I am following chapter 4 of this paper (1).
Background information: The stochastic differential equations (eq’s 4.2 to 4.5) are given by:
$$ \begin{aligned} & d S_t=\sqrt{\nu_t} d W_t \\ & d \nu_t=\theta\left(\alpha-\nu_t\right) d t+\xi \sqrt{\nu_t} d B_t \\ & d X_t=\left(S_t+\delta_t^a\right) d N_t^a-\left(S_t-\delta_t^b\right) d N_t^b \\ & d q_t=d N_t^b-d N_t^a \end{aligned} $$
where $N_t$ are independent poisson distributions with intensities $\kappa$.
The goal is to maximise $V(t,q,\nu,X,S) = \sup_{\delta^{b*},\delta^{a*}}\mathbb{E}\left[-e^{-\gamma\left(X_T+q_T S_T-l\left(\left|q_T\right|\right)\right)}\right]$
So we’re finding the optimal bid-ask quotes ($\delta$), that maximise this function.
Using the HJB, the author comes to this solution (eq’s 4.7):
$$ \begin{aligned} 0= & V_t(t, q, \nu, X, S)+\frac{1}{2} \nu V_{S S}(t, q, \nu, X, S)+\theta(\alpha-\nu) V_\nu(t, q, \nu, X, S) \\ & +\frac{1}{2} \xi^2 \nu V_{\nu \nu}(t, q, \nu, X, S)+\rho \xi \nu V_{S \nu}(t, q, \nu, X, S) \\ & + \sup _{\delta_t^b}\left[\left[V\left(t, q+1, \nu+d \nu, X-S+\delta^b, S\right)-V(t, q, \nu, X, S)\right] \Lambda^b\left(\delta_t^b\right)\right] \\ & + \sup _{\delta_t^a}\left[\left[V\left(t, q-1, \nu+d \nu, X+S+\delta^a, S\right)-V(t, q, \nu, X, S)\right] \Lambda^a\left(\delta_t^a\right)\right] \end{aligned} $$
with terminal condition $V(T,q,\nu,X,S) = -e^{-\gamma(X+qS -l\left(\left(\left|q_T\right|\right)\right)}$. (I removed the indicators as they’re not relevant). $\Lambda(\delta) = Ae^{-k\delta}$ for a constant A.
Choosing the ansatz solution:
$$V(t,q,\nu,X,S) = - e^{-\gamma(X+qS)}U(t,q\nu)$$
The author reduces the $V$ pde to (eq 4.10):
$$ \begin{aligned} 0= & U_t(t, q, \nu)+\frac{1}{2} \nu \gamma^2 q^2 U(t, q, \nu)+(\theta(\alpha-\nu)-\rho \xi \nu \gamma q) U_\nu(t, q, \nu) \\ & +\frac{1}{2} \xi^2 \nu U_{\nu \nu}(t, q, \nu)+\inf _{\delta_t^a}\left[\Lambda^a\left(\delta^a\right)\left[U(t, q-1, \nu) e^{-\gamma \delta^a}-U(t, q, \nu)\right]\right] \\ & +\inf _{\delta_t^b}\left[\Lambda^b\left(\delta^b\right)\left[U(t, q+1, \nu) e^{-\gamma \delta^b}-U(t, q, \nu)\right]\right] \end{aligned} $$
He then lets $\tau = T - t$ and using the implicit scheme, gets (eq 4.20):
$$ f_{j, q}^m=\Lambda^a\left(\delta^{\left(a^*\right)_{j, q}^m}\right)\left(U_{j, q-1}^m e^{-\gamma \delta^{\left(a^*\right)_{j, q}^m}}-U_{j, q}^m\right)+\Lambda^b\left(\delta^{\left(b^*\right)_{j, q}^m}\right)\left(U_{j, q+1}^m e^{-\gamma \delta^{\left(b^*\right)}{ }_{j, q}^m}-U_{j, q}^m\right) $$
$$ \begin{aligned} & U_{j-1, q}^{m+1}\left(-\frac{[\theta(\alpha-\nu)-\rho \xi \nu \gamma q]}{2(\Delta \nu)}+\frac{\xi^2 \nu}{2(\Delta \nu)^2}\right) \\ & +U_{j, q}^{m+1}\left(-\frac{1}{\Delta \tau}+\frac{\nu \gamma^2 q^2}{2}-\frac{\xi^2 \nu}{(\Delta \nu)^2}\right) \\ & +U_{j+1, q}^{m+1}\left(\frac{[\theta(\alpha-\nu)-\rho \xi \nu \gamma q]}{2(\Delta \nu)}+\frac{\xi^2 \nu}{2(\Delta \nu)^2}\right)=-\frac{U_{j, q}^m}{\Delta \tau}-f_{j, q}^m \end{aligned} $$
He then goes on to prove that numerical discretisation is stable and converges to the viscosity solution.
If we slightly change the problem, as given in chapter 4.4 (page 27) of this paper (2).
When maximising over $E(X - \frac{l}{2} \int_t^T (dI_t)^2)$ (quadratic utility), where $I$ is the inventory, the HJB in paper (2) (eq 4.16) is:
$$ \begin{aligned} & V_t+\theta(\alpha-\nu) V_\nu+\frac{1}{2} \nu V_{s s}+\xi \rho \nu V_{s \nu}+\frac{1}{2} \xi^2 \nu V_{\nu \nu}-\frac{\gamma}{2} \nu \xi^2 C_\nu^2\left(q_t^o\right)^2+\max _{\left(\delta_t^{a, o}, \delta_t^{b, o}\right) \in \mathcal{A}}\left\{\Lambda(\delta^a) \left[\delta_t^{a, o}+\right.\right. \\ & \left.\left.V\left(s, \nu, q_t^o-1, t\right)-V\left(s, \nu, q_t^o, t\right)\right]+\Lambda(\delta^a)\left[\delta_t^{b, o}+V\left(s, \nu, q_t^o+1, t\right)-V\left(s, \nu, q_t^o, t\right)\right]\right\}=0 \end{aligned} $$
with the terminal condition: $V\left(s, \nu, q^o, T\right)=0$.
This quadratic utility $V$ is functionally analogues to the exponential utility from paper 1. This also has a nice ansatz and is stable as shown in this paper (3).
The difference is that the derivative of the call option wrt volatility ($C_{\nu}$) is in the PDE of $V$.
My question: With this slight change (a $C_{\nu}$ term), but using a similar ansatz solution that satisfies that terminal, can it be inferred that the same discretisation is also stable and converges to the viscosity solution? My main goal is that I need to reduce the dimensions of $V(s,\nu,q^o,t)$ to only $V(\nu,q^0,t)$ like in paper 1, so that I can solve the PDE.
As a reminder, choosing the same utility still yields a nice ansatz solution to reduce the dimensions as shown in paper (3), the only main difference is that we get an extra deterministic term $C_\nu$ in the HJB. Does the presence of $C_\nu$ mean that after reducing the solutions, we do not have convergence anymore?