The time-fractional Heston model is governed by the following PDE:

${\partial }_t^{\alpha }u+\underset{a\in \mathcal{A}}{\sup }\mathscr{H}\left(x,u,\nabla u,\text{Δ}u,a\right)=0,0<\alpha <1,$(1)

where:

$\begin{array}{c}\mathscr{H}\left(x,u,\nabla u,\text{Δ}u,a\right)=\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}u+\kappa \left(\theta -v\right){\partial }_{v}u\\ +\lambda \left(\mathbb{E}\left[u\left(x+\xi ,v\right)\right]-u\left(x,v\right)\right)-ru+I\left(u\right).\end{array}$(2)

with:

${\partial }_t^{\alpha }u\left(x,t\right)=\frac{1}{\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t\frac{\partial u\left(x,s\right)}{\partial s}\frac{\text{d}s}{{\left(t-s\right)}^{\alpha }}}.$(3)

Theorem 3.1 (Existence of Viscosity Solutions) Let $u_0\left(x,v\right)\in C^{\alpha }\left(x,v\right)$ be the initial condition. There exists a viscosity solution $u\in C\left(\left[0,T\right];C^{\alpha }\left(x,v\right)\right)$ to the time-fractional HJB equation:

${\partial }_t^{\alpha }u+\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)=0,\left(x,v,t\right)\in {ℝ}^2\times \left[0,T\right],0<\alpha <1,$(4)

where:

$\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)=\frac{1}{2}v{\sigma }^2{x}^2{\partial }_{xx}u+\kappa \left(\theta -v\right){\partial }_{v}u+\lambda \left(\mathbb{E}\left[u\left(x+\xi ,v\right)\right]-u\left(x,v\right)\right).$(5)

Proof. We introduce a small regularization parameter $ϵ>0$ and consider the regularized equation:

$ϵ\text{Δ}{u}^{ϵ}+{\partial }_t^{\alpha }{u}^{ϵ}+\mathscr{H}\left(x,u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)=0,\left(x,v,t\right)\in {ℝ}^2\times \left[0,T\right],$(6)

where $\text{Δ}{u}^{ϵ}={\partial }_{xx}{u}^{ϵ}+{\partial }_{vv}{u}^{ϵ}$.

Using the Caputo fractional derivative:

${\partial }_t^{\alpha }{u}^{ϵ}\left(t\right)=\frac{1}{\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t\frac{{\partial }_s{u}^{ϵ}\left(s\right)}{{\left(t-s\right)}^{\alpha }}\text{d}s},$(7)

The regularized equation becomes:

$ϵ\text{Δ}{u}^{ϵ}+\frac{1}{\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t\frac{{\partial }_s{u}^{ϵ}\left(s\right)}{{\left(t-s\right)}^{\alpha }}\text{d}s}+\mathscr{H}\left(x,u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)=0.$(8)

Next, multiply the regularized equation by $u^{ϵ}$ and integrate over ${ℝ}^2\times \left[0,t\right]$:

${\displaystyle {\int }_{{ℝ}^2}ϵ\text{Δ}{u}^{ϵ}\cdot u^{ϵ}\text{d}x\text{d}v}+{\displaystyle {\int }_{{ℝ}^2}{\partial }_t^{\alpha }{u}^{ϵ}\cdot u^{ϵ}\text{d}x\text{d}v}+{\displaystyle {\int }_{{ℝ}^2}\mathscr{H}}\left(x,u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)\cdot u^{ϵ}\text{d}x\text{d}v=0.$(9)

Respectively, for the First term (integration by parts):

${\displaystyle {\int }_{{ℝ}^2}ϵ\text{Δ}{u}^{ϵ}\cdot u^{ϵ}\text{d}x\text{d}v}=-ϵ{\displaystyle {\int }_{{ℝ}^2}{\left|\nabla u^{ϵ}\right|}^2\text{d}x\text{d}v}.$(10)

${\displaystyle {\int }_{{ℝ}^2}{\partial }_t^{\alpha }{u}^{ϵ}\cdot u^{ϵ}\text{d}x\text{d}v}\le C{\Vert u^{ϵ}\Vert }_{L^2}^2.$(11)

${\displaystyle {\int }_{{ℝ}^2}\mathscr{H}}\left(x,u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)\cdot u^{ϵ}\text{d}x\text{d}v\le C\left({\Vert \nabla u^{ϵ}\Vert }_{L^2}^2+{\Vert u^{ϵ}\Vert }_{L^2}^2\right).$(12)

Combining all terms:

$-ϵ{\Vert \nabla u^{ϵ}\Vert }_{L^2}^2+C{\Vert u^{ϵ}\Vert }_{L^2}^2+C{\Vert \nabla u^{ϵ}\Vert }_{L^2}^2\le 0.$(13)

Simplifying:

${\Vert \nabla u^{ϵ}\Vert }_{L^2}^2\le C{\Vert u^{ϵ}\Vert }_{L^2}^2.$(14)

Furthermore,

From the a priori estimates, $\left\{u^{ϵ}\right\}$ is uniformly bounded in $H^1\left({ℝ}^2\right)$. By the Rellich-Kondrachov compactness theorem, there exists a subsequence $u^{ϵ}\to u$ strongly in $L^2$ as $ϵ\to 0$. Passing to the limit in Equation (8):

${\partial }_t^{\alpha }u+\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)=0.$(15)

Finally,

To confirm $u$ as a viscosity solution: For test functions $\varphi$, if $u-\varphi$ has a local maximum, we verify:

${\partial }_t^{\alpha }\varphi +\mathscr{H}\left(x_0,\varphi ,\nabla \varphi ,\text{Δ}\varphi \right)\le 0.$(16)

${\partial }_t^{\alpha }\varphi +\mathscr{H}\left(x_0,\varphi ,\nabla \varphi ,\text{Δ}\varphi \right)\ge 0.$(17)

Thus, $u$ is a viscosity solution of the time-fractional HJB equation.

