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User:Bmuperle/Continuous Time Finance

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This is an outline of the lecture contents.

Basic sources of reference are Shreve [1] and Shiryaev [2]. For no-arbitrage theory we recommend Föllmer and Schied [3] and Delbaen and Schachermayer [4]. 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 [5].

The main reference for models with jumps is Rama Cont and Tankov [6]. Mathematical texts on the same subject are Protter [7] and Jacod and Shiryaev [8].

From Wiener process to jump diffusions

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Overview

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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

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  • 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

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Consider a stochastic process with continuous time. The following definition extends the notion of a random walk.

Definition: The process is a Levy process if it has independent and stationary increments, and if as .

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 , i.e. and . The covariance is .

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \dfrac{X_t}{t}\stackrel{P}{\to}E(X_1)\quad\text{as $t\to\infty$}.}

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

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

It follows that centering an integrable Levy process (subtracting ) preserves the Levy property.

Distributions of Levy processes

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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.

Definition: A family is called a convolution semigroup if it satisfies
(1) (where denotes the convolution of probability distributions), and
(2) , .

Theorem: For every Levy process the family of distributions is a convolution semigroup.

Theorem: For every convolution semigroup there is a Levy process such that .

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.

Theorem: Let where is a Poisson process.
(1) , .
(2) Let and such that . Then the distributions of tend to the distributions of a Brownian motion.

Path properties

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

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.

Another path property is the counting-process property.

Definition: A process is a counting process if its paths are increasing steps functions such that , and for all .

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

Filtrations and martingales

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Let be a stochastic process. Then is called the past of the process at time . The family of pasts is called the history of the process.

Definition: Any increasing family of sigma-fields is called a filtration. A process is adapted to the filtration if is -measurable for every .

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 and its history . The model is then a filtered probability space 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 is a Levy process w.r.t. a given filtration if it is adapted to the filtration and if for every its increments are independent of .

A Wiener process w.r.t. a filtration is a continuous Levy process w.r.t. that filtration having increments .

\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 is a martingale w.r.t. a filtration if it is adapted to the filtraton, integrable and satisfies for all . }}

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

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 is a Poisson process with intensity then (the compensated Poisson process) is a martingale.

{{User:Bmuperle/Theorem Let be a square integrable Levy martingale with variance . Then is a martingale. }}

{{User:Bmuperle/Proof \ldots }}

If is a Wiener process then is a martingale.

{{User:Bmuperle/Theorem Let be a Levy process such that is finite for . Then

:

is a martingale. }}

{{User:Bmuperle/Proof \ldots }}

If is a Wiener process then is a martingale.

\begin{corollary} An exponential Brownian motion satisfies iff . \end{corollary}

References

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  1. ^ Steven E. Shreve (3 June 2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer. ISBN 978-0-387-40101-0. Retrieved 24 January 2013.
  2. ^ Albert N. Shiryaev (1 February 1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific. ISBN 978-981-02-3605-2. Retrieved 24 January 2013.
  3. ^ Hans Föllmer; Alexander Schied (15 January 2011). Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter. ISBN 978-3-11-021804-6. Retrieved 25 January 2013.
  4. ^ Freddy Delbaen; Walter Schachermayer (19 November 2010). The Mathematics of Arbitrage. Springer. ISBN 978-3-642-06030-4. Retrieved 25 January 2013.
  5. ^ Paul Wilmott (11 January 2007). Paul Wilmott on Quantitative Finance, 3 Volume Set. John Wiley & Sons. ISBN 978-0-470-06077-3. Retrieved 25 January 2013.
  6. ^ Rama Cont; Peter Tankov (26 October 2012). Financial Modelling with Jump Processes, Second Edition. CRC PressINC. ISBN 978-1-4200-8219-7. Retrieved 24 January 2013.
  7. ^ Philip Protter (24 May 2005). Stochastic Integration and Differential Equations: Version 2.1. Springer. ISBN 978-3-540-00313-7. Retrieved 24 January 2013.
  8. ^ Jean Jacod; Albert N. Shiryaev (31 December 1987). Limit theorems for stochastic processes. Springer-Verlag. ISBN 978-3-540-17882-8. Retrieved 24 January 2013.