"An energy-harvesting circuit based on graphene could be incorporated into a chip to provide clean, limitless, low-voltage power for small devices or sensors," said Paul Thibado, professor of physics and lead researcher in the discovery. The Finance command was introduced in Maple 15.įor more information on Maple 15 changes, see Updates in Maple 15. Upper Saddle River, New Jersey: Prentice Hall, 2003.
Hull, J., Options, Futures, and Other Derivatives, 5th. Glasserman, P., Monte Carlo Methods in Financial Engineering. 3, timesteps = 10, replications = 10, thickness = 3, axes = BOXED, gridlines W3 ≔ eval y t, dsolve &DifferentialD &DifferentialD t y t = sin t, y 0 = 0 W2 ≔ BrownianMotion 0, f, 0.001, t, scheme = unbiased Compare the simulated results with the corresponding solution of an ordinary (non-stochastic) differential equation. For presentation purposes let us consider a Brownian motion with very low volatility and time-dependent drift. Here is the same using different discretization schemes. 3, timesteps = 100, replications = 10, axes = BOXED, gridlines = true Y ≔ BrownianMotion 0, 0.0, R t 2, t Here are examples involving stochastic volatility. Statistics Histogram V, bincount = 100 V ≔ SampleValues X → X 1, 2 + X 2, 2, W, 0. You can simulate values for any path function given as a Maple procedure. Here is an example of a two-dimensional Brownian motion. Z ≔ BrownianMotion 0.0, &mu, &sigma : Here is the same example but with drift and volatility given in the form of Maple procedures. Y ≔ BrownianMotion 0.0, &mu, &sigma, t : Here is an example of a one-dimensional Brownian motion with time-dependent parameters given in algebraic form. Μ u &ExponentialE _X1 u + σ u 2 &ExponentialE _X1 u 2 W ≔ BrownianMotion 1, &mu t, &sigma t, t T, timesteps = 100, replications = 10, axes = BOXED, gridlines = trueĬompute the drift and diffusion for functions of X. This scheme is appropriate in the case of a time-dependent drift and/or volatility.įirst consider the case of a one-dimensional Brownian motion with constant drift and volatility.
When scheme is set to unbiased the transition density will be used to simulate a value X t + dt given X t. By default the standard Euler scheme is used. The scheme option specifies the discretization scheme used for simulation of this process. The drift parameter Mu must be given as a Vector and the covariance matrix Sigma must be a given as a symmetric matrix (see Matrix ). In this case the drift vector and the covariance matrix are time-independent. W t is the standard m -dimensional Wiener process. The BrownianMotion( X 0, Mu, Sigma) defines an n -dimensional Brownian motion with drift Mu and covariance Sigma. Note that stochastic drift is not supported. In addition to this, the volatility can involve other (one-dimensional) stochastic variables. Similar to the drift, the volatility parameter can be constant or time dependent. Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift. If mu is given as an algebraic expression, then the parameter t must be passed to specify which variable in mu should be used as a time variable. Time-dependent drift can be given either as an algebraic expression or as a Maple procedure. In the simplest case of a constant drift, mu is a real number (i.e. The parameter x 0 defines the initial value of the underlying stochastic process. This is a stochastic process X t, which is governed by the stochastic differential equation (SDE) The BrownianMotion( x 0, mu, sigma) and BrownianMotion( x 0, mu, sigma, t) commands create a new one-dimensional Brownian motion process. Scheme = unbiased or Euler - This option specifies which discretization scheme should be used for simulating this process. (optional) equation(s) of the form option = value where option is scheme specify options for the BrownianMotion command Define one- or multi-dimensional Brownian motion processīrownianMotion( x 0, mu, sigma, opts )īrownianMotion( x 0, mu, sigma, t, opts )Īlgebraic expression, operator or procedure drift parameterĪlgebraic expression, operator, procedure or a one-dimensional stochastic process volatility
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