User:Bmuperle/Continuous Time Finance

This is an outline of the lecture contents.

Basic sources of reference are Shreve and Shiryaev . For no-arbitrage theory we recommend Föllmer and Schied and Delbaen and Schachermayer . Have a look into these books, read the introductions of each chapter.

Financial engineering without martingales has to be common body of knowledge for every quant. A popular reference for this stuff is Wilmott .

The main reference for models with jumps is Rama Cont and Tankov . Mathematical texts on the same subject are Protter and Jacod and Shiryaev .

From Wiener process to jump diffusions

Overview

Concepts

• Levy process (square integrable), cadlag property
• Wiener process, generalized Brownian motion
• counting process, Poisson process, compensated Poisson process
• filtration (past, history), independence of the past
• geometric Brownian motion, geometric Poisson process
• compound Poisson process (CPP), jump diffusion

.

Facts

• mean and variance of Levy processes, distribution of Levy processes (Gaussian case, Poisson case), covariance structure of a Levy process, law of large numbers for Levy processes, F-transform of Wiener process and Poisson process, normal approximation of the Poisson process

• independence of the past-property of Levy processes, martingale properties of Levy processes (and of squares), martingale properties of geometric Wiener process and geometric Poisson process.

• CPP are Levy processes, Fourier transform of CPP, moments of CPP, martingale properties of CPP, linear combinations of Levy processes

Basics about Levy processes

Consider a stochastic process $(X_{t})_{t\geq 0}$ with continuous time. The following definition extends the notion of a random walk.

Explanation: Random walk
A random walk $(X_{n})$ is the sequence of partial sums of $X_{n}=U_{1}+U_{2}+\cdots +U_{n}$ of i.i.d. random variables $(U_{k})$. A random walk has independent and stationary increments. Expectations and variances are proportional to $n$.

Definition: The process $(X_{t})$ is a Levy process if it has independent and stationary increments, and if $X_{t}{\stackrel {P}{\to }}X_{0}$ as $t\downarrow 0$.

Remark
The latter condition is a continuity condition which excludes cases where the process jumps immediately after having started.

Levy processes are named in honour of the famous French mathematician Paul Lévy.

Theorem: Mean and variance of a square integrable Levy process are proportional to $t$, i.e. $E(X_{t})=tE(X_{1})$ and $V(X_{t})=tV(X_{1})$. The covariance is ${\text{Cov}}(X_{s},X_{t})=\min(s,t)V(X_{1})$.

Proof
Under construction.

Theorem: Every square integrable Levy process satisfies the law of large numbers:

Failed to parse (syntax error): {\displaystyle \dfrac{X_t}{t}\stackrel{P}{\to}E(X_1)\quad\text{as $t\to\infty$}.}

Proof
Under construction.

Note that any linear function is a Levy process (with variance zero).

Theorem: Any linear combination of independent Levy processes is a Levy process.

Proof
Under construction.

It follows that centering an integrable Levy process (subtracting $tE(X_{1})$) preserves the Levy property.

Distributions of Levy processes

Which probability distributions are possible for Levy processes ? It will turn out that we can characterize the distributions of the increments of Levy processes by a simple property.

Explanation: Convolution
Let $\mu$ and $\nu$ be probability distributions. The convolution $\mu *\nu$ is the distribution ${\mathcal {L}}(Z|P)$ of a random variable $Z=X+Y$ where $X$ and$Y$ are independent and $\mu ={\mathcal {L}}(X|P),\;\nu ={\mathcal {L}}(Y|P)$.

Definition: A family $(\mu _{t})_{t\geq 0}$ is called a convolution semigroup if it satisfies
(1) $\mu _{s}*\mu _{t}=\mu _{s+t}$ (where $*$ denotes the convolution of probability distributions), and
(2) $\lim _{t\to 0}\mu _{t}(|x|>\epsilon )=0$, $\epsilon >0$.

Theorem: For every Levy process $(X_{t})$ the family of distributions ${\mathcal {L}}(X_{t}|P)$ is a convolution semigroup.