Corollary 3.1 (Uniqueness) The viscosity solution of (77) is unique under the comparison principle:

$u_1\le u_2\text{implies}{u}_1=u_2.$(18)

Proof. Define $w\left(x,t\right)=u_1\left(x,t\right)-u_2\left(x,t\right)$. Then $w$ satisfies:

${\partial }_t^{\alpha }w+\mathscr{H}\left(x,u_1,\nabla u_1,\text{Δ}{u}_1\right)-\mathscr{H}\left(x,u_2,\nabla u_2,\text{Δ}{u}_2\right)=0.$(19)

By the monotonicity of $\mathscr{H}$ with respect to $u$, we have:

$\mathscr{H}\left(x,u_1,\nabla u_1,\text{Δ}{u}_1\right)-\mathscr{H}\left(x,u_2,\nabla u_2,\text{Δ}{u}_2\right)\ge {\mathscr{H}}_u\left(x,u,\nabla u,\text{Δ}u\right)\cdot \left(u_1-u_2\right),$(20)

where ${\mathscr{H}}_u$ is the partial derivative of $\mathscr{H}$ with respect to $u$.

Substitute this inequality into the equation for $w$:

${\partial }_t^{\alpha }w+{\mathscr{H}}_u\left(x,u,\nabla u,\text{Δ}u\right)\cdot w\le 0.$(21)

Assume $w\left(x,t\right)$ achieves its maximum at $\left(x_0,t_0\right)$. At this point, the following conditions hold:

$\nabla w\left(x_0,t_0\right)=0,\text{Δ}w\left(x_0,t_0\right)\le 0.$(22)

From the viscosity inequality for $u_1$ and $u_2$, we have:

${\partial }_t^{\alpha }{u}_1+\mathscr{H}\left(x,u_1,\nabla u_1,\text{Δ}{u}_1\right)\le 0,{\partial }_t^{\alpha }{u}_2+\mathscr{H}\left(x,u_2,\nabla u_2,\text{Δ}{u}_2\right)\ge 0.$(23)

Subtract these two inequalities:

${\partial }_t^{\alpha }w+\left(\mathscr{H}\left(x,u_1,\nabla u_1,\text{Δ}{u}_1\right)-\mathscr{H}\left(x,u_2,\nabla u_2,\text{Δ}{u}_2\right)\right)\le 0.$(24)

By the monotonicity of $\mathscr{H}$, we replace the second term:

${\partial }_t^{\alpha }w+{\mathscr{H}}_u\left(x,u,\nabla u,\text{Δ}u\right)\cdot w\le 0.$(25)

At $\left(x_0,t_0\right)$, where $w$ achieves its maximum:

${\partial }_t^{\alpha }w\left(x_0,t_0\right)\ge 0.$(26)

Thus:

$0\ge {\partial }_t^{\alpha }w\left(x_0,t_0\right)+{\mathscr{H}}_u\left(x,u,\nabla u,\text{Δ}u\right)\cdot w\left(x_0,t_0\right)\ge {\mathscr{H}}_u\left(x,u,\nabla u,\text{Δ}u\right)\cdot w\left(x_0,t_0\right).$(27)

Since ${\mathscr{H}}_u>0$ by assumption on $\mathscr{H}$, it follows that:

$w\left(x_0,t_0\right)\le 0.$(28)

Because $w$ achieves its maximum at $\left(x_0,t_0\right)$, and $w\left(x_0,t_0\right)\le 0$, we conclude:

$w\left(x,t\right)\le 0\forall \left(x,t\right).$(29)

Now repeat the argument for $-w\left(x,t\right)$, where $-w$ achieves its maximum. The same steps show:

$-w\left(x,t\right)\le 0\forall \left(x,t\right).$(30)

Therefore:

$w\left(x,t\right)=0\forall \left(x,t\right).$(31)

Thus:

$u_1\left(x,t\right)=u_2\left(x,t\right).$(32)

Proposition 3.1 (Higher-Order Regularity) If $u_0\left(x,v\right)\in C^{2+\alpha }\left(x,v\right)$, then the viscosity solution $u\left(x,v,t\right)\in C^{2+\alpha }\left(x\right)\cap C^{1+\alpha }\left(t\right)$.

