Brownian motion in continuous time

The main component of the asset price in this context is a Brownian motion in continuous time. This can be intuitively thought of as the limit of the random walks

$$\Delta _{\delta }B\left( t\right) =B\left( t+\delta \right) -B\le... ...ght) =\sqrt{\delta }\varepsilon \left( t+\delta \right) +o\left( \delta \right)$$   ,

where the innovations $\varepsilon \left( t\right)$ satisfy the usual conditions, and the limit is taken as $\delta \rightarrow 0$. For example one can visualize how the convergence is achieved when the number of steps go from $40$ to $2000$ in figure 7.1. The mean is taken to be equal to one, and the volatility equal to 2.

Figure 7.1: Brownian and Geometric Brownian Motion paths. One can also observe the expected value and the $\pm 40 \%$ intervals. Samples of 2000 and 40 points are generated to show the convergence from discrete to continuous time.

Now the process $B\left( t\right)$, thought of as a function of time has two very important properties:

• $B\left( t\right)$ is almost surely continuous for all $t$, and

• $B\left( t\right)$ is almost surely nowhere differentiable for all $%% t$.

The Brownian motion $B\left( t\right)$ is generating a filtration $%% \left\{ \mathscr{F}\left( t\right) \right\} _{t\geq 0}$ in a similar way that a discrete time random walk generated the filtration $\left\{ \mathscr{F}_{t}\right\} _{t=0}^{n}$. It includes the information that one obtains by observing the Brownian motion up to time $t$. It is easy to show that $B\left( t\right)$ forms a martingale, $\mathbf{E}\left[ B\left( t+s\right) \vert\mathscr{F}\left( t\right) \right] =B\left( t\right)$. In addition,

$$B\left( t\right) -B\left( s\right) \sim \mathcal{N}\left( 0,\left\vert t-s\right\vert \right)$$   .

Using the Brownian motion one can define other simple processes, called stochastic differential equations [SDEs], of the form

$$dX\left( t\right) =\alpha dt+\beta dB\left( t\right)$$   .

The parameter $\alpha$ is the drift while the parameter $\beta$ is the volatility of the process.

If one wants to solve the above process assuming $X\left( 0\right) =0$, and to find the value of $X\left( t\right)$, elementary calculus would suggest a solution of the form

$$X\left( t\right) =\int_{0}^{t}\alpha ds+\int_{0}^{t}\beta dB\left( s\right)$$   .

At this point the problems start to begin: since the function $B\left( t\right)$ is discontinuous everywhere, one cannot approach the second integral of the above expression in the Riemann sense^7.1. Such integrals called Itô integrals; their general form is

$$I\left( t\right) =\int_{0}^{t}\eta \left( s\right) dB\left( s\right)$$   ,

where $\eta \left( t\right)$ is $\mathscr{F}\left( t\right)$-measurable [or adapted] and square integrable $\int_{0}^{t}\eta ^{2}\left( s\right) ds<\infty$ [just a technical condition]. Itô integrals have the following properties:

Adapted-ness

$I\left( t\right)$ is $\mathscr{F}\left( t\right)$ adapted;

Linearity

For every $\alpha _{1},\alpha _{2}\in \mathbbm{R}$, and $%% \eta _{1}\left( t\right) ,\eta _{2}\left( t\right)$ $\mathscr{F}\left( t\right)$-measurable functions $\int_{0}^{t}\left[ \alpha _{1}\eta _{1}\left( s\right) +a_{2}\eta _{2}\left( s... ...t( s\right) +\alpha _{2}\int_{0}^{t}\eta _{2}\left( s\right) dB\left( s\right)$;

Martingale

$I\left( t\right)$ is a martingale [w.r.t. its filtration $\mathscr{F}\left( t\right)$]

Continuity

$I\left( t\right)$ is a continuous function of the upper limit of the integration, $t$.

Itô isometry

$\mathbf{E}I^{2}\left( t\right) =\mathbf{E}%% \int_{0}^{t}\eta ^{2}\left( s\right) ds$ [hence the technical condition].

Example 41   The process $X\left( t\right)$ above will have

$$\mathbf{E}X\left( t\right) = \mathbf{E}\left[ \int_{0}^{t}\alpha ds+\int_{0}^{t}\beta dB\left( s\right) \right]$$

$$= \alpha t$$

$$\mathbf{E}\left[ X\left( t\right) -\alpha t\right] ^{2} = \mathbf{E}%% \int_{0}^{t}\beta ^{2}ds$$

$$= \beta ^{2}t$$

We have already said that if $t\geq s$ we have $\mathbf{E}\left[ B\left( t\right) -B\left( s\right) \right] ^{2}=t-s$ [which is just the variance]. We can also show that $\mathbf{Var}\left[ B\left( t\right) -B\left( s\right) \right] ^{2}=2\left[ t-s\right] ^{2}$. This can be rewritten as

$$\mathbf{E}\left[ \Delta _{\delta }B\left( t\right) \right] ^{2} = \delta$$

$$\mathbf{Var}\left[ \Delta _{\delta }B\left( t\right) \right] ^{2} = o\left( \delta \right)$$

This can also be expressed as $\left[ \Delta _{\delta }B\left( t\right) %% \right] ^{2}=\delta +o\left( \delta \right)$. Now if we take the limit $%% s\rightarrow t$ [or $\delta \rightarrow 0$] we will have

$$\mathbf{E}\left[ dB\left( t\right) \right] ^{2} = dt$$

$$\mathbf{Var}\left[ dB\left( t\right) \right] ^{2} =$$ 0.

Therefore although $dB\left( t\right)$ represents an instantaneous random movement, $\left[ dB\left( t\right) \right] ^{2}$ turns out to be deterministic, and in fact equal to $dt$. We write this relationship informally as

$$dB\left( t\right) \times dB\left( t\right) =dt$$.