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Specifying an array indicates a static non-time-varying parametric specification.

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MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. 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.

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. 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.

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.

Name is a property name and Value is its corresponding value. Name must appear inside single quotes ''. The BM object has the following Properties:. 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. 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. If you specify Mu as a function of time and state, it calculates the expected instantaneous rate of drift. A real-valued scalar observation time t. Sigma represents the parameter V , specified as an array or deterministic function of time. In this case, each row of Sigma corresponds to a particular state variable.

Each column corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty. Although the gbm constructor enforces no restrictions on the sign of Sigma volatilities, they are specified as positive values. Starting time of first observation, applied to all state variables, specified as a scalar. Initial values of state variables, specified as a scalar, column vector, or matrix. 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. A Correlation matrix represents a static condition. As a deterministic function of time, Correlation allows you to specify a dynamic correlation structure. Drift rate component of continuous-time stochastic differential equations SDEs , specified as a drift object or function accessible by t , X t.

The drift class allows you to create drift-rate objects using drift of the form:. The drift-rate function, F t,X t. The intercept term, A t,X t , of F t,X t. The first order term, B t,X t , of F t,X t. A and B enable you to query the original inputs. The function stored in Rate fully encapsulates the combined effect of A and B. However, specifying either A or B as a function allows you to customize virtually any drift rate specification.

You can express drift and diffusion classes in the most general form to emphasize the functional t , X t interface. Diffusion rate component of continuous-time stochastic differential equations SDEs , specified as a drift object or function accessible by t , X t. The diffusion class allows you to create diffusion-rate objects using diffusion:. Each diagonal element of D is the corresponding element of the state vector raised to the corresponding element of an exponent Alpha , which is an NVARS -by- 1 vector-valued function.

Alpha and Sigma are also accessible using the t , X t interface. The diffusion-rate function, G t,X t. The state vector exponent, which determines the format of D t,X t of G t,X t. The volatility rate, V t,X t , of G t,X t. Alpha and Sigma enable you to query the original inputs. The combined effect of the individual Alpha and Sigma parameters is fully encapsulated by the function stored in Rate. The Rate functions are the calculation engines for the drift and diffusion objects, and are the only parameters required for simulation.

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