Proof. Let first determine the Spatial Regularity Estimate (x-derivatives)

The time-fractional Hamilton-Jacobi-Bellman equation is given by:

${\partial }_t^{\alpha }u+\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)=0,$(33)

where:

$\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)=\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}u+\kappa \left(\theta -v\right){\partial }_{v}u+\lambda \left(\mathbb{E}\left[u\left(x+\xi ,v\right)\right]-u\left(x,v\right)\right).$

Differentiate (33) twice with respect to $x$:

${\partial }_t^{\alpha }\left({\partial }_{xx}u\right)+{\mathscr{H}}_{xx}=0,$(34)

where:

${\mathscr{H}}_{xx}=\frac{{\sigma }^{2}v{x}^2}{2}{\partial }_{xxxx}u+{\sigma }^{2}vx{\partial }_{xxx}u+\kappa \left(\theta -v\right){\partial }_{xxv}u.$

The leading order term ${\partial }_{xx}u$ satisfies the elliptic regularity estimate:

${\Vert {\partial }_{xx}u\Vert }_{L^p}\le C\left({\Vert u\Vert }_{L^p}+{\Vert \nabla u\Vert }_{L^p}+{\Vert \text{Δ}u\Vert }_{L^p}\right),p>1.$(35)

Using the Sobolev embedding theorem:

$W^{2,p}\subset C^{2-\frac{n}{p}}\text{for}p>n,$

where $n=1$ (log-price $x$), choose $p>1$ large enough to satisfy:

$\alpha =1-\frac{1}{p}.$

Thus:

${\partial }_{xx}u\in C^{\alpha }\left(x\right)\Rightarrow u\in C^{2+\alpha }\left(x\right).$

Next, let determine the Temporal Regularity Estimate (t-derivatives)

The Caputo fractional derivative is expressed using the Duhamel principle:

${\partial }_t^{\alpha }u=\frac{1}{\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\alpha }\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)\text{d}s}.$(36)

Since $\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)\in C^{\alpha }\left(x\right)$ from Step 1:

$\left|\mathscr{H}\left(x,u,\nabla u,\text{Δ}u\right)\right|\le C\left(1+\left|\nabla u\right|+\left|\text{Δ}u\right|\right).$

The fractional temporal estimate for the Caputo derivative gives:

${\Vert {\partial }_t^{\alpha }u\Vert }_{L^{\infty }}\le C{\Vert \mathscr{H}\Vert }_{C^{\alpha }}.$(37)

From (36) and the regularity of $\mathscr{H}$, we obtain:

$u\in C^{1+\alpha }\left(t\right).$

Combining the spatial regularity ${\partial }_{xx}u\in C^{\alpha }\left(x\right)$ and temporal regularity ${\partial }_t^{\alpha }u\in C^{\alpha }\left(t\right)$, we conclude:

$u\in C^{2+\alpha }\left(x\right)\cap C^{1+\alpha }\left(t\right).$

We extend the system to a nonlinear regime, where the Denjoy-type theorem guarantees the continuity of invariant measures. The topological entropy $h_{\text{top}}$ is computed as:

$h_{\text{top}}=\underset{ϵ\to 0}{\text{lim}}\frac{\log N\left(ϵ,T\right)}{T},$(38)

Quantifying the system’s complexity and market unpredictability.

Theorem 3.2 For the nonlinear system:

${\partial }_t^{\alpha }u+\mathscr{H}\left(u,\nabla u,\text{Δ}u\right)=0,$(39)

There exists a continuous invariant measure $\mu$, and the system satisfies the Denjoy-type theorem.

Proof. The equation is:

${\partial }_t^{\alpha }u+\mathscr{H}\left(u,\nabla u,\text{Δ}u\right)=0,\left(x,t\right)\in {ℝ}^n\times \left[0,T\right],$(40)

where:

$\mathscr{H}\left(u,\nabla u,\text{Δ}u\right)=\frac{1}{2}v{\sigma }^2{x}^2{\partial }_{xx}u+\kappa \left(\theta -v\right){\partial }_{v}u+\lambda \mathbb{E}\left[u\left(x+\xi ,v\right)\right]-\lambda u\left(x,v\right).$(41)

The fractional derivative is in the Caputo sense:

${\partial }_t^{\alpha }u\left(x,t\right)=\frac{1}{\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t\frac{{\partial }_{s}u\left(x,s\right)}{{\left(t-s\right)}^{\alpha }}\text{d}s}.$(42)

Next, the regularize the problem:

$ϵ\text{Δ}{u}^{ϵ}+{\partial }_t^{\alpha }{u}^{ϵ}+\mathscr{H}\left(u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)=0,ϵ>0.$(43)

Multiply through by $u^{ϵ}$ and integrate over a domain $\Omega \subset {ℝ}^n$:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}{\partial }_t^{\alpha }{u}^{ϵ}\text{d}x}+{\displaystyle {\int }_{\Omega }{u}^{ϵ}\mathscr{H}\left(u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)\text{d}x}+ϵ{\displaystyle {\int }_{\Omega }{u}^{ϵ}\text{Δ}{u}^{ϵ}\text{d}x}=0.$(44)

Moreover, substituting the Caputo derivative:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}{\partial }_t^{\alpha }{u}^{ϵ}\text{d}x}=\frac{1}{\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }\frac{u^{ϵ}{\partial }_s{u}^{ϵ}}{{\left(t-s\right)}^{\alpha }}\text{d}x\text{d}s}}.$(45)

Using Fubini’s theorem:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}{\partial }_t^{\alpha }{u}^{ϵ}\text{d}x}=\frac{1}{2\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t\frac{\partial }{\partial s}}\left({\displaystyle {\int }_{\Omega }{\left|u^{ϵ}\right|}^2\text{d}x}\right)\frac{1}{{\left(t-s\right)}^{\alpha }}\text{d}s.$(46)

