Optimal control theory, a core area of applied mathematics, focuses on determining strategies that optimize a performance criterion under system dynamics. The two main approaches are dynamic programming, introduced by Bellman [1], and the maximum principle, developed by Pontryagin et al. [2]. This study examines an optimal control problem under a jump-uncertain stochastic framework, using dynamic programming to derive the value function and connect it with the uncertain stochastic maximum principle. As an application, we analyze a portfolio selection problem in a financial market with jump uncertainty, extending Chirima et al. [3], who considered models without jumps.

Dynamic programming solves optimal control problems recursively through a value function that satisfies the Hamilton–Jacobi–Bellman (HJB) equation. In deterministic systems, this yields a first-order partial differential equation, while in stochastic settings with Brownian motion, it becomes a second-order non-linear partial differential equation. Liu's uncertainty theory adapts dynamic programming using belief-based measures, resulting in uncertain differential equations (UDEs) and a modified HJB equation with the Liu expected value. In jump-diffusion models, the HJB includes an integro-differential term that accounts for sudden events, offering more realistic dynamics.

Dynamic programming has been applied in many settings. Bellman's work [1] laid the foundation for deterministic control; stochastic extensions were developed by Kushner [4] and Fleming and Rishel [5]. Liu [6] extended the framework to uncertain systems, and Yong and Zhou [7] unified dynamic programming and maximum priciple for stochastic control. Øksendal and Sulem [8] applied dynamic programming to jump-diffusions in finance, while Chirima et al. [3] studied uncertain stochastic systems without jumps.

The maximum principle provides necessary conditions for optimality by introducing adjoint (costate) variables and requiring the maximization of the Hamiltonian function. In deterministic settings, this formulation typically leads to a two-point boundary value problem. In stochastic control, Peng [9] developed a backward stochastic differential equation (BSDE) formulation, which was subsequently extended by Yong and Zhou [7] to broader stochastic contexts. For systems governed by jump-diffusion dynamics, additional terms are incorporated to represent the effects of Poisson, Lévy, or uncertain jumps. In uncertain systems, the principle has been adapted to uncertain differential equations, featuring modified adjoint processes and transversality conditions [10]. Within uncertain stochastic frameworks, uncertain stochastic adjoint equations have been introduced to model both randomness and belief-based uncertainty [11].

Since its introduction by Pontryagin et al. [2], the maximum principle has evolved into a versatile tool for optimal control across diverse settings. Peng's BSDE-based formulation [9] and the extensions by Yong and Zhou [7] advanced its applicability in stochastic environments. Ge and Zhu [10] broadened its scope to uncertain systems, while Chikodza and Hlahla [11] applied it within uncertain stochastic frameworks. In the realm of jump-diffusion control, Framstad et al. [12] demonstrated its effectiveness and established its connection with dynamic programming. These developments collectively underscore the maximum principle's flexibility and its unifying role in addressing optimal control problems across a wide range of mathematical models.

A fundamental link exists between the maximum principle and dynamic programming, the two main approaches in optimal control theory. Dynamic programming characterizes the optimal value function through the Hamilton–Jacobi–Bellman (HJB) equation. When the value function is sufficiently smooth, maximum principle adjoints can be expressed as gradients of the value function, showing consistency between the two methods. Yong and Zhou [7] established this equivalence in stochastic systems, and Framstad et al. [12] extended it to jump-diffusion models, affirming its validity even with discontinuities.

Financial markets often experience sudden, unpredictable events—known as jumps—that cause sharp changes in asset prices. These deviations differ from the continuous variations captured by Brownian models. For instance, Brent crude rose over 13 percent intraday on 13 June 2025 following Israeli airstrikes on Iranian facilities and climbed further after renewed missile attacks [13–15]. Such movements underscore the need to model jumps explicitly.

Ignoring jumps can misrepresent risk and lead to inaccurate pricing, underestimated Value-at-Risk, and suboptimal investment decisions. Models that incorporate jump processes—such as Poisson, Lévy, or V-uncertain terms—better capture market realities and improve portfolio and hedging strategies. Thus, jump modeling is essential for robust financial decision-making.

