I am trying to simulate the instanteneous volatility of a Heston process.
My equations are the following :
wealth process: $$dX_t = r_t X_t + \theta \sqrt {V_t} u_t\, dt + u_t \,dW_{1t}$$
Volatility: $$dV_t = (\kappa \phi - \lambda V_t) \,dt + \sigma \sqrt {V_t} \, dB_t $$
With, I start my simulations with a 2D brownian motion : $(W_1, W_2)$ and another "corrolated" Brownian motion $B_t = \rho \, d \widetilde{W}_{1t} + \sqrt{1- \rho^2} \, dW_{2t} $
My problem lies in the $d \widetilde{W}_{1t}$. Its definition is :
$$ \widetilde{W}_{1t} = W_{1t} + 2 \theta \int_0^t \sqrt {V_s} \, ds. $$
So I know how to simulate the wealth process, it s a classical "flow".
The volatility follows the same pattern, iff the brownian motion $dB_t$ is a classical one. Here there is a drift movement which makes the whole simulation cyclic. I have no idea how to deal with it.
- Is it possible to simulate that ? Is my problem markovian ?
- How would one deal with that problem. I simply need a solution for $\tilde{W}_{1t} $, I'll deal with the rest.
Thank you
$$ d\widetilde{W}_{1t} = d W_{1t} + 2 \theta \sqrt {V_s} dt $$ Then one can replace $$ dB_t = \rho d W_{1t} + 2 \theta \sqrt {V_s} dt + \sqrt{1- \rho^2} dW_{2t} $$