Numerical Methods in Computational Finance - Daniel J. Duffy

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      System (3.14) is sometimes called the Lotka–Volterra equations, which are an example of a more general Kolmogorov model to model the dynamics of ecological systems with predator-prey interactions, competition, disease and mutualism (Lotka (1956)).

      3.3.4 Logistic Function

      (3.15)f left-parenthesis x right-parenthesis equals StartFraction upper L Over 1 plus e Superscript minus k left-parenthesis x minus x 0 right-parenthesis Baseline EndFraction comma

      where

x 0 equals x-value of sigmoid's midpoint
upper L equals curve's maximum value
k equals steepness of the curve.

      A special case is when k equals 1 comma x 0 equals 0 comma upper L equals 1, resulting in the standard logistic function defined by the equation:

f left-parenthesis x right-parenthesis equals StartFraction 1 Over 1 plus e Superscript negative x Baseline EndFraction period

      We can verify from this equation that the logistic function satisfies the non-linear initial value problem:

      (3.16)StartLayout 1st Row 1st Column StartFraction italic d f Over italic d x EndFraction equals f left-parenthesis 1 minus f right-parenthesis comma 2nd Column f left-parenthesis 0 right-parenthesis equals one half period EndLayout

      The logistic function models processes in a range of fields such as artificial neural networks (learning algorithms, where it is called an activation function), economics, probability and statistics, to name a few.

      A random process is a family of random variables defined on some probability space and indexed by the parameter t where t belongs to some index set. A random process is a function of two variables:

StartSet xi left-parenthesis t comma x right-parenthesis colon t element-of upper T comma x element-of upper S EndSet

      where T is the index set and S is the sample space. For a fixed value of t, the random process becomes a random variable, while for a fixed sample point x in S the random process is a real-valued function of t called a sample function or a realisation of the process. It is also sometimes called a path.

      The index set T is called the parameter set, and the values assumed by xi left-parenthesis t comma omega right-parenthesis are called the states; finally, the set of all possible values is called the state space of the random process.

      The index set T can be discrete or continuous; if T is discrete, then the process is called a discrete-parameter or discrete-time process (also known as a random sequence). If T is continuous, then we say that the random process is called continuous-parameter or continuous-time. We can also consider the situation where the state is discrete or continuous. We then say that the random process is called discrete-state (chain) or continuous-state, respectively.

      3.4.1 Stochastic Differential Equations (SDEs)

      We introduce the scalar random processes described by SDEs of the form:

      where:

        random process

        transition (drift) coefficient

        diffusion coefficient

        Brownian process

        given initial condition

      defined in the interval [0, T]. We assume for the moment that the process takes values on the real line. We know that this SDE can be written in the equivalent integral form:

      This is a non-linear equation, because the unknown random process appears on both sides of the equation and it cannot be expressed in a closed form. We know that the second integral:

integral Subscript 0 Superscript t Baseline sigma left-parenthesis s comma xi left-parenthesis s right-parenthesis right-parenthesis italic d upper W left-parenthesis s right-parenthesis

      is a continuous process (with probability 1) provided sigma left-parenthesis s comma xi left-parenthesis s right-parenthesis right-parenthesis is a bounded process. In particular, we restrict the scope to those functions for which:

sup Underscript StartAbsoluteValue x EndAbsoluteValue less-than-or-equal-to upper C Endscripts left-parenthesis StartAbsoluteValue mu left-parenthesis s comma x right-parenthesis EndAbsoluteValue plus StartAbsoluteValue sigma left-parenthesis s comma x right-parenthesis EndAbsoluteValue right-parenthesis less-than infinity comma t element-of left-parenthesis 0 comma upper T right-bracket and for every upper C greater-than 0 period

      We now discuss existence and uniqueness theorems. First, we define some conditions on the coefficients in Equation

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