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.

Concepts
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 * 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 $$(X_t)_{t\ge 0}$$ with continuous time. The following definition extends the notion of a random walk.

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

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

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.

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

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.

Path properties
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.

Another path property is the counting-process property.

Filtrations and martingales
Let $$(X_t)$$ be a stochastic process. Then $$\mathcal{F}^X_t:=\sigma(X_s:s\le t)$$ is called the past of the process at time $$t$$. The family of pasts $$(\mathcal{F}^X_t)_{t\ge 0}$$ is called the history of the process.

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.

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}

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.

If $$(W_t)$$ is a Wiener process then $$W_t^2-t$$ is a martingale.

If $$(W_t)$$ is a Wiener process then $$S_t=e^{\sigma W_t-\sigma^2t/2}$$ 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}