Else, let expand the Hamiltonian term:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}\mathscr{H}\left(u^{ϵ},\nabla u^{ϵ},\text{Δ}{u}^{ϵ}\right)\text{d}x}={\displaystyle {\int }_{\Omega }{u}^{ϵ}\left(\frac{1}{2}v{\sigma }^2{x}^2{\partial }_{xx}{u}^{ϵ}\right)\text{d}x}+{\displaystyle {\int }_{\Omega }{u}^{ϵ}\kappa \left(\theta -v\right){\partial }_v{u}^{ϵ}\text{d}x}.$(47)

Integrate the second-order term by parts:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}v{\sigma }^2{x}^2{\partial }_{xx}{u}^{ϵ}\text{d}x}=-{\displaystyle {\int }_{\Omega }v{\sigma }^2{x}^2{\left|{\partial }_x{u}^{ϵ}\right|}^2\text{d}x}.$(48)

Similarly, for the mean-reversion term:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}\kappa \left(\theta -v\right){\partial }_v{u}^{ϵ}\text{d}x}=-\frac{\kappa }{2}{\displaystyle {\int }_{\Omega }\left(\theta -v\right){\left|{\partial }_v{u}^{ϵ}\right|}^2\text{d}x}.$(49)

Finally, the jump term becomes:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}}\left(\lambda \mathbb{E}\left[u^{ϵ}\left(x+\xi ,v\right)\right]-\lambda u^{ϵ}\left(x,v\right)\right)\text{d}x.$(50)

Taking the expectation:

${\displaystyle {\int }_{\Omega }{u}^{ϵ}\mathbb{E}\left[u^{ϵ}\left(x+\xi ,v\right)\right]\text{d}x}={\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{ℝ}{u}^{ϵ}\left(x,v\right)u^{ϵ}\left(x+\xi ,v\right)f_{\xi }\left(\xi \right)\text{d}\xi \text{d}x}}.$(51)

Combining all terms, we get:

$\begin{array}{l}\frac{1}{2\text{Γ}\left(1-\alpha \right)}{\displaystyle {\int }_0^t\frac{\partial }{\partial s}}\left({\displaystyle {\int }_{\Omega }{\left|u^{ϵ}\right|}^2\text{d}x}\right)\frac{1}{{\left(t-s\right)}^{\alpha }}\text{d}s+ϵ{\displaystyle {\int }_{\Omega }{\left|\nabla u^{ϵ}\right|}^2\text{d}x}-{\displaystyle {\int }_{\Omega }v{\sigma }^2{x}^2{\left|{\partial }_x{u}^{ϵ}\right|}^2\text{d}x}\\ -\frac{\kappa }{2}{\displaystyle {\int }_{\Omega }\left(\theta -v\right){\left|{\partial }_v{u}^{ϵ}\right|}^2\text{d}x}+\lambda {\displaystyle {\int }_{\Omega }{u}^{ϵ}\left(\mathbb{E}\left[u^{ϵ}\left(x+\xi ,v\right)\right]-u^{ϵ}\right)\text{d}x}=0.\end{array}$(52)

As $ϵ\to 0$, compactness in $L^2$ implies:

$\underset{ϵ\to 0}{\text{lim}}{\displaystyle {\int }_{\Omega }{\left|\nabla u^{ϵ}\right|}^2\text{d}x}<\infty .$(53)

Therefore, let define the invariant measure $\mu$ as:

${\displaystyle {\int }_{\Omega }\mathscr{H}\left(u,\nabla u,\text{Δ}u\right)\text{d}\mu }=0.$(54)

Theorem 3.3 (Topological Entropy) The topological entropy $h_{top}$ is given by:

$h_{\text{top}}=\underset{ϵ\to 0}{\text{lim}}\frac{\log N\left(ϵ,T\right)}{T},$

where $N\left(ϵ,T\right)$ is the minimum number of $ϵ$-balls required to cover all trajectories over time $T$.

Proof. Let define progressively the following: Let ${\left\{x_t\right\}}_{t\in \left[0,T\right]}$ be a solution of the time-fractional Heston model:

${\partial }_t^{\alpha }{x}_t=f\left(x_t\right)+J_t+I_t,0<\alpha <1.$(55)

Define the metric on the space of trajectories $X_T$ as:

$d_T\left(x,y\right)=\underset{0\le t\le T}{\text{sup}}\Vert x_t-y_t\Vert .$(56)

The solution $x_t$ satisfies:

$x_t=x_0+\frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^{\infty }{\left(t-s\right)}^{\alpha -1}}\left[f\left(x_s\right)+J_s+I_s\right]\text{d}s.$(57)

For two trajectories $x_t$ and $y_t$, subtract their governing equations:

${\partial }_t^{\alpha }\left(x_t-y_t\right)=f\left(x_t\right)-f\left(y_t\right)+\left(J_t-{J^{\prime }}_t\right)+\left(I_t-{I^{\prime }}_t\right).$(58)

Integrating in the Caputo sense, we have:

$x_t-y_t=\frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}}\left[f\left(x_s\right)-f\left(y_s\right)\right]\text{d}s+\text{Δ}{J}_t+\text{Δ}{I}_t.$(59)

Assume $f$ is Lipschitz continuous with constant $L>0$:

$\Vert f\left(x\right)-f\left(y\right)\Vert \le L\Vert x-y\Vert .$(60)

Then:

$\Vert x_t-y_t\Vert \le \frac{L}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}\Vert x_s-y_s\Vert \text{d}s}+\Vert \text{Δ}{J}_t\Vert +\Vert \text{Δ}{I}_t\Vert .$(61)

Applying the fractional Grönwall inequality, we obtain:

$\Vert x_t-y_t\Vert \le \left(\Vert x_0-y_0\Vert +\Vert \text{Δ}{J}_t\Vert +\Vert \text{Δ}{I}_t\Vert \right)E_{\alpha }\left(L{t}^{\alpha }\right),$(62)

where $E_{\alpha }$ is the Mittag-Leffler function:

$E_{\alpha }\left(z\right)=\underset{k=0}{\overset{\infty }{{\displaystyle \sum }}}\frac{z^k}{\text{Γ}\left(\alpha k+1\right)}.$(63)

For $ϵ>0$, the number of $ϵ$-balls required to cover $X_T$ satisfies:

$N\left(ϵ,T\right)~\text{exp}\left(C{T}^{\alpha }\right),$(64)

where $C$ depends on $L$, $\Vert J_t\Vert$, and $\Vert I_t\Vert$. Substituting $N\left(ϵ,T\right)$ into the entropy definition:

$h_{\text{top}}=\underset{ϵ\to 0}{\text{lim}}\frac{\log N\left(ϵ,T\right)}{T}.$(65)

Using the asymptotic form of $N\left(ϵ,T\right)$, we find:

$h_{\text{top}}=\underset{ϵ\to 0}{\text{lim}}\frac{\text{log}\left[\text{exp}\left(C{T}^{\alpha }\right)\right]}{T}.$(66)

Simplifying:

$h_{\text{top}}=C{T}^{\alpha -1}.$(67)

The topological entropy of the time-fractional Heston model with jumps and inertia is given by:

$h_{\text{top}}=C{T}^{\alpha -1},0<\alpha <1.$(68)

Remark 3.1 (Application of Topological entropy to real option assessment and risk management) Consider Equation (1) and assume the trajectory space be equipped with the metric:

$d_T\left(u_1,u_2\right):=\underset{0\le t\le T}{\text{sup}}\Vert u_1\left(x,v,t\right)-u_2\left(x,v,t\right)\Vert .$(69)

Assuming Lipschitz continuity of $\mathscr{H}$, we easily get

$\Vert u_1\left(x,v,t\right)-u_2\left(x,v,t\right)\Vert \le C_0{E}_{\alpha }\left(L{t}^{\alpha }\right),$(70)

where $E_{\alpha }$ is the Mittag-Leffler function, and $C_0:=\Vert u_1\left(x,v,0\right)-u_2\left(x,v,0\right)\Vert +\Vert \text{Δ}J\Vert +\Vert \text{Δ}I\Vert$.

The minimal number of $ϵ$-balls covering the solution space satisfies

$N\left(ϵ,T\right)~\text{exp}\left(C{T}^{\alpha }\right),C=C\left(\sigma ,\lambda ,\Vert \text{Δ}J\Vert ,\Vert \text{Δ}I\Vert \right),$(71)

Yielding the topological entropy

$h_{\text{top}}:=\underset{ϵ\to 0}{\text{lim}}\frac{\log N\left(ϵ,T\right)}{T}=C{T}^{\alpha -1}.$(72)

We define the entropy-based risk functional

$ℛ\left(T\right):=h_{\text{top}}=C{T}^{\alpha -1}.$(73)

Thus, the entropy sensitivity of the solution is

$\frac{\partial u}{\partial ℛ}=\frac{\partial u}{\partial T}\cdot \frac{\text{d}ℛ}{\text{d}T}=\frac{\partial u}{\partial T}\cdot C\left(\alpha -1\right)T^{\alpha -2}.$(74)

taking the limits

$\underset{\alpha \to 1^{-}}{\text{lim}}ℛ\left(T\right)=C,\underset{\alpha \to 0^{+}}{\text{lim}}ℛ\left(T\right)=0,\underset{T\to 0^{+}}{\text{lim}}\frac{\text{d}ℛ}{\text{d}T}=\infty .$(75)

Therefore, in the evaluation of real options and in risk management, the entropy $ℛ\left(T\right)$ acts as a non-linear, memory-sensitive quantifier of information volatility. It also captures both the diversity and the instability of future scenarios governed by the underlying fractional dynamics of (1).

Recall that the time-fractional Heston model with jumps and inertia is given by:

${\partial }_t^{\alpha }u+{ℒ}_{H}u+\mathcal{J}\left[u\right]+I\left(u\right)=0,0<\alpha <1,$(76)

where:

${ℒ}_{H}u=\frac{1}{2}v{\sigma }^2{x}^2{\partial }_{xx}u+\kappa \left(\theta -v\right){\partial }_{v}u-ru,$

$\mathcal{J}\left[u\right]=\lambda {\displaystyle {\int }_{ℝ}\left(u\left(x+\xi ,v,t\right)-u\left(x,v,t\right)\right)\text{d}\mu }\left(\xi \right),$

and $I\left(u\right)$ models inertia.