Proof

Since $X_{s+t}-X_{0}=(X_{s+t}-X_{s})+(X_{s}-X_{0})$ it is clear that the distributions of $X_{s}-X_{0}$ and $X_{s+t}-X_{s}$ add to the distribution of $X_{s+t}-X_{0}$. The increments $X_{s}-X_{0}$ and $X_{s+t}-X_{s}$ are independent random variables. Hence $X_{s+t}-X_{0}$ is the sum of independent random variables and its distribution is the convolution of the distributions of $X_{s}-X_{0}$ and $X_{s+t}-X_{s}$: ${\mathcal {L}}(X_{s+t}-X_{0}|P)={\mathcal {L}}(X_{s+t}-X_{s}|P)*{\mathcal {L}}(X_{s}-X_{0}|P)$ Recall that the increments of a Levy process are stationary. This implies, that the distributions of increments do not change if the interval is translated: ${\mathcal {L}}(X_{t}-X_{0}|P)={\mathcal {L}}(X_{a+t}-X_{a}|P)$ for every $a\geq 0$ Let us denote $\mu _{t}:={\mathcal {L}}(X_{t}-X_{0}|P)$. Then we obtain $\mu _{s}*\mu _{t}=\mu _{s+t}$

Theorem: For every convolution semigroup $(\mu _{t})_{t\geq 0}$ there is a Levy process such that $\mu _{t}={\mathcal {L}}(X_{t}|P)$.

Proof
Cite Protter.

When we are looking for possible distributions of Levy processes then we need families of distributions satisfying the convolution property of the preceding theorem.

Example: Brownian motion, Wiener process
The family of normal distributions $\mu _{t}:=N(at,\sigma ^{2}t)$ is a convolution semigroup. Proof Under construction. A corresponding Levy process $(X_{t})$ is called a Brownian motion. The Brownian motion with $a=0$ and $\sigma ^{2}=1$ is called a Wiener process. The Fourier transform is $E(e^{iuX_{t}})=e^{t(iau-\sigma ^{2}u^{2}/2)}$. Proof Under construction.

Example: Poisson process
The family of Poisson distributions $\mu _{t}:=\pi (\lambda t)$ is a convolution semigroup. Proof Under construction. A corresponding Levy process is called a Poisson process. The Fourier transform is $E(e^{iuX_{t}})=e^{\lambda t(e^{iu}-1)}$. Proof Under construction.

The following assertion is an easy application of Fourier transforms which shows that compensated (centered) Poisson processes with many small jumps look like Brownian motions.

Theorem: Let $X_{t}=h(N_{t}-\lambda t)$ where $(N_{t})$ is a Poisson process.
(1) $E(X_{t})=0$, $V(X_{t})=\lambda h^{2}t$.
(2) Let $\lambda _{n}\to \infty$ and $h_{n}\to 0$ such that $\lambda _{n}h_{n}^{2}\to \sigma ^{2}>0$. Then the distributions of $X_{nt}$ tend to the distributions of a Brownian motion.

Proof
Under construction.

Path properties

Definition: A stochastic process satisfies the cadlag property if all paths of the process are continuous from right and have limits from left.

Remark
There is no difference between having the cadlag property for all paths or for almost all paths (i.e. with probability 1).

Notation: Jumps
Assume that $(X_{t})$ is a stochastic process with cadlg paths. Denote $\Delta X_{t}:=X_{t}-X_{t-}$. If $\Delta X_{t}=0$ then the path is continuous at $t$. Otherwise there is a jump at $t$ with jump height $\Delta X_{t}$.

In general, Levy processes can be constructed such that their paths have the cadlag property. For a proof cite Protter.

Brownian motions and Poisson processes have very special path properties.

Theorem: A Levy process has continuous paths (with probability 1) iff it is a Brownian motion.

Proof
Any Brownian motion can be constructed with continuous paths: This is an advanced probabilistic construction (cite Bauer). Every Brownian motion has continuous paths with probability 1: This is an advanced measure theoretic result (cite Bauer). Every Levy process with continuous paths is a Brownian motion: We will indicate a simple proof later.

Another path property is the counting-process property.

