Stochastic Analysis

A path of the Wiener process (blue) and a stochastic integral calculated with it (green)

The stochastic analysis is a branch of mathematics , more precisely the probability theory . She deals with the generalization of concepts, statements and models of analysis to stochastic processes , i.e. to functions whose values ​​are random. The focus of stochastic analysis is the formulation and investigation of stochastic integrals and, based on them, of stochastic differential equations .

Historically, the subject goes back to the work of the Japanese mathematician Kiyoshi Itō from 1944. The main impetus for this was the mathematical description of the physical phenomenon of Brownian motion by Albert Einstein and Norbert Wiener . This model, the Wiener process , with its numerous remarkable properties and generalizations, forms a starting point for stochastic analysis. Applications of the subject can be found in biology , physics and engineering , but above all in financial mathematics . A first high point was the groundbreaking Black-Scholes model published in 1973 for the valuation of options on a share , the price development of which is described by a stochastic differential equation.

introduction

The terminology of stochastic analysis is particularly elementary and clear in its main area of ​​application, financial mathematics. For this purpose, the following describes the processes involved in trading a financial instrument , which for the sake of simplicity is called a share and which fulfills some idealized assumptions. In particular, it should be possible to buy and sell the financial instrument in any quantity at any time.

From price gain in stock trading to the stochastic integral

The gain (or loss) on owning stock shares within a given period of time obviously depends on how many shares you own in the stock and how its price, the market price , changes over that period. For example, if you own shares in a share with an initial price and the price rises, for example within a day, the price gain is rounded ${\displaystyle H_{1}=4{,}27}$ ${\displaystyle S_{0}=106{,}12\,\mathrm {\euro} }$ ${\displaystyle S_{1}=107{,}58\,\mathrm {\euro} }$

${\displaystyle H_{1}\cdot (S_{1}-S_{0})=4{,}27\cdot (107{,}58\,\mathrm {\euro} -106{,}12\,\mathrm {\euro} )=6{,}23\,\mathrm {\euro} }$.

After this day, additional shares of the stock could be bought or sold so that you now own shares, for example if you buy an entire share. If the share price is noticeable by the following day , the total profit can be seen through within the two days ${\displaystyle H_{2}}$ ${\displaystyle H_{2}=5{,}27}$ ${\displaystyle S_{2}=105{,}96\,\mathrm {\euro} }$

${\displaystyle H_{1}\cdot (S_{1}-S_{0})+H_{2}\cdot (S_{2}-S_{1})=-2{,}30\,\mathrm {\euro} }$

determine. A negative profit, i.e. a loss, occurred in the period under review. In general, the total profit by time segment is the sum of the partial profits of each individual segment: ${\displaystyle G_{n}}$ ${\displaystyle n}$

${\displaystyle G_{n}=\sum _{k=1}^{n}H_{k}\cdot \Delta S_{k}=H_{1}\cdot (S_{1}-S_{0})+H_{2}\cdot (S_{2}-S_{1})+\ldots +H_{n}\cdot (S_{n}-S_{n-1})}$.

Here referred to the price change in th period and indicates how many shares of the stock, the shareholder has in this period. ${\displaystyle \Delta S_{k}=S_{k}-S_{k-1}}$ ${\displaystyle k}$ ${\displaystyle H_{k}}$

Analysis comes into play with these elementary considerations when not just a sequence of discrete trading times , but all times from a time interval are considered. At any point in time, a share has a price that is generally “constantly changing”. The shares held can also be viewed in an idealized manner as a continuously changing process, the so-called portfolio process . The total profit during the time interval can then no longer be determined by the simple summation of a finite number of partial profits shown above. Instead, it results from the integration of the portfolio process, weighted with the changes in the share price, as a formula ${\displaystyle t}$ ${\displaystyle [0,T]}$ ${\displaystyle t}$ ${\displaystyle S_{t}}$ ${\displaystyle H_{t}}$ ${\displaystyle G_{T}}$ ${\displaystyle [0,T]}$

${\displaystyle G_{T}=\int _{0}^{T}H_{t}\,\mathrm {d} S_{t}}$

written. A basic task of stochastic analysis is to mathematically define such stochastic integrals for integrands and integrators as general as possible and to investigate their properties. ${\displaystyle H_{t}}$ ${\displaystyle S_{t}}$

From a stock price model to stochastic differential equations

The considerations that lead to the above representation of the profit as a stochastic integral are so general that they can be carried out for any variable with time. In the specific example of stock trading, the question arises as to what a mathematical model for the price development over time could look like. Real price developments seem to move up and down "randomly". From the point of view of mathematics, it therefore makes sense to model each as a random variable , i.e. as a function that assigns the value to each result of an (abstract) random experiment . The share price then depends both on the point in time and on the random result : it is a stochastic process . ${\displaystyle S_{t}}$ ${\displaystyle S_{t}}$ ${\displaystyle \omega }$ ${\displaystyle S_{t}(\omega )}$ ${\displaystyle t}$ ${\displaystyle \omega }$

The Australian share index All Ordinaries showed pronounced exponential growth over the period from 1875 to 2013. In a logarithmic representation, it follows roughly a straight line.

