Stochastic Differential Equations

The previous article on Brownian motion and the Wiener Process introduced the *standard Brownian motion*, as a means of modeling asset price paths. However, a standard Brownian motion has a non-zero probability of being negative. This is clearly not a property shared by real-world assets - stock prices cannot be less than zero. Hence, although the stochastic nature of a Brownian motion for our model should be retained, it is necessary to adjust exactly how that randomness is distributed. In particular, the concept of *geometric Brownian motion* (GBM) will now be introduced, which will solve the problem of negative stock prices.

However, before the geometric Brownian motion is considered, it is necessary to discuss the concept of a *Stochastic Differential Equation* (SDE). This will allow us to formulate the GBM and solve it to obtain a function for the asset price path.

Now that we have defined Brownian motion, we can utilise it as a building block to start constructing *stochastic differential equations* (SDE). We need SDE in order to discuss how functions $f = f(S)$ and their derivatives with respect to $S$ behave, where $S$ is a stock price determined by a Brownian motion.

Some of the rules of ordinary calculus do not work as expected in a stochastic world. We need to modify them to take into account both the random behaviour of Brownian motion as well as its non-differentiable nature. We will begin by discussing *stochastic integrals*, which will lead us naturally to the concept of an SDE.

Definition (Stochastic Integral)

Astochastic integralof the function $f=f(t)$ is a function $W=W(t)$, $t\in[0,T]$ given by:

\begin{eqnarray*} W(t) = \int^t_0 f(s) dB(s) = \lim_{n\rightarrow \infty} \sum_{k=1}^N f(t_{k-1})\left(B(t_k)-B(t_{k-1})\right) \end{eqnarray*}where $t_k = \frac{kt}{N}$.

Note that the function $f$ is *non-anticipatory*, in the sense that it is evaluated within the summation sign at time $t_{k-1}$. This means that it has no information as to what the random variable at $X(t_k)$ is. Supposing that $f$ represented some portfolio allocation based on $B$, then if it were not evaluated at $t_{k-1}$, but rather at $t_k$, we would be able to anticipate the future and modify the portfolio accordingly.

The previous expression provided for $W(t)$ is an integral expression and thus is well-defined for a non-differentiable variable, $B(t)$, due the property of finiteness as well as the chosen mean and variance. However, we wish to be able to write it in differential form:

\begin{eqnarray*} dW = f(t)dB \end{eqnarray*}It should be stressed that this is shorthand notation for the integral form. Indeed, to divide through by $dB$ would necessitate the definition of $\frac{dW}{dB}$ - a differential operator on a non-smooth function $W$.

One can consider the term $dB$ as being a normally distributed random variable with zero mean and variance $dt$. The formal definition is provided:

Definition (Stochastic Differential Equation)

Let $B(t)$ be a Brownian motion. If $W(t)$ is a sequence of random variables, such that for all $t$, \begin{eqnarray*} W(t+\delta t)-W(t)-\delta t \mu (t, W(t)) - \sigma(t, B(t)) (B(t+\delta t)-B(t)) \end{eqnarray*} is a random variable with mean and variance that are $o(\delta t)$, then: \begin{eqnarray*} d W = \mu(t, W(t)) dt + \sigma(t, W(t)) dB \end{eqnarray*} is astochastic differential equationfor $W(t)$.

A sequence of random variables given by the above is termed an *Ito drift-diffusion process*, or simply an *Ito process* or a *stochastic process*.

It can be seen that $\mu$ and $\sigma$ are both functions of $t$ and $W$. $\mu$ has the interpretation of a non-stochastic *drift* coefficient, while $\sigma$ represents the coefficient of *volatility* - it is multiplied by the stochastic $dB$ term. Hence, stochastic differential equations have both a non-stochastic and stochastic component.

In the following section on geometric Brownian motion, a stochastic differential equation will be utilised to model asset price movements.

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