Definition: A process $(N_{t})$ is a counting process if its paths are increasing steps functions such that $N_{0}=0$, $\Delta N_{t}=1$ and $P(N_{t}<\infty )=1$ for all $t\geq 0$.

Theorem: A Levy process is a counting process (with probability 1) iff it is a Poisson process.

Proof
A Poisson process can be constructed as a counting process: This is a simple probabilistic construction.⁄. Every Poisson process is a counting process: This is again an advanced measure theoretic result (cite Bauer). Every Levy counting process is a Poisson process: We will indicate a simple proof later.

Filtrations and martingales

Let $(X_{t})$ be a stochastic process. Then ${\mathcal {F}}_{t}^{X}:=\sigma (X_{s}:s\leq t)$ is called the past of the process at time $t$. The family of pasts $({\mathcal {F}}_{t}^{X})_{t\geq 0}$ is called the history of the process.

Definition: Any increasing family $({\mathcal {F}}_{t})$ of sigma-fields is called a filtration. A process $(X_{t})$ is adapted to the filtration if $X_{t}$ is ${\mathcal {F}}_{t}$-measurable for every $t\geq 0$.

The history of a process is a filtration. Each process is adapted to its own history.

Usually, a stochastic model starts with a basic stochastic process $(\xi _{t})$ and its history $({\mathcal {F}}_{t})$. The model is then a filtered probability space $(\Omega ,({\mathcal {F}}_{t}),P,(\xi _{t}))$ describing the evolution of observable information in the course of time.

Any further processes depending on the same observational history have to be adapted to the filtration of the model.

Definition: A Levy process $(X_{t})$ is a Levy process w.r.t. a given filtration $({\mathcal {F}}_{t})$ if it is adapted to the filtration and if for every $s\geq 0$ its increments $X_{t}-X_{s}$ are independent of ${\mathcal {F}}_{s}$.

A Wiener process w.r.t. a filtration $({\mathcal {F}}_{t})$ is a continuous Levy process $(W_{t})$ w.r.t. that filtration having increments $W_{t}-W_{s}\sim N(0,t-s)$.

\begin{lemma} Every Levy process is a Levy process w.r.t. its own history. \end{lemma}

{{User:Bmuperle/Proof \ldots }}

{{User:Bmuperle/Definition A process $(X_{t})$ is a martingale w.r.t. a filtration $({\mathcal {F}}_{t})$ if it is adapted to the filtraton, integrable and satisfies $E(X_{t}|{\mathcal {F}}_{s})=X_{s}$ for all $s<t$. }}

{{User:Bmuperle/Theorem A Levy process (w.r.t. a filtration $({\mathcal {F}}_{t})$) is a martingale iff it is centered. }}

Proof
\ldots

In the following we tacitely assume an underlying filtered probability space. All process properties are to be understood w.r.t. the given filtration.

Any Wiener process is a martingale. If $(N_{t})$ is a Poisson process with intensity $\lambda$ then $N_{t}-\lambda t$ (the compensated Poisson process) is a martingale.

{{User:Bmuperle/Theorem Let $(M_{t})$ be a square integrable Levy martingale with variance $\sigma ^{2}$. Then $M_{t}^{2}-\sigma ^{2}t$ is a martingale. }}

{{User:Bmuperle/Proof \ldots }}

If $(W_{t})$ is a Wiener process then $W_{t}^{2}-t$ is a martingale.

{{User:Bmuperle/Theorem Let $(M_{t})$ be a Levy process such that $E(e^{uM_{t}})=e^{t\phi (u)}$ is finite for $u\in \Re ,\,t\geq 0$. Then

:$S_{t}=e^{uM_{t}-t\phi (u)}$

is a martingale. }}

{{User:Bmuperle/Proof \ldots }}

If $(W_{t})$ is a Wiener process then $S_{t}=e^{\sigma W_{t}-\sigma ^{2}t/2}$ is a martingale.

\begin{corollary} An exponential Brownian motion $S_{t}=\exp(at+\sigma W_{t})$ satisfies $E(S_{t})=e^{\mu t}$ iff $a=\mu -\sigma ^{2}/2$. \end{corollary}