In reality, stock prices appear random, but they are not completely random. Over longer periods "undisturbed" price movements often have as a basic trend an exponential growth in so rise, for example, from year to year by a few percent. This can usually be seen even more clearly in the case of stock indices where a number of individual prices are averaged, such as - in a particularly pronounced way - in the adjacent chart for the Australian All Ordinaries stock index . A completely undisturbed exponential growth of would mean that the change in a short period of time is proportional to and to : ${\displaystyle S_{t}}$ ${\displaystyle \Delta S_{t}=S_{t+\Delta t}-S_{t}}$ ${\displaystyle \Delta t}$ ${\displaystyle S_{t}}$ ${\displaystyle \Delta t}$

${\displaystyle \Delta S_{t}=\mu \cdot S_{t}\cdot \Delta t}$

at a growth rate . With a savings balance, this would correspond to exponential growth from compound interest . In the case of stocks, however, this growth law is apparently overlaid by a complicated random movement in reality. Statistics and probability theory suggest that in the case of random disturbances that are composed of many small individual changes, a normal distribution is the simplest model. It also shows that the variance of the disturbances is proportional to the observed period . The Wiener process has all of these desired properties and is therefore suitable as a model for the temporal development of the random component of the share price. Overall, these considerations lead to the following model equation for the change in a time interval : ${\displaystyle \mu }$ ${\displaystyle \Delta t}$ ${\displaystyle W_{t}}$ ${\displaystyle \Delta S_{t}}$ ${\displaystyle \Delta t}$

${\displaystyle \Delta S_{t}=\mu \cdot S_{t}\cdot \Delta t+\sigma \cdot S_{t}\cdot \Delta W_{t}}$.

Here describe the change of the Wiener process in the period and a proportionality constant that models how much the random component affects the change of course, the so-called volatility . ${\displaystyle \Delta W_{t}=W_{t+\Delta t}-W_{t}}$ ${\displaystyle \sigma }$

One possible path to solving the stochastic differential equation for stock price modeling

The further procedure of classical analysis would now be to let the time interval converge to zero and thus obtain an ordinary differential equation for the function sought. This is not possible here in this way, because as it turns out, neither the paths of the Wiener process nor those of the stock price process sought can be differentiated , so they cannot be investigated by classical differential calculus. Instead, you get a stochastic differential equation , mostly in the form ${\displaystyle \Delta t}$

${\displaystyle \mathrm {d} S_{t}=\mu \cdot S_{t}\cdot \mathrm {d} t+\sigma \cdot S_{t}\cdot \mathrm {d} W_{t}}$

is written. The differentials used are a pure part of notation and it is the task of stochastic analysis to first give such equations a mathematical meaning. This is achieved through the observation that, because of the fundamental theorem of analysis, relationships between a differentiable function and its derivative can also be written with the aid of its integral. Accordingly, a stochastic differential equation is only an intuitive notation of an integral equation with a stochastic integral. Stochastic analysis deals, among other things, with the questions under which conditions stochastic differential equations can be uniquely solved and which analytical and probabilistic properties the solutions have. For some simple types of stochastic differential equations, there are methods to calculate the solutions explicitly. For example, the equation for the stock price derived above has the so-called geometric Brownian motion as a solution . As is already the case with ordinary differential equations, the solutions of many stochastic differential equations can only be calculated numerically .

history

The beginnings: Mathematical models for Brownian motion

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Brownian motion of fat droplets in milk

As Brownian movement , the physical phenomenon is known that small particles suspended in a liquid or in a gas move "trembling" in an irregularly and randomly appearing Art. It is named after the botanist Robert Brown , who first observed and described it in 1827 during the microscopic examination of pollen grains in a drop of water. In the period that followed, experiments showed that the physical cause of the Brownian movement lies in the thermal movement of the liquid molecules . These constantly hit the much larger particles, causing them to move irregularly.

The French mathematician Louis Bachelier investigated an early application of the Brownian movement in financial mathematics, which at the time was not scientifically recognized. In 1900 he tried to model share prices on the Paris stock exchange with the help of random movements and thus derive price formulas for warrants . He anticipated numerous ideas from the famous Black-Scholes model that was only introduced 73 years later.

Albert Einstein , 1905

Brownian motion came back into the light of the scientific community, as Albert Einstein in 1905, so in his annus mirabilis , presented a mathematical model of the physical phenomenon. He assumed that Brownian movement, in modern language, is a stochastic process with continuous paths and independent, normally distributed increases, that is, fulfills fundamental and from a physical point of view meaningful conditions. However, he did not presuppose other, physically just as necessary conditions, especially not that a particle can only cover a finite distance in a fixed period of time, the so-called rectifiability of the paths. Since in 1905 the theoretical foundation of probability theory by Émile Borel and Henri Lebesgue had only just begun, Einstein could not prove that his model actually existed as a mathematical object .

A mathematical construction of Einstein's model was only possible in 1923 by the American mathematician Norbert Wiener . He used an approach to measure theory developed by Percy John Daniell in 1913 and the theory of Fourier series . When examining the properties of this model, which was also called the Wiener process in his honor , it was found that its paths are not rectifiable. In retrospect it turned out that Einstein had selected exactly the “right” properties in his model; with the additional assumption of rectifiability, it would not have existed mathematically at all.

In 1931, the Soviet mathematician Andrei Nikolajewitsch Kolmogorow found a way to study stochastic processes using analysis. He looked at generalizations of the Wiener process, the so-called Markov processes , and developed a theory to describe them. He showed that these processes can be broken down into a non-random drift term and a purely stochastic part. Kolmogorow thus established a connection between their probability distributions and certain partial differential equations , the Kolmogorow equations . He used methods of classical analysis; a generalization of integral and differential calculus to a “stochastic analysis” has not yet been found with him.

