# sde

Stochastic Differential Equation (`SDE`) model

## Description

Creates and displays general stochastic differential equation (`SDE`) models from user-defined drift and diffusion rate functions.

Use `sde` objects to simulate sample paths of `NVars` state variables driven by `NBROWNS` Brownian motion sources of risk over `NPeriods` consecutive observation periods, approximating continuous-time stochastic processes.

An `sde` object enables you to simulate any vector-valued SDE of the form:

`$d{X}_{t}=F\left(t,{X}_{t}\right)dt+G\left(t,{X}_{t}\right)d{W}_{t}$`

where:

• Xt is an `NVars`-by-`1` state vector of process variables.

• dWt is an `NBROWNS`-by-`1` Brownian motion vector.

• F is an `NVars`-by-`1` vector-valued drift-rate function.

• G is an `NVars`-by-`NBROWNS` matrix-valued diffusion-rate function.

## Creation

### Syntax

``SDE = sde(DriftRate,DiffusionRate)``
``SDE = sde(___,Name,Value)``

### Description

example

````SDE = sde(DriftRate,DiffusionRate)` creates a default `SDE` object.```

example

````SDE = sde(___,Name,Value)` creates a `SDE` 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`.The `SDE` 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 ```

### Input Arguments

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`DriftRate` is a user-defined drift-rate function and represents the parameter F, specified as a vector or object of class `drift`.

`DriftRate` is a function that returns an `NVars`-by-`1` drift-rate vector when called with two inputs:

• A real-valued scalar observation time t.

• An `NVars`-by-`1` state vector Xt.

Alternatively, `DriftRate` can also be an object of class `drift` that encapsulates the drift-rate specification. In this case, however, `sde` uses only the `Rate` parameter of the object. For more information on the `drift` object, see `drift`.

Data Types: `double` | `object`

`DiffusionRate` is a user-defined drift-rate function and represents the parameter G, specified as a matrix or object of class `diffusion`.

`DiffusionRate` is a function that returns an `NVars`-by-`NBROWNS` diffusion-rate matrix when called with two inputs:

• A real-valued scalar observation time t.

• An `NVars`-by-`1` state vector Xt.

Alternatively, `DiffusionRate` can also be an object of class `diffusion` that encapsulates the diffusion-rate specification. In this case, however, `sde` uses only the `Rate` parameter of the object. For more information on the `diffusion` object, see `diffusion`.

Data Types: `double` | `object`

## Properties

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Starting time of first observation, applied to all state variables, specified as a scalar

Data Types: `double`

Initial values of state variables, specified as a scalar, column vector, or matrix.

If `StartState` is a scalar, `sde` applies the same initial value to all state variables on all trials.

If `StartState` is a column vector, `sde` applies a unique initial value to each state variable on all trials.

If `StartState` is a matrix, `sde` applies a unique initial value to each state variable on each trial.

Data Types: `double`

Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes), specified 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. If `Correlation` is not a symmetric positive semidefinite matrix, use `nearcorr` to create a positive semidefinite matrix for a correlation matrix.

A `Correlation` matrix represents a static condition.

As a deterministic function of time, `Correlation` allows you to specify a dynamic correlation structure.

Data Types: `double`

User-defined simulation function or SDE simulation method, specified as a function or SDE simulation method.

Data Types: `function_handle`

Drift rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The drift rate specification supports the simulation of sample paths of `NVars` state variables driven by `NBROWNS` Brownian motion sources of risk over `NPeriods` consecutive observation periods, approximating continuous-time stochastic processes.

The `drift` class allows you to create drift-rate objects (using `drift`) of the form:

`$F\left(t,{X}_{t}\right)=A\left(t\right)+B\left(t\right){X}_{t}$`

where:

• `A` is an `NVars`-by-`1` vector-valued function accessible using the (t, Xt) interface.

• `B` is an `NVars`-by-`NVars` matrix-valued function accessible using the (t, Xt) interface.

The displayed parameters for a `drift` object are:

• `Rate`: The drift-rate function, F(t,Xt)

• `A`: The intercept term, A(t,Xt), of F(t,Xt)

• `B`: The first order term, B(t,Xt), of F(t,Xt)

`A` and `B` enable you to query the original inputs. The function stored in `Rate` fully encapsulates the combined effect of `A` and `B`.

