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Figure 11 The time sheet object encapsulates the properties within the object and exposes them through methods This depiction indicates that an object s property values should not be directly accessed by something outside the object For example, the time 01fig01eps 62765779 pickup 093098 period should not be adjusted except through the get time period method, and the employee name would not be retrieved except through the get employee name method This concept of hiding the internal data is a part of what is called encapsulation and is a key premise of object-oriented programming (See the What Are Classes section for a coding example demonstrating this feature of encapsulation) Every object is of a speci c type For example, the object de ned as Sam Smith the carpenter at 3322 Main Street is an employee object type, and the May 15th time sheet for Jessica Jones is a time sheet object type All the objects of a particular type have the same set of properties and methods From a programming perspective, you can de ne your own object types using classes, as described in the What Are Classes section Or you can use the object types provided in the NET Framework In either

Another useful tool for making calculations with (8.53) is this: Consider W(t)Vdt W(tf).Jdt'. Due to the differentials, the variance and all higher moments of W(t)VdtW(tf).Jdt' will be ofhigherorderthan dt and dtf. In other words, any fluctuation in W(t)VdtW(tf).Jdt' will be negligible. Thus, W(t)VdtW(tf).Jdt' can be regarded as a sure quantity (deterministic) equal to its mean. As (W(t)W(tf ) vanishes for t =f tf and is unity for t = tf, we have

(8.56)

1

Now let us show that Ito's calculus has a different rule for chain differentiation (change of variables) than ordinary calculus. Let f be an arbitrary function of X(t): What is the expression for

df(X(t)) = f(X(t + dt - f(X(t))

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case, you rst create an object of the desired type, and then you can set or get the object s properties and call its methods The following code creates an object using the NET Framework Timer object type It then sets the object s Interval property to de ne how often the timer goes off and executes its Start method to start the timer

public int compareTo(Object o) { return compareTo((String) o); }

(8.57)

according to Ito's calculus Assuming that f is continuous and reasonably smooth,14 we can expand f(X (t + dt in a Taylor series and neglect all terms of order higher than dt:

Dim myTimer As New Timer myTimerInterval = 1000 myTimerStart()

df(X(t))

The New keyword in the rst line of code creates a new Timer object and assigns a reference to that new object to the myTimer object variable To access a property or method of the new Timer object, use the object variable and a period () and then the name of the property or method

dX(t)

X X(t)

A similar idiom is sometimes used for equals methods:

NOTE: When you type the object variable name and a period in Visual Studio, you see a list of the properties, methods, and events that are appropriate for that object This demonstrates the List Members feature of Intellisense Intellisense provides auto-completion and display of class documentation within Visual Studio as you are coding Other Intellisense features are demonstrated in later chapters

{dX(t)}2

X(t)

Everything in Visual Basic can be accessed as an object: forms, strings, even integers Check out this code:

A (X(t), t)

fxIX(t)

To provide consistency in your user interface, you often want a set of controls to have the same behavior For example, suppose your application needs to change the background color of the control that has focus To implement this requirement, you need to de ne Enter and Leave event handlers for each control on the form A set of controls can share an event handler by simply adding another control and event name to the Handles clause like this:

+ VD(X(t),t)

public boolean equals(Object o) { return o instanceof String && equals((String) o); }

+ ~D (X(t), t)

Private Sub ProcessEnter(ByVal sender As Object, _ ByVal e As SystemEventArgs) _ Handles NameTextBoxEnter, DescriptionTextBoxEnter DirectCast(sender, Control)BackColor = ColorBurlyWood End Sub

W(t)Vdt,

X(t)

Instead of typing each of these, you can have Visual Studio automatically add the Handles clause for the other controls by using the Events view of the Properties window (see Figure 45)

IX(t)}

(8.58)

To associate an event handler with additional controls and events from the Forms Designer:

where we have used (8.53) in dX(t) == X(t + dt) - X(t) and kept terms only up to the order of dt. Equation (8.58) is known as Ito's formula or Ito's rule. It differs from the usual rule for chain differentiation in ordinary calculus by the presence of the second term inside the braces on the second line of (8.58). Like the chain rule in ordinary calculus, this is a very important tool in Ito calculus. Next, we use it to derive a differential equation for the probability distribution associated with X (t). If we take the ensemble average of (8.58), the term on the last line vanishes, so that dividing by dt, we find that

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