Itō's approach and further development

Kiyoshi Itō , 1970

At the beginning of stochastic analysis is a work by the Japanese mathematician Kiyoshi Itō (1915–2008) from 1944 with the simple title Stochastic Integral. Itō, who is considered to be the founder of the field, constructed a general stochastic differential equation for the investigation of Markov processes. He gave the occurring stochastic differentials meaning by constructing a new integral term for stochastic processes, the Itō integral . In his subsequent work he also made a connection to Kolmogorov's results by proving that the solutions of his stochastic differential equation satisfy the Kolmogorov equations. In 1951 he published one of the most fundamental results of stochastic analysis, the Itō formula . This generalizes the chain rule , the product rule , the substitution rule and the partial integration of classical analysis to stochastic differentials or integrals of so-called Itō processes .

Itō's work thus represented the starting point of a rapid development in the field, which continues today. As it turned out, however, in 2000, the Franco-German mathematician Wolfgang Döblin had anticipated many of Ito's ideas as early as 1940. Because Döblin, who fought for France in World War II, sent his work in a sealed envelope to the Paris Académie des Sciences , burned his notes and then committed suicide to avoid capture by the German Wehrmacht, nobody knew about it for 60 years Döblin's results.

Paul-André Meyer , 1969

In the period following Itō's work, the focus of mathematical research was on generalizations of his results. In 1953, Joseph L. Doob expanded in his influential book on stochastic processes Itō's integral from the Wiener process to processes with uncorrelated increases. Doob also dealt with the question of how his decomposition theorem for discrete processes can be generalized to the time-continuous case. This problem was solved by Paul-André Meyer in 1962 . Meyer showed that such a decomposition, the Doob-Meyer decomposition , is only possible under an additional condition, which he called "class (D)". In 1970, Catherine Doléans-Dade and Meyer generalized the Itō integral to so-called semimartingals as integrators, i.e. to stochastic processes that are composed of a local martingale and a process with locally finite variation . In a certain way, semimartingals are the most general class of stochastic processes for which a meaningful integral term can be defined.

From the point of view of the applications of stochastic analysis, the historically most significant result is the Black-Scholes model for the valuation of financial options published by Fischer Black , Myron S. Scholes and Robert C. Merton in 1973 after the preparatory work of the economist Paul A. Samuelson and the mathematician Henry McKean . Therein the price of the underlying stock is given by the stochastic differential equation derived in the introductory section of this article. The Itō formula, applied to the price of an option as a function of time and share price, then leads to a certain partial differential equation, the Black-Scholes equation, together with an economic argument by means of a hedge . This can be solved explicitly and thus results in formulas for the value of call and put options on the share. Scholes and Merton received the Alfred Nobel Memorial Prize for Economics (“ Nobel Prize in Economics”) in 1997 for their achievements ; Black had died two years earlier.

Further developments in stochastic analysis are, for example, stochastic differential geometry , which deals with stochastic processes on manifolds , and the Malliavin calculus (after Paul Malliavin ), a generalization of the calculus of variations on functional stochastic processes. Another current research topic is theory and applications of stochastic partial differential equations .

Basic terms, statements and methods

The Wiener process and its analytical properties

→ Main article : Wiener Trial

An example path of the Wiener process with an enlarged section; it is so “jagged” on all time scales that it cannot be differentiated at any point.

The Wiener process stands not only historically as a model for the Brownian movement at the beginning of stochastic analysis; Because of its numerous mathematically interesting properties, as the "basic type" of a random process, it is a central object of investigation in the field. The Wiener process can be defined as a stochastic process with independent , stationary and normally distributed increases, which almost certainly has continuous paths and is standardized to. The definition gives rise to numerous important properties that can naturally be generalized to more general classes of stochastic processes. The Wiener process is a typical representative of the Gauss processes , the Markov processes and the Lévy processes . ${\displaystyle (W_{t})_{t\geq 0}}$ ${\displaystyle W_{0}=0}$

The Wiener Trial is also a so-called martingale . Thus, given the current value , the increase has the conditional expected value zero. To put it simply and clearly: It behaves like the profit of a player who takes part in a fair game of chance, i.e. a game in which profits and losses are balanced out on average. The term martingale is a central concept of modern probability theory, because on the one hand stochastic models often lead to a martingale or can be transformed into one, on the other hand numerous theorems can be proven for this process class, such as Doob's maximum inequality , the optional sampling theorem or the martingale convergence theorems . ${\displaystyle W_{t+\Delta t}-W_{t}}$ ${\displaystyle W_{t}}$

The paths of the Wiener process are almost certainly not differentiable at any point , so they are clearly “jagged” on all time scales so that a tangent cannot be created anywhere . This means that they evade differential calculus in the classic sense. However, by definition they are continuous, so that they can appear as the integrand of a classical integral without any problems. The paths of the Wiener process are also of infinite variation , so the total sums of their changes over a finite time interval are unlimited. The consequence of this is that an integral with the Wiener process as an integrator cannot be interpreted path-wise as a classic Stieltjes integral ; a new “stochastic” integral term is therefore necessary for this. ${\displaystyle t}$

Stochastic integrals

→ Main article : Stochastic integration

In the introductory section of this article, it was shown how a stochastic integral can be understood as a limit value of sums at which a time step size converges to zero. This consideration can be made more precise mathematically and in this way leads to a possible definition of an integral with the Wiener process as integrator, the Itō integral . In modern textbooks, however, a somewhat more abstract construction is usually used, which is more similar to the usual procedure for defining the Lebesgue integral : First, an integral for elementary processes , i.e. for piecewise constant processes, is defined as an integrand. This concept of integral is then continued step- by- step to more general integrands with the help of tightness arguments . In this general form there are stochastic integrals

${\displaystyle \int _{0}^{T}H_{t}\,\mathrm {d} X_{t}}$

for semimartingals as integrators and adapted stochastic processes with càglàd paths possible. Both processes can therefore also have jump points . ${\displaystyle (X_{t})}$ ${\displaystyle (H_{t})}$

Analogous to the concept of the integral function in classical analysis, one also considers the integral as a function of the upper limit of integration in stochastic analysis and thus again receives a stochastic process

${\displaystyle G_{t}=\int _{0}^{t}H_{s}\,\mathrm {d} X_{s}}$

or in differential notation

${\displaystyle \mathrm {d} G_{t}=H_{t}\,\mathrm {d} X_{t}}$.

