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**Superclasses: **

Brownian motion models

The `bm`

constructor
creates and displays Brownian motion (sometimes called *arithmetic
Brownian motion* or *generalized Wiener process*) `bm`

objects
that derive from the `sdeld`

(SDE
with drift rate expressed in linear form) class. Use `bm`

objects
to simulate sample paths of `NVARS`

state variables
driven by `NBROWNS`

sources of risk over `NPERIODS`

consecutive
observation periods, approximating continuous-time Brownian motion
stochastic processes. This enables you to transform a vector of `NBROWNS`

uncorrelated,
zero-drift, unit-variance rate Brownian components into a vector of `NVARS`

Brownian
components with arbitrary drift, variance rate, and correlation structure.

The `bm`

constructor
allows you to simulate any vector-valued BM process of the form:

$$d{X}_{t}=\mu (t)dt+V(t)d{W}_{t}$$

*X*is an_{t}`NVARS`

-by-`1`

state vector of process variables.*μ*is an`NVARS`

-by-`1`

drift-rate vector.*V*is an`NVARS`

-by-`NBROWNS`

instantaneous volatility rate matrix.*dW*is an_{t}`NBROWNS`

-by-`1`

vector of (possibly) correlated zero-drift/unit-variance rate Brownian components.

`BM = bm(Mu,Sigma)`

constructs
a default `bm`

object.

`BM = bm(Mu,Sigma,`

constructs
a `Name,Value`

)`bm`

object with additional options specified by
one or more `Name,Value`

pair arguments.

`Name`

is a property name and `Value`

is
its corresponding value. `Name`

must appear inside
single quotes (`''`

). You can specify several name-value
pair arguments in any order as `Name1,Value1,…,NameN,ValueN`

.

For more information on constructing a `bm`

object,
see `bm`

.

The following figure illustrates the inheritance relationships among SDE classes.

For more information, see SDE Class Hierarchy.

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.

Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.

When you invoke these parameters with inputs, they behave like
functions, giving the impression of dynamic behavior. The parameters
accept the observation time *t* and a state vector *X _{t}*,
and return an array of appropriate dimension. Even if you originally
specified an input as an array,

`bm`

treats it as
a static function of time and state, by that means guaranteeing that
all parameters are accessible by the same interface.Ait-Sahalia, Y. “Testing Continuous-Time Models of the
Spot Interest Rate.” *The Review of Financial Studies*,
Spring 1996, Vol. 9, No. 2, pp. 385–426.

Ait-Sahalia, Y. “Transition Densities for Interest Rate
and Other Nonlinear Diffusions.” *The Journal of
Finance*, Vol. 54, No. 4, August 1999.

Glasserman, P. *Monte Carlo Methods in Financial Engineering.* New
York, Springer-Verlag, 2004.

Hull, J. C. *Options, Futures, and Other Derivatives*,
5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.

Johnson, N. L., S. Kotz, and N. Balakrishnan. *Continuous
Univariate Distributions.* Vol. 2, 2nd ed. New York, John
Wiley & Sons, 1995.

Shreve, S. E. *Stochastic Calculus for Finance II:
Continuous-Time Models.* New York: Springer-Verlag, 2004.

`diffusion`

| `drift`

| `interpolate`

| `sdeld`

| `simByEuler`

| `simulate`

- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Pricing American Basket Options by Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- Class Attributes (MATLAB)
- Property Attributes (MATLAB)
- SDEs
- SDE Models
- SDE Class Hierarchy
- Performance Considerations

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