Liu's uncertainty theory, introduced in 2007 [16], models belief-based uncertainty that arises from subjective judgment rather than randomness. Unlike probability theory, which relies on frequencies, Liu uncertainty theory uses axiomatic uncertain measures to define uncertain variables and processes, such as the canonical Liu process and uncertain differential equations. It is especially useful where probabilities are unreliable or unknown, such as expert-based financial forecasts.

In this framework, Zhu [17] applied dynamic programming to uncertain optimal control and solved a portfolio selection problem. Deng and Zhu [18] addressed uncertain systems with jumps driven by both Liu and V-jump processes, applying dynamic programming to pension fund control. Zhu [19] later introduced optimistic and expected value-based models for uncertain dynamic systems. These studies highlight Liu uncertainty theory's role in addressing ambiguity beyond classical stochastic tools.

Uncertain stochastic systems combine randomness and belief-based uncertainty, capturing dynamics influenced by both Brownian motion and Liu-uncertain processes. This hybrid framework is effective in contexts where some variables are stochastic, while others reflect subjective expectations. Researchers have extended both dynamic programming and maximum principle to such systems, developed Itô–Liu calculus, and solved problems in portfolio optimization, inventory control, and differential games.

Fei [20] studied uncertain stochastic systems with Markov switching, derived a generalized HJB equation, and solved optimal consumption and investment problems. In related work, Fei [21] proved the existence and uniqueness of backward uncertain stochastic differential equations (BUSDEs). More recently, Chirima et al. [3] used dynamic programming to address uncertain stochastic control without jumps in portfolio models, while Hlahla and Chikodza [11] developed a maximum principle involving both Brownian and Liu-uncertain drivers. These contributions show that Liu uncertain stochastic systems offer a practical and flexible framework for modeling decisions under both randomness and human ambiguity.

Portfolio selection involves deciding how to allocate wealth across financial assets to balance expected return and risk. The goal is to either maximize return for a given risk level or minimize risk for a target return. For example, investing equally in a risk-free bond yielding 2% and a stock expected to return 8 percent yields an average return of 5% with lower risk than full stock investment. By adjusting asset proportions, investors create portfolios offering different risk-return combinations, forming the efficient frontier–portfolios that optimize return for each risk level.

Harry Markowitz [22] established the foundation of portfolio theory with the mean–variance optimization model, showing that diversification reduces risk without lowering expected return. Merton [23] later extended this to continuous time, using stochastic calculus to derive dynamic investment strategies. His model showed that, under certain assumptions, optimal investment proportions depend on market parameters and risk preferences, not current wealth. Together, Markowitz and Merton laid the foundation for modern quantitative finance and the analysis of optimal investment under uncertainty.

More recent research addresses complex uncertainties. Deterministic models assume known returns, while stochastic models use processes such as Brownian motion to reflect randomness. Framstad et al. [12] included jump-diffusion to capture price shocks. In contrast, Liu's uncertainty theory models belief-based uncertainty with uncertain variables. Zhu [17] applied uncertain differential equations to portfolio problems under this framework. Hybrid models now combine stochastic and uncertain elements. Chirima et al. [3] used dynamic programming for portfolio selection in a setting with both randomness and belief-based uncertainty but no jumps. These advances reflect a growing need for models that integrate stochastic processes, Liu uncertainty, and jump components in financial decision-making.

The central contribution of this article is the study of an optimal control problem within a jump–uncertain stochastic framework using the dynamic programming approach. First, we establish an existence and uniqueness theorem for jump–uncertain stochastic differential equations, thereby extending the results of Chen and Liu [24], who proved a similar result in a Liu–uncertain framework without jumps, and Chirima et al. [25], who established it in an uncertain–stochastic setting without jumps. Second, we prove the principle of optimality and derive the corresponding Bellman optimality equation in the jump–uncertain stochastic framework, extending the work of Chirima et al. [3], who considered the case without jumps. Third, we extend the work of Framstad et.al [12] on the connection between the maximum principle and dynamic programming in jump–diffusion models, by providing a rigorous link between the dynamic programming principle and the uncertain stochastic maximum principle in a jump–uncertain setting. Finally, we demonstrate the applicability of these theoretical results through a portfolio selection problem, thereby providing a unified framework for optimal portfolio choice that integrates theoretical foundations with computational insights.