It can be shown that it is also a semi-martingale. If the integrator is even a martingale, for example the Wiener process, and the integrand satisfies certain restrictions, then it is also a martingale. In this case, the stochastic integral can be understood as a time-continuous martingale transformation. ${\displaystyle (G_{t})}$ ${\displaystyle (X_{t})}$ ${\displaystyle (G_{t})}$

The Itō formula

→ Main article : Lemma of Itō

For the Itō integral and its generalizations, some of the usual calculation rules of analysis only apply in a modified form. This is clearly due to the fact that, due to the infinite variation of the Wiener process, in the case of small changes in time, not only the associated changes must be taken into account, but also their squares , which themselves are of the order of magnitude . The resulting "new" calculation rules are summarized in the Itō formula. Even in its simplest form, it is clear how the chain rule for the Wiener process must be changed: If a twice continuously differentiable function, then for the process in differential notation ${\displaystyle (W_{t})}$ ${\displaystyle \Delta t}$ ${\displaystyle \Delta W_{t}}$ ${\displaystyle (\Delta W_{t})^{2}}$ ${\displaystyle \Delta t}$ ${\displaystyle h}$ ${\displaystyle X_{t}=h(W_{t})}$

${\displaystyle \mathrm {d} X_{t}=h'(W_{t})\,\mathrm {d} W_{t}+{\frac {1}{2}}h''(W_{t})\,\mathrm {d} t}$.

Compared to the classical chain rule, you get an additional term that contains the second derivative of . The Itō formula can be generalized to vector-valued semimartingales. ${\displaystyle h}$

The additional terms of this Itō calculus sometimes make concrete calculations time-consuming and confusing. Therefore, for some application tasks, another stochastic integral term is available, the Stratonowitsch integral (after Ruslan Stratonowitsch ). Its main advantage over the Itō integral is that the calculation rules essentially correspond to those of classical analysis. In particular, in physical problems, a formulation with the Stratonowitsch integral is therefore often more natural. However, it does not have some important mathematical properties of the Itō integral; in particular, the integral process is not a martingale. However, every statement for Stratonowitsch integrals can be formulated with Itō integrals and vice versa. Both terms are only two representations of the same state of affairs.

Stochastic differential equations

→ Main article : Stochastic differential equation

A general form of a stochastic differential equation is

${\displaystyle \mathrm {d} X_{t}=a(t,X_{t})\,\mathrm {d} t+b(t,X_{t})\,\mathrm {d} W_{t}}$

with the Wiener process and given functions and ; the stochastic process is wanted. As always, this differential notation is only an abbreviation for stochastic integrals: A process is a solution if it has the integral equation ${\displaystyle (W_{t})}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle (X_{t})}$ ${\displaystyle (X_{t})}$

${\displaystyle X_{t}=X_{0}+\int _{0}^{t}a(s,X_{s})\,\mathrm {d} s+\int _{0}^{t}b(s,X_{s})\,\mathrm {d} W_{s}}$

Fulfills. At the beginning of the theoretical investigation of these equations there is the question of the existence and uniqueness of the solutions. The conditions for this result in a similar way as in the classical analysis of ordinary differential equations. Analogous to Picard-Lindelöf's theorem , a stochastic differential equation has a uniquely determined solution that exists for all if the coefficient functions and Lipschitz are continuous and linearly bounded . ${\displaystyle (X_{t})}$ ${\displaystyle t\geq 0}$ ${\displaystyle a}$ ${\displaystyle b}$

For some simple types of stochastic differential equations the solution can be given explicitly. In particular, linear equations can be solved analogously to the classical methods - modified with regard to the Itō formula - by an exponential approach (with a stochastic exponential ) and variation of the constants . However, analytical and stochastic properties of the solution can often already be derived from the differential equation itself. A theoretically important general result is that solutions of stochastic differential equations are always Markov processes .

Exact solution (black) of a stochastic differential equation and Euler-Maruyama approximation (red)

Furthermore, there is the possibility of solving equations numerically and thus determining the quantities you are looking for using a Monte Carlo simulation . The simplest and practically most important numerical method for solving stochastic differential equations, the Euler-Maruyama method , is a direct generalization of the explicit Euler method . As in the derivation of the equation, short time steps are considered and the differential of the Wiener process is replaced by the increase , i.e. by a normally distributed random variable with an expected value of zero and variance . As in the numerics of ordinary differential equations, there are numerous further developments of the Euler-Maruyama method that have a higher order of convergence , i.e. provide more precise approximations of the solution for a given step size. A simple example is the Milstein method . In contrast to the case of ordinary differential equations, the importance of methods with a high order of convergence is rather small for most practical applications. On the one hand, this is due to the fact that such methods are numerically very complex and therefore computationally intensive. On the other hand, most applications require the rapid calculation of a large number of individual paths in a simulation; the accuracy with which a single path is calculated then does not play an essential role, because the end result is dominated by the error of the Monte Carlo simulation. ${\displaystyle \Delta t}$ ${\displaystyle \Delta W_{t}}$ ${\displaystyle \Delta t}$

Applications

Financial math

The application of probabilistic methods to problems in financial mathematics has led to a fruitful interaction between mathematics and economics in the last few decades. Models that describe the development of economic variables over time at discrete points in time, such as the well-known binomial model by Cox, Ross and Rubinstein , can be set up and examined using elementary probability calculations. For questions with a continuously varying time parameter, however, terms and sentences from stochastic analysis are required. The development of financial-mathematical variables such as share prices, prices of derivatives , exchange rates or interest rates is modeled through time-continuous stochastic processes , the changes of which are given by suitable stochastic differential equations.

