Explicit Finite-Difference Schemes for a Nonlinear HJB Equation

# Numerical scheme for this HJB equation


# Numerical scheme for this HJB equation












Without dwelling on details on how to obtain the HJB equation for this problem, I would like to know if the scheme I wrote for solving it numerically is viable or did I miss something.

I need to solve the following equation :

$$\begin{aligned} 0 = & \; \partial_{t}v(t,v,\mathcal{V}) + a_{\mathbb{P}}(t,v)\partial_{v}v(t,v,\mathcal{V}) + \frac{1}{2}v\xi^{2}\partial^{2}_{vv}v(t,v,\mathcal{V}) \\ & + \mathcal{V}\frac{a_{\mathbb{P}}(t,v) - a_{\mathbb{Q}}(t,v)}{2\sqrt{v}} - \frac{\gamma \xi^{2}}{8} \mathcal{V}^{2} \\ & + \sum_{Nb = 1}^{Nb = N}\sum_{l = a,l=b} \int_{R^{+}} z \, \mathbb{1}_{| \mathcal{V} - \phi(l)z\mathcal{V}^{I}| \leq \tilde{V}} H^{Nb,l}(\frac{v(t,v,\mathcal{V}) - v(t,v,\mathcal{V} - \phi(j)z\mathcal{V}^{i})}{z}) \mu(z)^{Nb,l} \end{aligned}$$

and terminal condition $v(T,v,\mathcal{V}) = 0$ ($\xi$, $\gamma$ are constant parameters).

H corresponds to a hamiltonian, $H(p) = \sup_{\delta > \delta_{\infty}} \Lambda(\delta)(\delta - p)$, $\delta_{\infty}$ is a constant and $\Lambda(\delta)$ is defined such that there exists a maximiser to obtain H.

I want to calculate this value function over a grid $[0,T] \times [ 0.1,0.2] \times [ -\tilde{V},+\tilde{V}]$. I denoted the time step $dt$, variance step $dv$, Vega $dV$. I denote $v^{k}_{i,j} = v(t_{k},v_{i},V_{j})$ I approximate the partial derviatives as follow :

$\partial_{t}v(t_{k},v_{i},\mathcal{V}_{j}) = \frac{v^{k}_{i,j} -v^{k-1}_{i,j}}{dt}$

$\partial_{v}v(t_{k},v_{i},\mathcal{V}_{j}) = \frac{v^{k}_{i+1,j} -v^{k}_{i-1,j}}{2dv}$

$\partial^{2}_{v}v(t_{k},v_{i},\mathcal{V}_{j}) = \frac{v^{k}_{i-1,j} -2 v^{k}_{i,j} + v^{k}_{i+1,j} }{dv^{2}}$

Then the scheme calculates the previous time step in function of the current time step.

Namely $$\begin{aligned} v^{k-1}_{i,j} = & \, v^{k}_{i,j} + dt \cdot \Big( a_{\mathbb{P}}(t_{k},v_{i}) \frac{v^{k}_{i,j} -v^{k}_{i-1,j}}{dv} \\ & + \frac{1}{2}v_{i}\xi^{2}\frac{v^{k}_{i-1,j} -2 v^{k}_{i,j} + v^{k}_{i+1,j} }{dv^{2}} + \mathcal{V}_{j} \frac{a_{\mathbb{P}}(t_{k},v_{i})- a_{\mathbb{Q}}(t_{k},v_{i})}{2 \sqrt{v_{j}}} \\ & - \frac{\gamma \xi^{2}}{8} \mathcal{V}_{j}^{2} + \sum_{Nb = 1}^{Nb = N}\sum_{l = a,l=b} \int_{R} z \mathbb{1}_{| \mathcal{V_{j}} - \phi(l)z\mathcal{V}^{I}| \leq \tilde{V}} H^{Nb,l}(\frac{v^{k}_{i,j} - v^{k}_{i,j-\phi(l)} }{z}) \mu(z)^{Nb,l} \Big) \end{aligned}$$

I don't know much about numerically methods for HJB equations so I just derived some explicit Euler scheme, is this correct or there are conditions for convergence and all ? In a previous discussion, it was understood that there may be a CFL condition underneath but I do not know more about it