I'm a beginner with FE. My application is the pricing of financial derivatives where the space is five dimensional. So, adding time, the problem has six dimensions.
I tried to look around (Fenics, escript, deal.II,...), but my understanding is that those software are limited to 3 + 1 (3d space + 1d time). Is this correct?
My targeted language are Python or C++.
Description of my problem
I would like to price an investment product where, each month, the investor has the freedom to re-invests or not.
I would like to do so with stochastic volatility, stochastic interest rate, and stochastic mortality.
The stochastic PDEs look like this
\begin{align}
dS_t &= \mu^S_t d_t + \sqrt{\sigma_t} dB^S_t &\text{(stock)}\\
d\sigma_t &= \mu^\sigma_t dt + \nu^\sigma_t dB^\sigma_t & \text{(volatility)} \\
dr_t &= \mu^r_t dt + \nu^r_t dB^r_t & \text{(interest rate)} \\
dq_t &= \mu^q_t dt + \nu^q_t dB^q_t & \text{(mortality)}
\end{align}
Where $\mu^S_t$ is a time-dependent constant associated to the stock price $S$, and $B^S_t$ is an independent Levy process which creates noise in the stock price $S$. Similarly for the other quantities: $\nu^\sigma_t$ is a time-dependent quantity associated to the volatility $\sigma$.
Let $C_\tau$ denotes the admissible investments at time $\tau$.
The stochastic control problem looks like
$$
V_\tau = max \left\{ c \in C_\tau : P(\text{death})E(r_\tau f(S_{\tau+1})) + P(alive)E(r_\tau V_{\tau+1})\right\}.
$$
The above PDEs are continuous, but the value of the product $V_\tau$ is solved only at predefined $\tau$-times, say each month.
I guess Monte-Carlo can always brute force my problem, but it is very slow.
Deterministic form of the stochastic PDEs
For this part, assume that the value of the option
$$
V : (t, S_t, \sigma_t, r_t, q_t, c_t) \mapsto (t, V_t),
$$
is define on the natural time $t$, not the $\tau$-times, with $c_t$ the investment at time $t$.
Define the differential operator
\begin{align}
L_t &= \partial_{r,S} + \partial_{r,\sigma} + \partial_{\sigma,S} \\
L^S_t &= \sigma_t \partial_S + r_t \partial_{S,S} \\
L^r_t &= \partial_r + \partial_{r,r} \\
L^\sigma_t &= \partial_\sigma + \partial_{\sigma,\sigma} \\
L^q_t &= \partial_q + \partial_{q,q}
\end{align}
where time-dependent constant $\{\mu^S_t,\ldots\}$ are ignored.
The deterministic PDE is then
$$
\partial_t V_t +\left(L_t+ L^S_t + L^\sigma_t + L^r_t+L^q_t\right)V_t = 0,
$$
which can adapted to the optimal control problem on the $\tau$-times.