Valuation of derivatives

From a mathematical description of a share price one might at first “naively” expect that its main task is to forecast the further price development. However, this is not the case when modeling using a stochastic differential equation , because, as we have seen, it is assumed that the course changes are random and “only” describes this randomness of all possible developments in terms of probability theory. A correctly asked and central question, however, is how the price of a derivative, for example a purchase option on a share, can be calculated in such a random model .

Price V of a European call option based on the Black-Scholes model as a function of the share price S and the remaining term T

A derivative is generally a financial instrument that results in a future payout. The amount of this payout depends on another economic variable, such as a share price. Since this changes randomly up to the time of payment, the payment of the derivative is also random. The key question is how the price of such a derivative can be determined. An obvious but wrong idea is to average all possible payouts with their respective probabilities, i.e. to form the expected value of the payout. Economic considerations show, for example, the seemingly paradoxical fact that the price of a call option on a share does not depend on the probability of the share rising or falling (for an elementary example, see also risk-neutral valuation # example ).

Correct approaches to valuing derivatives, on the other hand, are hedging and risk-neutral valuation . The basic idea of ​​hedging is this: if there is a self-financing trading strategy that does not use the derivative, but in all cases delivers the same payout as the derivative, then the price of the derivative must be the same as the price of the trading strategy. Risk-neutral valuation, on the other hand, is based on the principle of so-called freedom from arbitrage , to put it simply: the derivative has the right price if you cannot make a risk-free profit when trading the derivative and the other financial instruments . For example, if it were possible to always make a positive profit by buying the derivative, then its price would be too low. In reality, the price of the derivative would rise until a market equilibrium occurs.

A change in the probability measure, represented here by different color intensities, can, for example, transform a Wiener process with drift (left) into a martingale (right).

These economic considerations can all be formalized mathematically, for which purpose the methods of stochastic analysis are required in the case of continuous time. Under certain technical prerequisites for the market model, it can be proven that both the hedging approach and the determination of the arbitrage-free price are clearly feasible and deliver the same results. A fairly general mathematical approach represents the arbitrage-free price of a derivative at a certain point in time by means of a conditional expected value . As noted above, however, the real probability measure , which determines the development of the share price, cannot be used for this. Instead, a change to the risk-neutral probability measure is considered. Under this measure, also known as the equivalent martingale measure, the discounted share price is a martingale , which, in simple terms, behaves like the winnings of a player who takes part in a fair game of chance. The conversion formulas for the solutions of the stochastic differential equations are provided by Girsanow's theorem . The problem of determining the arbitrage-free price as a conditional expected value depending on time and the initial value can also be reduced to solving a (non-stochastic) partial differential equation with the aid of Feynman-Kac's theorem .

Interest rate models

→ Main article : Interest structure model

In all economic considerations that include prices at different points in time, the time value of money must be taken into account, the main causes of which are interest rates and inflation . In the simplest case, also in the Black-Scholes model, a fixed-income security with a risk-free, constant interest rate is considered. All future payments, such as with a derivative on a share, then only have to be divided by an exponential factor according to the constant interest rate in order to obtain their current value.

In reality, however, interest rates fluctuate similarly to stock prices, which suggests that they should also be modeled as stochastic processes in the sense of stochastic analysis. With the so-called short rate models, there are numerous approaches to describe the development of an instantaneous interest rate over time (“short rate”) using stochastic differential equations. The current interest rate modeled in this way can then be used to display the prices of interest-bearing securities such as zero-coupon bonds or floaters . As in the case of stocks, price formulas for interest rate derivatives such as caps , swaps and swaptions are obtained through risk-neutral valuation . ${\displaystyle r_{t}}$

Interest rates like the EURIBOR here often show random movements around a mean value, to which they keep returning.

Examples of simple models are the Ho Lee model , which assumes a scaled Wiener process, and the Dothan model , which uses a geometric Brownian motion for it. Such approaches lead to mathematically simple derivations, but for various reasons describe reality only poorly. For the Dothan model, for example, it can be shown mathematically that it would be possible to earn an infinite amount of money in a finite period of time. Real interest rates predominantly show a behavior corresponding to the so-called mean reversion effect , i.e., despite their seemingly random fluctuations, they always return to a mean value. Two important basic types of differential equation models that simulate this effect are the Vasicek model and the popular Cox-Ingersoll-Ross model . Another approach to modeling interest rates is used by the group of HJM models , which describe not the current interest rate but the forward rate (“forward rate”). ${\displaystyle r_{t}}$

Physics, chemistry and engineering

The movement of physical bodies is described by Newton's laws : A force that acts on a body causes its speed to change at constant mass . If the force depends in a given way on the location and the speed of the body, an equation results which relates the acceleration , the speed and the location of the body to one another. Since the acceleration is the derivative of the speed and the speed is the derivative of the location, this is an ordinary differential equation. If the force also has a random component, it becomes a stochastic differential equation.

