alt="d xi left-parenthesis t right-parenthesis equals mu left-parenthesis t comma xi left-parenthesis t right-parenthesis right-parenthesis italic d t plus sigma left-parenthesis t comma xi left-parenthesis t right-parenthesis right-parenthesis italic d upper W left-parenthesis t right-parenthesis"/>
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:
is a continuous process (with probability 1) provided
Using this fact, we shall see that the solution of Equation (3.17) is bounded and continuous with probability 1.
We now discuss existence and uniqueness theorems. First, we define some conditions on the coefficients in Equation (3.17):
C1: such that .
C2: , such that
C3: and are defined and measurable with respect to their variables where .
C4: and are continuous with respect to their variables for .
Condition C2 is called a Lipschitz condition in the second variable, while condition C1 constrains the growth of the coefficients in Equation (3.17). We assume throughout that the random variable
Theorem 3.2 Assume conditions C1, C2 and C3 hold. Then Equation (3.17) has a unique continuous solution with probability 1 for any initial condition
Theorem 3.3 Assume that conditions C1 and C4 hold. Then the Equation (3.17) has a continuous solution with probability 1 for any initial condition
We note the difference between the two theorems: the condition C2 is what makes the solution unique. Finally, both theorems assume that
We now define another condition on the diffusion coefficient in Equation (3.17).
C5:
Theorem 3.4 Assume conditions C4, C1 and C5 hold. Then the Equation (3.17) has a continuous solution with probability 1 for any initial condition
For proofs of these theorems, see Skorokhod (1982), for example.
In some cases it is possible to find a closed-form solution of Equation (3.17) (or equivalently, Equation (3.18)). When the drift and diffusion coefficients are constant, we see that the exact solution is given by the formula:
(3.19)
Knowing the exact solution is useful, because we can test the accuracy of finite difference schemes against it, and this gives us some insights into how well these schemes work for a range of parameter values.
It is useful to know how the solution of Equation (3.17)
