Simulation Heston Model, markovianity

Question

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 \tilde{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.

1. Is it possible to simulate that ? Is my problem markovian ?
2. How would one deal with that problem. I simply need a solution for $\widetilde{W}_{1t} $, I'll deal with the rest.

Thank you

Answer 1

You should replace the differential of the correlated process dBt with its value in the volatility equation, then replace dW~t in the same equation with:
dW~t=dWt+ 2*theta*square_root(Vt)*dt
you will get an formula with Vt,W1t and W2t. You can then simulate the volatility.