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# bm

Construct Brownian motion models

## Synopsis

`BM = bm(Mu, Sigma)`

```BM = bm(Mu, Sigma, 'Name1', Value1, 'Name2', Value2, ...)```

`bm`

## Description

This constructor creates and displays Brownian motion (sometimes called arithmetic Brownian motion or generalized Wiener process) 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` method allows you to simulate any vector-valued BM process of the form:

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

where:

• Xt is an `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.

• dWt is an `NBROWNS`-by-`1` vector of (possibly) correlated zero-drift/unit-variance rate Brownian components.

## Input Arguments

Specify required input parameters as one of the following types:

• A MATLAB® array. Specifying an array indicates a static (non-time-varying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.

• A MATLAB function. Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function.

 Note:   You can specify combinations of array and function input parameters as needed. Moreover, a parameter is identified as a deterministic function of time if the function accepts a scalar time `t` as its only input argument. Otherwise, a parameter is assumed to be a function of time t and state X(t) and is invoked with both input arguments.

The required input parameters are:

 `Mu` `Mu` represents μ. If you specify `Mu` as an array, it must be an `NVARS`-by-`1` column vector representing the drift rate (the expected instantaneous rate of drift, or time trend). As a deterministic function of time, when `Mu` is called with a real-valued scalar time `t` as its only input, `Mu` must produce an `NVARS`-by-`NVARS` matrix. If you specify `Mu` as a function of time and state, it calculates the expected instantaneous rate of drift. This function must generate an `NVARS`-by-`1` column vector when invoked with two inputs:A real-valued scalar observation time t.An `NVARS`-by-`1` state vector Xt. `Sigma` `Sigma` represents the parameter V. If you specify `Sigma` as an array, it must be an `NVARS`-by-`NBROWNS` matrix of instantaneous volatility rates. In this case, each row of `Sigma` corresponds to a particular state variable. Each column of `Sigma` corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty. As a deterministic function of time, when `Sigma` is called with a real-valued scalar time `t` as its only input, `Sigma` must produce an `NVARS`-by-`NBROWNS` matrix. If you specify `Sigma` as a function of time and state, it must generate an `NVARS`-by-`NBROWNS` matrix of volatility rates when invoked with two inputs: A real-valued scalar observation time t.An `NVARS`-by-`1` state vector Xt.Although the constructor does not enforce restrictions on the sign of this argument, `Sigma` is specified as a positive value.

## Optional Input Arguments

Specify optional inputs as matching parameter name/value pairs as follows:

• Specify the parameter name as a character vector, followed by its corresponding value.

• You can specify parameter name/value pairs in any order.

• Parameter names are case insensitive.

• You can specify unambiguous partial character vector matches.

Valid parameter names are:

 `StartTime` Scalar starting time of the first observation, applied to all state variables. If you do not specify a value for `StartTime`, the default is `0`. `StartState` Scalar, `NVARS`-by-`1` column vector, or `NVARS`-by-`NTRIALS` matrix of initial values of the state variables. If `StartState` is a scalar, `bm` applies the same initial value to all state variables on all trials.If `StartState` is a column vector, `bm` applies a unique initial value to each state variable on all trials.If `StartState` is a matrix, `bm` applies a unique initial value to each state variable on each trial. If you do not specify a value for `StartState`, all variables start at `1`. `Correlation` Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes). Specify `Correlation` as an `NBROWNS`-by-`NBROWNS` positive semidefinite matrix, or as a deterministic function C(t) that accepts the current time t and returns an `NBROWNS`-by-`NBROWNS` positive semidefinite correlation matrix. A `Correlation` matrix represents a static condition.As a deterministic function of time, `Correlation` allows you to specify a dynamic correlation structure.If you do not specify a value for `Correlation`, the default is an `NBROWNS`-by-`NBROWNS` identity matrix representing independent Gaussian processes. `Simulation` A user-defined simulation function or SDE simulation method. If you do not specify a value for `Simulation`, the default method is simulation by Euler approximation (`simByEuler`).

## Output Arguments

 `BM` Object of class `BM` with the following displayed parameters: `StartTime`: Initial observation time`StartState`: Initial state at time `StartTime``Correlation`: Access function for the `Correlation` input argument, callable as a function of time`Drift`: Composite drift-rate function, callable as a function of time and state `Diffusion`: Composite diffusion-rate function, callable as a function of time and state `Simulation`: A simulation function or method`Mu`: Access function for the input argument `Mu`, callable as a function of time and state `Sigma`: Access function for the input argument `Sigma`, callable as a function of time and state

## Algorithm

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 Xt, 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.

## Examples

Creating Brownian Motion (BM) Models

## References

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.