The physical description of particles subject to Brownian motion can be greatly refined and generalized. For example, for particles on which a frictional force proportional to the speed ( Stokes law ) and possibly a constant force also act, the result is that the speed is a solution of the Ornstein-Uhlenbeck equation , a stochastic differential equation that can be explicitly solved . Physics usually uses a different representation for this type of equation, which goes back to Paul Langevin and is known as the Langevin equation . The differential equation is noted with a so-called white noise , which can be interpreted as a formal derivation of the nondifferentiable Wiener process. Such equations of motion can be generalized even further, for example to oscillations with random disturbances. Here the acceleration depends on the deflection, on the speed and additionally on a random normally distributed force, the standard deviation of which can also depend on the location and speed. Together with the fact that the velocity is the derivative of the location, a system of two stochastic differential equations results.

In general, when it comes to physical issues, one is not so much interested in the movement paths of individual particles as in the averaged behavior of very many particles, which is noticeable as diffusion . The temporal development of the associated density function fulfills a (non-stochastic) partial differential equation , the Fokker-Planck equation named after Adriaan Daniël Fokker and Max Planck . Further generalizations of diffusion result in applications in theoretical chemistry : If, in addition to the random physical movements of particles, one also takes into account the possibility of chemical reactions between them, non-linear reaction diffusion equations result instead of the linear Fokker-Planck equations . With their help, numerous interesting phenomena of pattern formation , such as oscillating reactions , chemical waves or morphogenesis , can be studied.

Many applications in engineering and technology can be traced back to the problem of stochastic filtering . A dynamic system , whose development over time is described by a differential equation, is observed. In general, it is not possible to observe all the variables in the system directly; In addition, both the system equation and the observation equation are subject to random disturbances: A system of stochastic differential equations results and the task is to infer the development of the system from the observations. In the time-discrete case, i.e. when there are only a finite number of observations at certain points in time, there is an important estimation method for this , the Kalman filter . In the continuous-time case, this can be generalized to the so-called Kalman-Bucy filter . The stochastic control theory goes one step further : Here, the differential equation that describes the system also depends on a freely selectable control function, possibly with the exception of secondary conditions. What is needed is an optimal control of the system. Central to the theory of such tasks is the Hamilton-Jacobi-Bellman equation , a partial differential equation that results from Bellman's principle of optimality through transition to continuous time. With the Itō formula, the Hamilton-Jacobi-Bellman equation can be transferred to stochastic differential equations.

biology

Simulation of a Galton-Watson process : after 50 time units, 26 of 50 populations have become extinct.

Numerous models in biology use the terms and methods of stochastic analysis. The so-called branching processes such as the Galton-Watson process are simple discrete stochastic models for the growth of a population : each individual gets a random number of offspring in a generation, independently of the others, and then dies. In 1951, William Feller presented the branching-diffusion model, a time-continuous version of a branching process in which the size of the population is given by a stochastic differential equation. Standard probabilistic methods can be used to derive formulas for the exponential growth of the expected value of the population and for the probability of extinction.

Another type of stochastic models for the development of populations are birth and death processes , in which the size of the population increases or decreases by one individual after random time intervals. In the simplest case, the time intervals are exponentially distributed with constant rates. The population size is now a discontinuous jump process that satisfies a stochastic integral equation with a local martingale as a random component. More complicated and more interesting than the case of constant, “unchecked” growth is what is known as slow growth, in which the growth rate converges towards zero for increasing population sizes. Cases of dying, linearly growing and faster than linearly growing populations can be distinguished here. This results in formulas for the probability of extinction and for the expected values ​​of various stopping  times - for example for the time until the population reaches a certain maximum size.

A time-continuous model for gene drift , i.e. for the temporal development of the frequencies of certain genes or alleles in a population, is the Fisher-Wright diffusion . In the simplest case, two types of individuals are considered in a population, the proportions of which fluctuate randomly and mutate into one another at given rates . The associated stochastic differential equation can be used to derive conditions as to whether a stationary distribution is established over time or whether a type will die out. In the second case there is an explicit formula for the expected time to extinction. Another model for gene drift is the Moran model . The basic idea here is that the population evolves as one randomly chosen individual dies and another reproduces. The transition to continuous time results in a stochastic integral equation again. Refinements of the model also take into account mutations and selection advantages .

There are numerous biological models that specifically describe the origin and development of cancer . Oncogene models, which apply the Moran model to mutated and non-mutated cells, are a simple approach . Generalizations of this are based on the Knudson hypothesis , according to which the cause of cancer development are independent multiple mutations. The two-hit model uses cells with zero, one or two mutations. Mathematically, the development of several cell types can be understood as a birth and death process in several dimensions. Another approach is the kinetic model by Garay and Lefever from 1978, which leads to an ordinary differential equation for the concentration of malignant cells in an organism. There are various approaches that take additional random fluctuations in concentration into account and thus lead to different stochastic differential equations in a known manner.

Periodic solutions of the classical Lotka-Volterra equations, represented in the prey-predator phase space

Biology also deals with models for the temporal development of several populations that influence each other. A well-known simple approach to this, the Lotka-Volterra model , looks at a “predator” and a “prey” population. The two populations satisfy a system of two ordinary differential equations in which the growth rates depend on the other population. A disadvantage of this simple modeling is that it does not represent an extinction of a population, because as can be proven mathematically, the solutions of the Lotka-Volterra equations are periodic and positive for all times. That changes if you also introduce suitable stochastic components. For the stochastic differential equation system obtained in this way, the expected time until the prey population becomes extinct can now be determined.