The Caputo time-fractional derivative is approximated using the Grünwald-Letnikov scheme:

${\partial }_t^{\alpha }u\left(t_n\right)\approx \frac{1}{{\left(\text{Δ}t\right)}^{\alpha }}\underset{k=0}{\overset{n}{{\displaystyle \sum }}}{\omega }_k\left[u^{n-k}-u^{n-k-1}\right],{\omega }_k={\left(-1\right)}^k\left(\begin{array}{l}\alpha \hfill \\ k\hfill \end{array}\right).$(77)

The finite difference approximation of (76) becomes:

$\frac{1}{{\left(\text{Δ}t\right)}^{\alpha }}\underset{k=0}{\overset{n}{{\displaystyle \sum }}}{\omega }_k\left[u^{n-k}-u^{n-k-1}\right]+{ℒ}_H{u}^n+\mathcal{J}\left[u^n\right]+I\left(u^n\right)=0.$(78)

Reorganizing for $u^n$, we obtain:

$u^n={\left(1+\frac{{\omega }_0}{{\left(\text{Δ}t\right)}^{\alpha }}\right)}^{-1}\left[-\underset{k=1}{\overset{n}{{\displaystyle \sum }}}\frac{{\omega }_k}{{\left(\text{Δ}t\right)}^{\alpha }}{u}^{n-k}-{ℒ}_H{u}^n-\mathcal{J}\left[u^n\right]-I\left(u^n\right)\right].$(79)

Theorem 3.4 (Backward Martingale Convergence) Let ${\left\{u^n\right\}}_{n\ge 0}$ be the finite difference approximation to the time-fractional Heston model (76). Then:

$\underset{n\to \infty }{\text{lim}}{u}^n\left(x,v\right)=u^{\text{*}}\left(x,v\right)\text{a}.\text{s}.\text{and}\text{in}{L}^2\text{-norm},$

where $u^{\text{*}}$ is the unique viscosity solution of the time-fractional Heston equation.

Proof. Define:

$R^n=\left({ℒ}_H{u}^n+\mathcal{J}\left[u^n\right]+I\left(u^n\right)\right)\text{Δ}t.$

From (79), we write:

$u^n={\mathbb{E}}^n\left[u^{n+1}+R^n\right].$(80)

Conditioning on the filtration ${ℱ}_n=\sigma \left(u^0,u^1,\cdots ,u^n\right)$, we have:

${\mathbb{E}}^n\left[u^{n+1}\right]=u^n.$(81)

Thus, $\left\{u^n\right\}$ is a backward martingale.

Next, taking the square norm:

${\Vert u^n\Vert }_{L^2}^2\le {\mathbb{E}}^n\left[{\Vert u^{n+1}\Vert }_{L^2}^2\right]+C\text{Δ}{t}^2{\Vert R^n\Vert }_{L^2}^2.$

Summing from $n=0$ to $N$:

$\underset{n=0}{\overset{N}{{\displaystyle \sum }}}{\Vert u^n\Vert }_{L^2}^2\le {\Vert u^0\Vert }_{L^2}^2+C\text{Δ}t\underset{n=0}{\overset{N}{{\displaystyle \sum }}}{\Vert R^n\Vert }_{L^2}^2.$

Since $R^n$ is uniformly bounded:

${\Vert R^n\Vert }_{L^2}^2\le C\forall n.$

Thus:

${\Vert u^n\Vert }_{L^2}^2\le C.$

Let discuss on the $L^2$-Convergence Subtract $u^n$ from $u^{n+1}$:

${\Vert u^{n+1}-u^n\Vert }_{L^2}^2\le C\text{Δ}{t}^2{\Vert R^n\Vert }_{L^2}^2.$(82)

Summing over $n$:

$\underset{n=0}{\overset{N}{{\displaystyle \sum }}}{\Vert u^{n+1}-u^n\Vert }_{L^2}^2\to 0\text{as}N\to \infty .$

Thus $u^n$ is Cauchy in $L^2$, and there exists $u^{\text{*}}$ such that:

$\underset{n\to \infty }{\text{lim}}{\Vert u^n-u^{\text{*}}\Vert }_{L^2}=0.$

Again, from Doob's theorem, the backward martingale $\left\{u^n\right\}$ converges almost surely:

$\underset{n\to \infty }{\text{lim}}{u}^n=u^{\text{*}}\text{a}.\text{s}.$

Substitute $u^{\text{*}}$ into the original Equation (76) and verify that it satisfies:

${\partial }_t^{\alpha }{u}^{\text{*}}+{ℒ}_H{u}^{\text{*}}+\mathcal{J}\left[u^{\text{*}}\right]+I\left(u^{\text{*}}\right)=0.$(83)

The uniqueness of viscosity solutions guarantees that $u^{\text{*}}$ is the unique limit.

Theorem 3.5 Let $V\left(x,v,t\right)$ be the value function of the following two-player zero-sum game:

$\underset{{\tau }_1}{\text{max}}\underset{{\tau }_2}{\text{min}}\mathbb{E}\left[P\left({\tau }_1,{\tau }_2\right)\right],$

where ${\tau }_1$ and ${\tau }_2$ are stopping times. There exists a Nash equilibrium solution $V\left(x,v,t\right)$ satisfying Equation (1).