When specified as MATLAB® double arrays, the inputs `A` and `B` are clearly associated with a linear drift rate parametric form. However, specifying either `A` or `B` as a function allows you to customize virtually any drift rate specification.

Note

You can express `drift` and `diffusion` classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components `A` and `B` as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

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

Data Types: `object`

Diffusion rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The diffusion rate specification supports the simulation of sample paths of `NVars` state variables driven by `NBROWNS` Brownian motion sources of risk over `NPeriods` consecutive observation periods, approximating continuous-time stochastic processes.

The `diffusion` class allows you to create diffusion-rate objects (using `diffusion`):

`$G\left(t,{X}_{t}\right)=D\left(t,{X}_{t}^{\alpha \left(t\right)}\right)V\left(t\right)$`

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, Xt) interface.

The displayed parameters for a `diffusion` object are:

• `Rate`: The diffusion-rate function, G(t,Xt).

• `Alpha`: The state vector exponent, which determines the format of D(t,Xt) of G(t,Xt).

• `Sigma`: The volatility rate, V(t,Xt), of G(t,Xt).

`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.

Note

You can express `drift` and `diffusion` classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components `A` and `B` as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

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

Data Types: `object`

## Object Functions

 `interpolate` Brownian interpolation of stochastic differential equations (SDEs) for `SDE`, `BM`, `GBM`, `CEV`, `CIR`, `HWV`, `Heston`, `SDEDDO`, `SDELD`, or `SDEMRD` models `simulate` Simulate multivariate stochastic differential equations (SDEs) for `SDE`, `BM`, `GBM`, `CEV`, `CIR`, `HWV`, `Heston`, `SDEDDO`, `SDELD`, `SDEMRD`, `Merton`, or `Bates` models `simByEuler` Euler simulation of stochastic differential equations (SDEs) for `SDE`, `BM`, `GBM`, `CEV`, `CIR`, `HWV`, `Heston`, `SDEDDO`, `SDELD`, or `SDEMRD` models

## Examples

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Construct an `SDE` object `obj` to represent a univariate geometric Brownian Motion model of the form: $d{X}_{t}=0.1{X}_{t}dt+0.3{X}_{t}d{W}_{t}$.

Create drift and diffusion functions that are accessible by the common $\left(t,{X}_{t}\right)$ interface:

```F = @(t,X) 0.1 * X; G = @(t,X) 0.3 * X;```

Pass the functions to `sde` to create an object (`obj`) of class `sde`:

`obj = sde(F, G) % dX = F(t,X)dt + G(t,X)dW`
```obj = Class SDE: Stochastic Differential Equation ------------------------------------------- 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 ```

`obj` displays like a MATLAB® structure, with the following information:

• The object's class

• A brief description of the object

• A summary of the dimensionality of the model

The object's displayed parameters are as follows:

• `StartTime`: The initial observation time (real-valued scalar)

• `StartState`: The initial state vector (`NVARS`-by-1 column vector)

• `Correlation`: The correlation structure between Brownian process

• `Drift`: The drift-rate function $F\left(t,{X}_{t}\right)$

• `Diffusion`: The diffusion-rate function $G\left(t,{X}_{t}\right)$

• `Simulation`: The simulation method or function.

Of these displayed parameters, only `Drift` and `Diffusion` are required inputs.

The only exception to the ($t$, ${X}_{t}$) evaluation interface is `Correlation`. Specifically, when you enter `Correlation` as a function, the SDE engine assumes that it is a deterministic function of time, $C\left(t\right)$. This restriction on `Correlation` as a deterministic function of time allows Cholesky factors to be computed and stored before the formal simulation. This inconsistency dramatically improves run-time performance for dynamic correlation structures. If `Correlation` is stochastic, you can also include it within the simulation architecture as part of a more general random number generation function.

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## Algorithms

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, `sde` treats it as a static function of time and state, by that means guaranteeing that all parameters are accessible by the same interface.

 Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies, vol. 9, no. 2, Apr. 1996, pp. 385–426.

 Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, vol. 54, no. 4, Aug. 1999, pp. 1361–95.

 Glasserman, Paul. Monte Carlo Methods in Financial Engineering. Springer, 2004.

 Hull, John. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.

 Johnson, Norman Lloyd, et al. Continuous Univariate Distributions. 2nd ed, Wiley, 1994.

 Shreve, Steven E. Stochastic Calculus for Finance. Springer, 2004.