literature

General textbooks

• Samuel N. Cohen, Robert J. Elliott: Stochastic Calculus and Applications . 2nd Edition. Springer, New York a. a. 2015, ISBN 978-1-4939-2866-8 .
• Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 .
• Richard Durrett : Stochastic Calculus - A Practical Introduction . CRC Press, Boca Raton et al. a. 1996, ISBN 0-8493-8071-5 .
• Wolfgang Hackenbroch, Anton Thalmaier: Stochastic Analysis - An Introduction to the Theory of Continuous Semimartingales . Springer Fachmedien, Wiesbaden 1994, ISBN 978-3-519-02229-9 .
• Ioannis Karatzas, Steven E. Shreve: Brownian Motion and Stochastic Calculus . 2nd Edition. Springer, New York 1998, ISBN 0-387-97655-8 .
• Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 .
• Jean-François Le Gall : Brownian Motion, Martingales, and Stochastic Calculus . Springer, Berlin / Heidelberg 2013, ISBN 978-3-319-31088-6 .
• Philip E. Protter: Stochastic Integrals and Differential Equations . 2nd (Version 2.1) edition. Springer, Berlin et al. 2005, ISBN 3-540-00313-4 .

Textbooks with a focus on financial mathematics

• Steven Shreve: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model . 1st edition. Springer, New York 2004, ISBN 978-0-387-24968-1 .
• Steven Shreve: Stochastic Calculus for Finance II: Continuous Time-Models . 1st edition. Springer, New York 2004, ISBN 978-1-4419-2311-0 .
• Thomas Björk: Arbitrage Theory in Continuous Time . 3. Edition. Oxford University Press, New York 2009, ISBN 978-0-19-957474-2 .
• Albrecht Irle : Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 .
• Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 .
• Monique Jeanblanc, Marc Yor , Marc Chesney: Mathematical Methods for Financial Markets . Springer, Dordrecht et al. 2009, ISBN 978-1-85233-376-8 .
• Michael Hoffmann: Stochastic Integration: An Introduction to Financial Mathematics . 1st edition. Springer Spectrum, Wiesbaden 2016, ISBN 978-3-658-14131-8 .

Web links

• Egbert Dettweiler: Basics of Stochastic Analysis. (PDF) April 2006, archived from the original on September 23, 2016 ; accessed on September 30, 2016 (lecture notes from TU Dresden ). 
• Ramon van Handel: Stochastic Calculus, Filtering, and Stochastic Control. (PDF) May 29, 2007, accessed September 30, 2016 (English, Princeton University lecture notes ). 
• Christoph Kuhn: Lecture notes "Financial Mathematics in Constant Time". (PDF) May 8, 2016, accessed on November 10, 2016 (lecture notes from Johann Wolfgang Goethe University Frankfurt am Main ). 

