6.3 Stochastic Differential Equations

Since the Wiener process fluctuates around its expectation 0 it can be approximated by means of symmetric random walks. As for random walks we are interested in stochastic processes in continuous time which are growing on average, i.e. which have a trend or drift. Proceeding from a Wiener process with arbitrary $ \sigma$ (see Section 5.1) we obtain the generalized Wiener process $ \{ X_t; \, \, t \ge 0 \}$ with drift rate $ \mu$ and variance $ \sigma^2\ :$

$\displaystyle X_t = \mu \cdot t + \sigma \cdot W_t \quad , \quad t \ge 0 \, .$ (6.3)

The general Wiener process $ X_t$ is at time $ t,$ N$ (\mu t, \,
\sigma^2 t)$-distributed. For its increment in a small time interval $ \Delta t$ we obtain

$\displaystyle X_{t+\Delta t} - X_t = \mu \cdot \Delta t + \sigma (W_{t + \Delta
t} - W_t ) \, .$

For $ \Delta t \rightarrow 0$ use the differential notation:

$\displaystyle dX_t = \mu \cdot dt + \sigma \cdot d W_t$ (6.4)

This is only a different expression for the relationship (5.3) which we can also write in integral form:

$\displaystyle X_t = \int^t_0 \mu \, ds + \int^t_0 \sigma dW_s$ (6.5)

Note, that from the definition of the stochastic integral it follows directly that $ \int^t_0 d\ W_s = W_t - W_0 = W_t \, .$

The differential notation (5.4) proceeds from the assumption that both the local drift rate given by $ \mu$ and the local variance given by $ \sigma^2$ are constant. A considerably larger class of stochastic processes which is more suited to model numerous economic and natural processes is obtained if $ \mu$ and $ \sigma^2$ in (5.4) are allowed to be time and state dependent. Such processes $ \{ X_t; \, \, t \ge 0 \} \, ,$ which we call Itô-processes, are defined as solutions of stochastic differential equations:

$\displaystyle dX_t = \mu (X_t, t) dt + \sigma (X_t, t) dW_t$ (6.6)

Intuitively, this means:

$\displaystyle X_{t + \Delta t} - X_t = \mu (X_t, t) \Delta t + \sigma (X_t, t) (W
_{t+\Delta t} - W_t), $

i.e. the process' increment in a small interval of length $ \Delta t$ after time $ t$ is $ \mu (X_t, t)
\cdot \Delta t$ plus a random fluctuation which is N$ (0,\
\sigma^2 (X_t, t) \cdot \Delta t)$ distributed. A precise definition of a solution of (5.6) is a stochastic process fulfilling the integral equation

$\displaystyle X_t - X_0 = \int^t_0 \mu (X_s, s) ds + \int^t_0 \sigma (X_s, s) dW _s$ (6.7)

In this sense (5.6) is only an abbreviation of (5.7). For $ 0 \le t' < t\ ,$ it follows immediately:

$\displaystyle X_t = X_{t'} + \int^t_{t'} \mu (X_s, s) ds + \int^t_{t'} \sigma (X_s, s)
dW_s \, .$

Since the increment of the Wiener process between $ t'$ and $ t$ does not dependent on the events which occurred up to time $ t'$, it follows that an Itô-process is Markovian.

Discrete approximations of (5.6) and (5.7) which can be used to simulate Itô-processes are obtained by observing the process between 0 and $ t$ only at evenly spaced points in time $ k \Delta t, \, k = 0, \ldots, n\ , \, \, \, n
\Delta t = t\ .$

With $ X_k = X_{k\Delta t}$ and $ Z_k = (W_{k\Delta t} -
W_{(k-1)\Delta t} ) / \sqrt{\Delta t}$ we get

$\displaystyle X_{k+1} - X_k = \mu (X_k, k) \cdot \Delta t + \sigma (X_k, k) \cdot
Z_{k+1} \cdot \sqrt{\Delta t} $

or rather with the abbreviations $ \mu _k (X) = \mu (X,k) \Delta t\ , \sigma _k (X) = \sigma (X,k)
\sqrt{\Delta t}\ :$

$\displaystyle X_n - X_0 = \sum^n_{k=1} \mu_{k-1} (X_{k-1}) + \sum^n_{k=1} \sigma
_{k-1} (X_{k-1} ) \cdot Z_k $

with identical independently distributed N$ (0,1)$-random variables $ Z_1, Z_2, \ldots \, . $
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