The remainder of this study is organized as follows. Section 2 reviews the fundamental concepts and theorems of jump–uncertain stochastic theory and establishes an existence and uniqueness result for jump–uncertain stochastic differential equations. Section 3 proves the principle of optimality and derives the associated Bellman optimality equation. Section 4 develops the maximum principle and shows that, under suitable conditions, the adjoint processes in the uncertain stochastic control problem can be expressed through derivatives of the value function V(t, x). Section 5 applies these theoretical results to a portfolio selection problem in a jump–uncertain stochastic financial market, thereby presenting a unified framework for optimal portfolio choice that integrates both theoretical foundations and computational insights. Finally, Section 6 concludes the study.

This study explores an optimal control problem applied to portfolio selection in a Liu-uncertain stochastic market with jumps. To support the analysis, this section reviews key concepts related to the chance space, defined as the product space (Γ,L,M)×(Ω,F,P). Here, (Ω,F,P) represents a classical probability space, where Ω is the sample space, F is a σ-algebra, and P is a probability measure. Meanwhile, (Γ,L,M) denotes an uncertain space, where Γ is the universal set, L is a σ-algebra, and M is an uncertain measure. For a comprehensive treatment of the chance space, we refer the reader to Hou [26], Liu [27], Fei [21], and Liu [28].

Definition 2.1 Hou [26] Let (Γ,L,M)×(Ω,F,P) be a chance space, and let Θ∈L×F be an event. Then, the chance measure of Θ is defined as

where (Ω,F,P) and (Γ,L,M) are a probability space and an uncertainty space in that order.

Hou [26] and Liu [27] proved that a chance measure satisfies normality, duality, monotonicity, and subadditivity properties, that is

(i) Normality Ch{Γ × Ω} = 1, Ch{∅} = 0.

(ii) Monotonicity Ch{Θ₁} ≤ Ch{Θ₂}, for any events Θ₁ ⊂ Θ₂.

(iii) Self- duality Ch{Θ} + Ch{Θ^c} = 1, for any eventΘ.

To illustrate this axiom, consider an event Θ∈L×F. When an observation is performed, the total chance that the event Θ either occurs or does not occur must be equal to 1. This reflects the intuitive requirement that one of the two mutually exclusive outcomes must happen with certainty.

(iv) Subadditivity Ch{⋃i=1∞Θi}≤∑i=1∞Ch{Θi}, for any countable sequence of events Θ₁, Θ₂, ⋯ .

Definition 2.2 Liu [28]. An uncertain random variable is a function ξ from a chance space (Γ,L,M)×(Ω,F,P) to the set of real numbers such that {ξ ∈ B} is an event in L×F for any Borel set B of real numbers.

Definition 2.3 Liu [28]. Let ξ be an uncertain random variable, then its chance distribution of ξ is defined by

for any x ∈ ℝ.

Definition 2.4 (i) Liu [28]. Let ξ be an uncertain random variable. Then, its expected value is defined by

provided that at least one of the two integrals is finite.

(ii) Fei [21]. The expected value of an uncertain random variable ξ is defined by

where E_p and EM are the expected values under the Liu uncertainty and the probability space, respectively.

The expected value of the uncertain random variable ξ is the probability expectation of the expected value of ξ under Liu uncertainty. In order to simplify the work and presentation, E_ch(.) shall be denoted by E(.).

Definition 2.5 Fei [21]. A hybrid process X_t is an uncertain stochastic process if X_t is an uncertain random variable for each t ∈ [0, T].

An uncertain stochastic process X(t) is said to be continuous if the sample paths of X(t) are all continuous functions of t for almost all (γ, ω) ∈ Γ × Ω.

Definition 2.6 Wu [29] (Jump Ito-Liu Integral) Suppose X(t) = (Y(t), Z(t)) is a jump-uncertain stochastic process, for any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_N+1 = b, the mesh is expressed as

The jump Ito-Liu integral of X(t) with respect to (W(t), C(t), N(t)) is defined as follows:

In the case above, X(t) is called jump Ito-Liu integrable.