Proof. For Player 1, who seeks to maximize the payoff function:

$V_1\left(x,v,t\right)=\underset{{\tau }_1}{\text{sup}}\mathbb{E}\left[P\left(x_{{\tau }_1},v_{{\tau }_1},{\tau }_1\right)|x_t=x,v_t=v\right],$

the associated HJB equation is:

${\partial }_t^{\alpha }{V}_1+\underset{a_1\in {\mathcal{A}}_1}{\text{sup}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}{V}_1+\kappa \left(\theta -v\right){\partial }_v{V}_1+\lambda \left(\mathbb{E}\left[V_1\left(x+\xi ,v\right)\right]-V_1\right)\right\}=0.$(84)

For Player 2, who seeks to minimize the same payoff:

$V_2\left(x,v,t\right)=\underset{{\tau }_2}{\text{inf}}\mathbb{E}\left[P\left(x_{{\tau }_2},v_{{\tau }_2},{\tau }_2\right)|x_t=x,v_t=v\right],$

the corresponding HJB equation is:

${\partial }_t^{\alpha }{V}_2+\underset{a_2\in {\mathcal{A}}_2}{\text{inf}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}{V}_2+\kappa \left(\theta -v\right){\partial }_v{V}_2+\lambda \left(\mathbb{E}\left[V_2\left(x+\xi ,v\right)\right]-V_2\right)\right\}=0.$(85)

Assume that a single value function $V\left(x,v,t\right)$ exists such that:

$V\left(x,v,t\right)=V_1\left(x,v,t\right)=V_2\left(x,v,t\right).$

Substitute $V$ into the HJB equations, resulting in:

${\partial }_t^{\alpha }V+\underset{a_1\in {\mathcal{A}}_1}{\text{sup}}\underset{a_2\in {\mathcal{A}}_2}{\text{inf}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}V+\kappa \left(\theta -v\right){\partial }_{v}V+\lambda \left(\mathbb{E}\left[V\left(x+\xi ,v\right)\right]-V\right)\right\}=0.$(86)

The supremum and infimum conditions for controls $a_1^{\text{*}}$ and $a_2^{\text{*}}$ are given by:

${\partial }_{a_1}ℒ\left[V\right]=0\text{and}{\partial }_{a_2}ℒ\left[V\right]=0,$

where the operator $ℒ$ is defined as:

$ℒ\left[V\right]=\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}V+\kappa \left(\theta -v\right){\partial }_{v}V+\lambda \left(\mathbb{E}\left[V\left(x+\xi ,v\right)\right]-V\right).$(87)

Finally, define a fixed-point operator $\mathcal{T}$ acting on $V$:

$\mathcal{T}\left[V\right]=V+\text{Δ}tℒ\left[V\right].$(88)

To prove existence:

By Banach’s Fixed-Point Theorem, if $\mathcal{T}$ is a contraction mapping:

$\Vert \mathcal{T}\left[V\right]-\mathcal{T}\left[W\right]\Vert \le C\Vert V-W\Vert ,C<1,$

then $V_n\to V^{\text{*}}$ as $n\to \infty$, where $V^{\text{*}}$ satisfies:

$\mathcal{T}\left[V^{\text{*}}\right]=V^{\text{*}}.$

At equilibrium, $V\left(x,v,t\right)$ satisfies:

$\mathbb{E}\left[V\left(x_{{\tau }_1},v_{{\tau }_1},{\tau }_1\right)\right]\le V\left(x,v,t\right),\left(\text{sub-martingale}\right)$(89)

$\mathbb{E}\left[V\left(x_{{\tau }_2},v_{{\tau }_2},{\tau }_2\right)\right]\ge V\left(x,v,t\right),\left(\text{super-martingale}\right).$(90)

Thus, the stopping times ${\tau }_1^{\text{*}}$ and ${\tau }_2^{\text{*}}$ achieve the equilibrium payoff:

$\underset{{\tau }_1}{\max }\underset{{\tau }_2}{\min }\mathbb{E}\left[P\left({\tau }_1,{\tau }_2\right)\right]=\mathbb{E}\left[P\left({\tau }_1^{*},{\tau }_2^{*}\right)\right].$(91)

An attempt to extend the above theorem to more complex game structures and multiple players is given by the result below:

Corollary 3.2 (Multi-Player Fractional Game Equilibrium) Let ${\left\{{\tau }_i\right\}}_{i=1}^N\subset {\mathcal{T}}_i$ be stopping times in an $N$-player zero-sum stochastic game with payoff functions

$J_i\left({\tau }_1,\cdots ,{\tau }_N\right)=\mathbb{E}\left[P_i\left(x_{{\tau }_1},\cdots ,x_{{\tau }_N};v_{{\tau }_1},\cdots ,v_{{\tau }_N};{\tau }_1,\cdots ,{\tau }_N\right)\right],$(92)

and assume ${\displaystyle {\sum }_{i=1}^N{J}_i}=0$. Then there exists an equilibrium profile $\left({\tau }_1^{\text{*}},\cdots ,{\tau }_N^{\text{*}}\right)\in {\mathcal{T}}_1\times \cdots \times {\mathcal{T}}_N$, and a common value function $V\left(x,v,t\right)$ solving the equation

${\partial }_t^{\alpha }V+\underset{a_1\in {\mathcal{A}}_1}{\text{sup}}\underset{a_2\in {\mathcal{A}}_2}{\text{inf}}\cdots \underset{a_N\in {\mathcal{A}}_N}{\text{sup}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}V+\kappa \left(\theta -v\right){\partial }_{v}V+\lambda \left(\mathbb{E}\left[V\left(x+\xi ,v\right)\right]-V\right)\right\}=0.$ (93)

