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Because base-level `sde`

objects accept drift and
diffusion objects in lieu of functions accessible by (*t*,
*X _{t}*), you can create

`sde`

objects with combinations of
customized drift or diffusion functions and objects. The `drift`

and `diffusion`

rate objects encapsulate
the details of input parameters to optimize run-time efficiency for any given
combination of input parameters.Although `drift`

and `diffusion`

objects differ in the
details of their representation, they are identical in their basic implementation
and interface. They look, feel like, and are evaluated as functions:

The

`drift`

object allows you to create drift-rate objects of the form:$$F(t,{X}_{t})=A(t)+B(t){X}_{t}$$

where:

`A`

is an`NVars`

-by-`1`

vector-valued function accessible using the (*t*,*X*) interface._{t}`B`

is an`NVars`

-by-`NVars`

matrix-valued function accessible using the (*t*,*X*) interface._{t}

Similarly, the

`diffusion`

object allows you to create diffusion-rate objects:$$G(t,{X}_{t})=D(t,{X}_{t}^{\alpha (t)})V(t)$$

where:

`D`

is an`NVars`

-by-`NVars`

diagonal matrix-valued function.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.`V`

is an`NVars`

-by-`NBrowns`

matrix-valued volatility rate function`Sigma`

.`Alpha`

and`Sigma`

are also accessible using the (*t*,*X*) interface._{t}

**Note**You can express

`drift`

and`diffusion`

objects in the most general form to emphasize the functional (*t*,*X*) interface. However, you can specify the components_{t}`A`

and`B`

as functions that adhere to the common (*t*,*X*) interface, or as MATLAB_{t}^{®}arrays of appropriate dimension.

In this example, you create `drift`

and `diffusion`

rate objects to create the
same model as in Example: Base SDE Models.

Create a drift-rate function `F`

and a diffusion-rate function
`G`

:

`F = drift(0, 0.1) % Drift rate function F(t,X)`

F = Class DRIFT: Drift Rate Specification ------------------------------------- Rate: drift rate function F(t,X(t)) A: 0 B: 0.1

`G = diffusion(1, 0.3) % Diffusion rate function G(t,X)`

G = Class DIFFUSION: Diffusion Rate Specification --------------------------------------------- Rate: diffusion rate function G(t,X(t)) Alpha: 1 Sigma: 0.3

Each object displays like a MATLAB structure and contains supplemental information, namely, the object's
class and a brief description. However, in contrast to the SDE representation, a
summary of the dimensionality of the model does not appear, because `drift`

and `diffusion`

objects create model
components rather than models. Neither `F`

nor `G`

contains enough information to characterize the dimensionality of a problem.

The `drift`

object's displayed parameters
are:

`Rate`

: The drift-rate function,*F(t,X*_{t})`A`

: The intercept term,*A(t,X*, of_{t})*F(t,X*_{t})`B`

: The first order term,*B(t,X*, of_{t})*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`

.

The `diffusion`

object's displayed
parameters are:

`Rate`

: The diffusion-rate function,*G(t,X*._{t})`Alpha`

: The state vector exponent, which determines the format of*D(t,X*of_{t})*G(t,X*._{t})`Sigma`

: The volatility rate,*V(t,X*, of_{t})*G(t,X*._{t})

Again, `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.

The `sdeddo`

object derives from the
base`sde`

object. To use this object, you
must pass drift and diffusion-rate objects to `sdeddo`

.

Create

`drift`

and`diffusion`

rate objects:F = drift(0, 0.1); % Drift rate function F(t,X) G = diffusion(1, 0.3); % Diffusion rate function G(t,X)

Pass these objects to the

`sdeddo`

object:`obj = sdeddo(F, G) % dX = F(t,X)dt + G(t,X)dW`

obj = Class SDEDDO: SDE from Drift and Diffusion Objects -------------------------------------------------- Dimensions: State = 1, Brownian = 1 -------------------------------------------------- StartTime: 0 StartState: 1 Correlation: 1 Drift: drift rate function F(t,X(t)) Diffusion: diffusion rate function G(t,X(t)) Simulation: simulation method/function simByEuler A: 0 B: 0.1 Alpha: 1 Sigma: 0.3

In this example, the object displays the additional parameters associated with input drift and diffusion objects.

`sde`

| `bm`

| `gbm`

| `merton`

| `bates`

| `drift`

| `diffusion`

| `sdeddo`

| `sdeld`

| `cev`

| `cir`

| `heston`

| `hwv`

| `sdemrd`

| `ts2func`

| `simulate`

| `simByEuler`

| `simBySolution`

| `simBySolution`

| `interpolate`

| `simByQuadExp`