Definition 2.7 Deng and Zhu [18]. An uncertain variable Z(r₁, r₂, t) is said to be an uncertain Z-jump uncertain variable with parameters r₁ and r₂ (0 < r₁ < r₂ < 1) for t > 0 if it has a jump uncertainty distribution

Definition 2.8 Deng and Zhu [18]. An uncertain process N_t is said to be a V-jump process with parameters r₁ and r₂ (0 < r₁ < r₂ < 1) for t ≥ 0 if (i) N₀ = 0, (ii) N_t has stationary and independent increments, (iii) every increment N_s+t − N_s is an uncertain Z-jump variable Z(r₁, r₂, t). Let N_t be an uncertain V-jump process, and ΔN_t = N_t+Δt − N_t. Then

Definition 2.9 Deng and Zhu [Jump–uncertain stochastic differential equation (jUSDE)] Suppose B_t is a one-dimensional Brownian motion, C_t is a one-dimensional Liu-canonical process, and N_t is a V–jump uncertain process with pararmeters 0 < r₁ < r₂ < 1, independent of (B_s, C_s), all defined on the chance space (Γ,L,M)×(Ω,F,P) with the natural filtration {Gt} generated by (B_s, C_s, Ns). For T > 0 and t ∈ [0, T], a process X_t is said to satisfy a jump–uncertain stochastic differential equation if

where α, σ, γ, η:[0, T] × ℝ → ℝ are progressively measurable with respect to {Gt} in t and continuous in x.

Theorem 2.1 (Existence and uniqueness for jump–uncertain stochastic differential equation) A jump-uncertain stochastic differential equation (Equation 6) admits a unique (up to indistinguishability) adapted solution X∈S*2([0,T]) if the coefficients α(t, x), σ(t, x), θ(t, x), and η(t, x) the coefficients of the jump-uncertain stochastic differential Equation 6 satisfies the Lipschitz condition

and linear growth condition

for L > 0 and K > 0 such that, for a.e. t ∈ [0, T] and all x ∈ ℝ.

Define

Furthermore, we assume the following

Proof of Theorem 2.1 To establish existence, we construct a Picard iteration scheme as in Chen and Liu [24], starting from a constant process and defining successive approximations by plugging the previous iterate into the integrals.

As in [24], set Xt(0)≡x and, for n ≥ 0,

By Theorem 2.1, each term is well–defined and X⁽ⁿ⁾ is adapted càdlàg.

We derive uniform L2 bounds for the iterates using inequalities tailored for each noise component—Burkholder–Davis–Gundy for the Brownian part, pathwise Lipschitz bounds for the Liu integral, and bounded- variation estimates for the V–jump part.

(i) For any square–integrable progressively measurable ψ,

(ii) Since C(γ) is Lipschitz with constant K_C(γ), for any integrable φ and 0 ≤ a < b ≤ T,

Hence, by Cauchy–Schwarz and Fubini,

(iii) Each sample path of N is of bounded variation. For any integrable ϕ and each γ,

for some finite pathwise constant K_N(γ). Consequently,

Using Minkowski inequality, Equations 8–10, and the linear growth condition, there exists C₁ > 0 (depending on K,T,E[KC2],E[KN2],C*) such that

Let un(t):=E[sup0≤r≤t|Xr(n)|2]. Then

To ensure these bounds remain finite, we apply the Gronwall's inequality:

Let D⁽ⁿ⁺¹⁾: = X⁽ⁿ⁺¹⁾ − X⁽ⁿ⁾. Subtract Equation 7 for consecutive indices:

Since the Lipschitz condition guarantees the iteration sequence is Cauchy in the space of adapted câdlàg processes with finite second moments, using Theorem 2.1 together with Equations 8–10 yields,

By dominated convergence, using the linear growth, and the continuity of the integrals with respect to S2 convergence Equations 8–10, pass to the limit in Equation 7 to conclude that X satisfies Equation 6. Moreover, Equation 11 gives X∈S*2([0,T]).

To establish uniqueness of the solution to Equation 6, let X,Y∈S*2([0,T]) be two solutions and define their difference by Z: = X − Y. Then, Z satisfies

Applying Theorem 2.1 together with the inequalities Equations 8–10, we obtain

By Gronwall's lemma, it follows that

which implies that Z_t = 0 almost surely for all t ∈ [0, T]. Therefore, X and Y are indistinguishable, and we conclude that Equation 6 admits a unique adapted solution (up to indistinguishability).

