu t - \half \sigma^2 + \sigma X_t}
\end{align*}

\subsection*{Removing the TVM}
\begin{align*}
  S^*(T)& = \dfrac{S(T)}{e^{rt}} \\
  S^*(t)& = S^*_0 e^{(\mu - r - \half \sigma^2) t + \sigma X_t} \\
  dS^* & = (\mu - r) S^* dt + \sigma S^* dX
\end{align*}

\subsection*{Self-financing Portfolios}

Trading Strategy:
\begin{align*}
  \phi_t = (\phi_t^S, \phi_t^B) \quad \textrm{processes}
\end{align*}

Self-financing portfolio: no in/out flows.
\begin{align*}
  \textrm{Value}: \quad V_t(\phi) = \phi_t^S S_t + \phi_t^B B_t \qquad \forall t \in [0,T] \\
  V_t(\phi) = V_0(\phi) + \int_0^t \phi_u^S dS_u + \int_0^t \phi_u^B dB_u
\end{align*}

Arbitrage opportunity:
\begin{align*}
  & V_0(\phi) = 0 \\
  & \textrm{with} \; P(V_T(\phi) > 0) > 0 \quad \textrm{and} \quad P(V_T(\phi) < 0) = 0
\end{align*}




}
\end{minipage}
\begin{minipage}{0.48\textwidth}
{\normalsize

\subsection*{Novikov Condition}
\begin{displaymath}
  E\left[ e^{\half \int_0^T \theta_s^2 ds} \right] < \infty
\end{displaymath}

\begin{align*}
  M_t^\theta = e^{(-\int_0^t \theta_s dXs - \half \int_0^t \theta_s^2 ds)} \quad 
  \textrm{is a Martingale}
\end{align*}


\subsection*{Girsanov's Theorem}
\begin{align*}
  \dfrac{\dQ}{\dP} & = e^{(-\int_0^t \theta_s dXs - \half \int_0^t \theta_s^2 ds)} \\
  X_t^\lQ &= X_t^\lP + \int_0^t \theta(s) ds
\end{align*}

\begin{itemize}
\item Provides an expression for the Radon-Nikod\'ym derivative.
\item Gives an explicit correspondence btw $\lP$ and $\lQ$ in terms of their Brownian motion.
\end{itemize}

\dots but does \textbf{not} tell you what $\theta$ is. We assume $\theta$ and check that it satisfies the Novikov condition. Then we have the RN derivative, and we can change measures!



\subsection*{Doléans/Stochastic Exponential}
\begin{align*}
  \mathcal{E}\left( \int_0^t \theta_s dX_s \right) & = \exp{\left( \int_0^t \theta_s dX_s - \half \int_0^t \theta_s^2 ds \right)} \\
  X_t^\lQ & = X_t^\lP - \int_0^t \theta(s) ds
\end{align*}


\subsection*{Feynman-Kač Equivalence}

\begin{align*}
\textrm{PDE:}\quad  & \parfrac{V}{t} + \mu \parfrac{V}{S} + \half \sigma^2 \parfrac[2]{V}{S} - rV = 0, \quad V(T,S) = G(S) \\
& dS_t = \mu(t,S_t) dt + \sigma(t,S_t) dX_t 
\end{align*}
\begin{center}
  $\Updownarrow$ 
\end{center}
\begin{align*}
\textrm{Expectation:}\quad & V(t,S_t) = e^{-r\tnt} \expect\left[ G(S_T) | \filt_t \right]
\end{align*}





}
\end{minipage}
\end{tabular}




















%-------------------------------------------------------------------------------------------------------------
% Copyright/distribution license.
{\vfill\hfill{\tiny Author: Martin Blais, 2009.
This work is licensed under the Creative Commons
``Attribution - Non-Commercial - Share-Alike'' license.}}
\end{document}


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