FxPaul

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Archive for May 27th, 2011

Closed-form solution of modified Ornstein-Uhlenbeck process

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Process definition

In this article we deduce the closed-from solution of the modified version of Ornstein-Uhlenbeck process:
dS = \theta(\mu - S) dt + \sigma S dW_t
where \theta – mean reversion parameter, \mu – mean and \sigma – volatility.

Integrating factor approach

There exists a general approach to non-linear stochastic differential equations of the form:
dX_t = f(t,X_t) dt + c(t) X_t dW_t \, , \, X_0 = x
where f and g are given continuous and deterministic functions.

The method consists of:

  1. Define the integrating factor:
    F_t = F_t(\omega) = \exp\left( - \int_0^t {c(s) dW_s} + \frac{1}{2}\int_0^t {c^2(s) dW_s} \right)
  2. So the original equation could be written as d\left(F_t X_t\right) = F_t f(t,X_t) dt
  3. Now define
    Y_t(\omega) = F_t(\omega) X_t(\omega)
    so that X_t = F^{-1}_t Y_t
  4. And it yields the deterministic differential equation for each \omega\in\Omega
    \frac{dY_t(\omega)}{dt} = F_t(\omega) f\left(t, F^{-1}_t(\omega) Y_t(\omega)\right); \, Y_0 = x

We can therefore solve it with \omega as a parameter to find Y_t(\omega) and then obtain X_t(\omega)

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Written by fxpaul

May 27, 2011 at 08:39

Posted in trading math