This study assumes that the uncertain stochastic optimal control model with jump is given by

where u_s is the control variable, and X_s represents the state variable. The functions f and G are the objective function and terminal utility function, respectively. Finally, the four functions α, σ, γ and η are functions of state X_s, control u_s and time s. B_s, C_s and N_s are the Brownian motion, Liu-uncertain canonical process, and uncertain V-jump process with parameters r₁, r₂ for s > 0, respectively.

Next, we present the principle of optimality and hence derive the equation of optimality.

Theorem 3.1 (Principle of optimality). For any (t, x) ∈ [0, T) × ℝ and Δt ∈ (0, T − t], the value function V(t, x) can be expressed as

where x + ΔX_t = X_t+Δt.

The proof of Theorem 3.1 follows the dynamic programming approach of Chirima et al. [3], which we extend to incorporate the Brownian–Liu–V-jump framework.

Proof of Theorem 3.1. Let X: = X^{t, x; u} and assume the standard admissibility conditions on the control process u (progressive measurability, integrability, and continuity) so that the system is well posed and satisfies the controlled Markov property. Then, the value function satisfies

For any admissible u, the performance functional can be decomposed as

By the Markov property and independence of future increments (B_s − B_t+Δt, C_s − C_t+Δt, N_s − N_t+Δt) from Gt+Δt, the conditional expectation equals the cost-to-go under the tail control ut+Δt:={us}s∈[t+Δt,T] starting from X_t+Δt, that is,

Hence,

Since V is the supremum over all admissible tail controls, it follows that

Substituting into Equation 15 and maximizing over all u on [t, T] yields

Conversely, fix ε > 0. For each y ∈ ℝ, let v^{ε, y} be an ε–optimal tail control on [t + Δt, T] such that

Using a measurable selection argument, one may construct the mapping y ↦ v^{ε, y} measurably. Define a combined control ū by setting ū_s = u_s for s ∈ [t, t + Δt) and u¯s=vε,Xt+Δts for s ∈ [t + Δt, T]. Admissibility of ū follows by construction. Applying Equation 15 with u replaced by ū gives

Maximizing over all admissible controls u on the interval [t, t + Δt] and then refining the approximation by taking ε arbitrarily small yields the reverse inequality. Taken together, the two inequalities establish the dynamic programming relation Equation 14, thereby completing the proof of Theorem 3.1.

Theorem 3.2 (Equation of optimality). Let V(t, x):[0, T] × ℝ be a twice continuously differentiable function. Then

where V_t(t, x) and V_x(t, x) are the partial derivatives of the function V(t, x) in t and x, respectively, and V_xx(t, x) is the partial derivative of V_x(t, x).

Proof of Theorem 3.2: For any Δt > 0, we have

By using the Taylor series expansion, we get

By substituting Equations 17 and 18 into Equation 13, we generate

We know that (ΔBt)2=Δt, and (ΔNt)2=(ΔCt)2=(Δt)2=ΔtΔBt=ΔtΔCt=ΔtΔNt=0. Based on the jump-uncertain stochastic differential equation, the constraint in Equation 12, ΔX_t = α(t, X_t, u_t)Δt + σ(t, X_t, u_t)ΔB_t + θ(t, X_t, u_t)ΔC_t + η(t, X_t, u_t)ΔN_t, we can generate Equations 20 and 21 as follows:

and

Substituting Equations 20, 21 into Equation 19 yields

Since E(ΔB_t) = E(ΔC_t) = 0, and from Equation 5

Dividing Equation 23 by Δt, we obtain

Thus, Theorem 3.2 has been proved.

In this section, we examine the relationship between the maximum principle and dynamic programming within a jump-uncertain stochastic control framework. Under appropriate regularity and non-degeneracy conditions adapted to the uncertain jump setting, we demonstrate that the value function V(t, x) is the unique classical solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Consequently, the adjoint processes arising in the maximum principle formulation can be identified with the derivatives of the value function, thereby establishing a rigorous correspondence between the two approaches.

Moreover, we verify that the optimal control derived from the HJB equation coincides with that obtained via the maximum principle, confirming their internal consistency in the context of uncertain jump-diffusion models. To illustrate this connection, we present a concrete example from a portfolio selection problem.