Sketch of Proof. Define individual value functions:

$V_i\left(x,v,t\right):=\underset{{\tau }_i\in {\mathcal{T}}_i}{\text{sup}}\underset{{\tau }_{-i}\in {\mathcal{T}}_{-i}}{\text{inf}}\mathbb{E}\left[P_i\left({\tau }_1,\cdots ,{\tau }_N\right)|x_t=x,v_t=v\right],\forall i=1,\cdots ,N.$(94)

First, assume symmetry:

$V_i\left(x,v,t\right)=V\left(x,v,t\right),\forall i.$(95)

Then, the HJB equation per player is given by:

${\partial }_t^{\alpha }V+H_i\left(x,v,\nabla V,{\nabla }^{2}V\right)=0,$(96)

$H_i:=\underset{a_i\in {\mathcal{A}}_i}{\text{sup}}\underset{a_{-i}\in {\mathcal{A}}_{-i}}{\text{inf}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}V+\kappa \left(\theta -v\right){\partial }_{v}V+\lambda \left(\mathbb{E}\left[V\left(x+\xi ,v\right)\right]-V\right)\right\}.$(97)

The compact equation is given by:

${\partial }_t^{\alpha }V+\underset{a_1\in {\mathcal{A}}_1}{\text{sup}}\underset{a_2\in {\mathcal{A}}_2}{\text{inf}}\cdots \underset{a_N\in {\mathcal{A}}_N}{\text{sup}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}V+\kappa \left(\theta -v\right){\partial }_{v}V+\lambda \left(\mathbb{E}\left[V\left(x+\xi ,v\right)\right]-V\right)\right\}=0.$ (98)

Secondly, consider the following fixed-point operator

$\mathcal{T}\left[V\right]:=V+\text{Δ}t\cdot \mathscr{H}\left[V\right],$(99)

$\mathscr{H}\left[V\right]:=\underset{a_1\in {\mathcal{A}}_1}{\text{sup}}\underset{a_2\in {\mathcal{A}}_2}{\text{inf}}\cdots \underset{a_N\in {\mathcal{A}}_N}{\text{sup}}\left\{\frac{1}{2}{\sigma }^{2}v{x}^2{\partial }_{xx}V+\kappa \left(\theta -v\right){\partial }_{v}V+\lambda \left(\mathbb{E}\left[V\left(x+\xi ,v\right)\right]-V\right)\right\}.$ (100)

Using contraction, we have the below property

$\Vert \mathcal{T}\left[V\right]-\mathcal{T}\left[W\right]\Vert \le \left(1+\text{Δ}t\cdot L\right)\Vert V-W\Vert ,$(101)

Taking $\text{Δ}t$ such that $C:=1+\text{Δ}t\cdot L<1\Rightarrow \mathcal{T}$ is a contraction.

Therefore,

$\exists !V^{\text{*}}\in \mathcal{X},\mathcal{T}\left[V^{\text{*}}\right]=V^{\text{*}}.$(102)

By the following sub and supermartingale definition

$\mathbb{E}\left[V^{\text{*}}\left(x_{{\tau }_i},v_{{\tau }_i},{\tau }_i\right)\right]\le V^{\text{*}}\left(x,v,t\right),\mathbb{E}\left[V^{\text{*}}\left(x_{{\tau }_j},v_{{\tau }_j},{\tau }_j\right)\right]\ge V^{\text{*}}\left(x,v,t\right),\forall i,j.$(103)

We ultimately get the equilibrium payoff

$\underset{{\tau }_1}{\text{sup}}\underset{{\tau }_2}{\text{inf}}\cdots \underset{{\tau }_N}{\text{sup}}\mathbb{E}\left[P\left({\tau }_1,\cdots ,{\tau }_N\right)\right]=\mathbb{E}\left[P\left({\tau }_1^{\text{*}},\cdots ,{\tau }_N^{\text{*}}\right)\right].$(104)

This study proposes an integrated methodology for pricing real options utilizing the time-fractional Heston model with jumps and inertia. Both continuous and discrete versions overcomes the flaws of classical stochastic volatility models, mainly their inability to incorporate: Memory effects represented by fractional derivatives, Sudden jumps accounted for by jump processes and inertia modeled as resistance to price changes. The combination of these features into a single unified model enables a more accurate representation of complex financial market dynamics. Specifically:

We presented an approach for assessing real options under the time-fractional Heston model with jumps and inertia (1) which opens several avenues for further research such as the fact to calibrate the model parameters to multiple assets with correlated volatilities, including the fractional order $\alpha$, jump intensity $\lambda$, and inertia terms, using real financial data. The necessity to incorporate machine learning techniques can enhance parameter estimation and volatility prediction. Moreover, investigate the relationship between topological entropy and systemic risk measures can enable to develop early warning systems based on entropy-based indicators to monitor financial market stability. Finally, extending the model to specific industries such as energy, infrastructure, and real estate markets, where investment decisions involve long-term uncertainty, inertia, and memory effects can strengthen the theoretical underpinning and broaden the scope of its application, guaranteeing the model remains relevant under volatile and uncertain market conditions.