Individual evidence

1. ^ Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , pp. 73 .
2. ^ Albrecht Irle : Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 , p. 181-182 .
3. ^ Albrecht Irle : Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 , p. 183 .
4. ^ ^{A b} Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , pp. 95 .
5. ^ John C. Hull: Options, Futures, and Other Derivatives . 7th edition. Pearson Studium, Munich 2009, ISBN 978-3-8273-7281-9 , pp. 332-334 .
6. ^ Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , pp. 101-102 .
7. ^ ^{A b} Robert Jarrow, Philip Protter: A short history of stochastic integration and mathematical finance the early years, 1880-1970 . In: IMS Lecture Notes Monograph . tape 45 , 2004, p. 1 ( PDF file ).
8. ^ Robert Jarrow, Philip Protter: A short history of stochastic integration and mathematical finance the early years, 1880-1970 . In: IMS Lecture Notes Monograph . tape 45 , 2004, p. 1–2 ( PDF file ).
9. ^ ^{A b} Robert Jarrow, Philip Protter: A short history of stochastic integration and mathematical finance the early years, 1880-1970 . In: IMS Lecture Notes Monograph . tape 45 , 2004, p. 2 ( PDF file ).
10. ^ Robert Jarrow, Philip Protter: A short history of stochastic integration and mathematical finance the early years, 1880-1970 . In: IMS Lecture Notes Monograph . tape 45 , 2004, p. 3–5 ( PDF file ).
11. ^ Bernard Bru, Marc Yor: Comments on the life and mathematical legacy of Wolfgang Doeblin . In: Finance and Stochastics . tape 6 , no. 1 , 2002, p. 3–47 ( PDF file ).
12. ^ Robert Jarrow, Philip Protter: A short history of stochastic integration and mathematical finance the early years, 1880-1970 . In: IMS Lecture Notes Monograph . tape 45 , 2004, p. 5–6 ( PDF file ).
13. ^ Robert Jarrow, Philip Protter: A short history of stochastic integration and mathematical finance the early years, 1880-1970 . In: IMS Lecture Notes Monograph . tape 45 , 2004, p. 12 ( PDF file ).
14. ^ Paul-André Meyer : Stochastic Processes from 1950 to the Present . In: Journ @ l Electronique d'Histoire des Probabilités et de la Statistique . tape 5 , no. 1 , 2009, p. 15 ( PDF file ).
15. ^ Ajay Shah: Black, Merton and Scholes: Their work and its consequences . In: Economic and Political Weekly . tape 32 , no. 52 , 1997, pp. 3337-3342 ( PDF file ).
16. ^ Paul-André Meyer : Stochastic Processes from 1950 to the Present . In: Journ @ l Electronique d'Histoire des Probabilités et de la Statistique . tape 5 , no. 1 , 2009, p. 22-25 ( PDF file ).
17. ↑ Pao-Liu Chow: Stochastic Partial Differential Equations . 2nd Edition. CRC Press, Boca Raton et al. 2015, ISBN 978-1-4665-7955-2 .
18. ↑ Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 , pp. 22 .
19. ↑ Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 , pp. 119-148 .
20. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-642-36017-6 , pp. 477, 507 f .
21. ^ Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , pp. 103 .
22. ↑ Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 , chap. 4 , p. 63-81 .
23. ↑ Philip E. Protter: Stochastic Integrals and Differential Equations . 2nd (Version 2.1) edition. Springer, Berlin et al. 2005, ISBN 3-540-00313-4 , pp. 56-59 .
24. ↑ Philip E. Protter: Stochastic Integrals and Differential Equations . 2nd (Version 2.1) edition. Springer, Berlin et al. 2005, ISBN 3-540-00313-4 , pp. 62 .
25. ^ Hui-Hsiung Kuo: Introduction to Stochastic Integration . Springer, New York 2006, ISBN 0-387-28720-5 , pp. 52-57 .
26. ↑ Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 , chap. 5 , p. 83-102 .
27. ↑ Philip E. Protter: Stochastic Integrals and Differential Equations . 2nd (Version 2.1) edition. Springer, Berlin et al. 2005, ISBN 3-540-00313-4 , pp. 78-82 .
28. ^ Samuel N. Cohen, Robert J. Elliott: Stochastic Calculus and Applications . 2nd Edition. Springer, New York a. a. 2015, ISBN 978-1-4939-2866-8 , pp. 361-363 .
29. ↑ Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 , pp. 109-113 .
30. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 131-134 .
31. ^ Samuel N. Cohen, Robert J. Elliott: Stochastic Calculus and Applications . 2nd Edition. Springer, New York a. a. 2015, ISBN 978-1-4939-2866-8 , pp. 427-450 .
32. ^ Søren Asmussen, Peter W. Glynn: Stochastic Simulation: Algorithms and Analysis . Springer, New York 2007, ISBN 978-0-387-30679-7 , pp. 274-305 .
33. ^ Carl Graham, Denis Talay: Stochastic Simulation and Monte Carlo Methods . Springer, Berlin / Heidelberg 2013, ISBN 978-3-642-39362-4 , pp. 155-195 .
34. ^ Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , Chapter 3. The discrete multi-period model. , S. 73-92 .
35. ^ Albrecht Irle: Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 , p. 9-38 .
36. ^ Albrecht Irle: Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 , p. 222-232 .
37. ^ J. Michael Steele: Stochastic Calculus and Financial Applications . Springer, New York 2001, ISBN 0-387-95016-8 , pp. 263-275 .
38. ^ Stefan Reitz: Mathematics in the modern financial world: derivatives, portfolio models and rating procedures . Vieweg + Teubner, Wiesbaden 2011, ISBN 978-3-8348-0943-8 , pp. 113 .
39. ↑ Damiano Brigo, Fabio Mercurio: Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit . 2nd Edition. Springer, Berlin / Heidelberg 2006, ISBN 978-3-540-22149-4 , pp. 1-22 .
40. ↑ Damiano Brigo, Fabio Mercurio: Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit . 2nd Edition. Springer, Berlin / Heidelberg 2006, ISBN 978-3-540-22149-4 , pp. 62-64 .
41. ^ Albrecht Irle: Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 , p. 291-294 .
42. ^ Albrecht Irle: Finanzmathematik: The valuation of derivatives . 3. Edition. Springer Spectrum, Wiesbaden 2012, ISBN 978-3-8348-1574-3 , p. 295-297 .
43. ↑ Damiano Brigo, Fabio Mercurio: Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit . 2nd Edition. Springer, Berlin / Heidelberg 2006, ISBN 978-3-540-22149-4 , pp. 183-194 .
44. ↑ Reinhard Mahnke, Jevgenijs Kaupuzs, Ihor Lubashevsky: Physics of Stochastic Processes: How Randomness Acts in Time . 2nd Edition. Wiley VCA, Weinheim 2009, ISBN 978-3-527-40840-5 , pp. 253-266 .
45. ↑ Thomas Deck: The Itô Calculus: Introduction and Applications . Springer, Berlin et al. 2006, ISBN 3-540-25392-0 , pp. 43 .
46. ↑ Reinhard Mahnke, Jevgenijs Kaupuzs, Ihor Lubashevsky: Physics of Stochastic Processes: How Randomness Acts in Time . 2nd Edition. Wiley VCA, Weinheim 2009, ISBN 978-3-527-40840-5 , pp. 145-157 .
47. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 402-409 .
48. ↑ Reinhard Mahnke, Jevgenijs Kaupuzs, Ihor Lubashevsky: Physics of Stochastic Processes: How Randomness Acts in Time . 2nd Edition. Wiley VCA, Weinheim 2009, ISBN 978-3-527-40840-5 , pp. 117-143 .
49. ^ Kurt Jacobs: Stochastic Processes for Physicists: Understanding Noisy Systems . Cambridge University Press, Cambridge et al. 2010, ISBN 978-0-521-76542-8 , pp. 116-126 .
50. ↑ Alan Bain, Dan Crisan: Fundamentals of Stochastic Filtering . Springer, New York 2009, ISBN 978-0-387-76895-3 .
51. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 395-401 .
52. ↑ Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications . 6th edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-540-04758-2 , pp. 85-114 .
53. ↑ Bernt Øksendal: Stochastic Differential Equations: An Introduction with Applications . 6th edition. Springer, Berlin / Heidelberg 2013, ISBN 978-3-540-04758-2 , pp. 243-268 .
54. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 353-357 .
55. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 359-369 .
56. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 357-358 .
57. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 369-375 .
58. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 377-379 .
59. ↑ Fima C. Klebaner: Introduction to Stochastic Calculus with Applications . 3. Edition. Imperial College Press, London 2012, ISBN 978-1-84816-831-2 , pp. 386-393 .

This article was added to the list of excellent articles on November 16, 2016 in this version .