math.OC 2026-03-26 1

Stochastic maximum principle for time-changed forward-backward stochastic control problem with Lévy noise

This paper establishes a stochastic maximum principle for optimal control problems governed by time-changed forward-backward stochastic differential equations with Lévy noise. The system incorporates a random, non-decreasing operational time (the inv…

Authors: Jingwei Chen, Jun Ye, Feng Chen

Sto c hastic maxim um principle for time-c hanged forw ard-bac kw ard sto c hastic con trol problem with L ´ evy noise Jingw ei Chen †∗ , Jun Y e ‡ , F eng Chen § Abstract This paper establishes a sto c hastic maximum principle for optimal control problems gov erned b y time-changed forward-bac kw ard sto c hastic differen tial equations with L´ evy noise. The system incorp orates a random, non-decreasing op erational time (the inv erse of an α -stable sub ordinator) to mo del phenomena lik e trapping even ts and subdiffusion. Using a dualit y transformation and the con vex v ariational method, we deriv e necessary and sufficient conditions for optimalit y , expressed through a no vel set of adjoint equations. Finally , the theoretical results are applied to solve an explicit cash managemen t problem under stochastic recursiv e utilit y . Keyw ords: Time-c hanged forward-bac kward stochastic con trol system, sto c hastic optimal control, sto c hastic maximum principle, L ´ evy noise AMS Sub ject Clasification: 49K15, 60H10 1 In tro duction Sto c hastic optimal control has emerged as a cornerstone to ol in mo dern mathematical finance, engineer- ing system optimization, and managemen t science, underpinning the analysis and solution of complex decision-making problems under uncertain ty . In classical con trol theory , the state dynamics of a con- trolled system are typically modeled via a forw ard sto chastic differen tial equation (SDE), whose solution captures the evolution of the system’s state ov er time in the presence of random p erturbations. How- ev er, in a broad class of adv anced applications, the state pro cess itself evolv es as an adapted sto c hastic pro cess whose dynamics must b e c haracterized by a backw ard sto chastic differen tial equation (BSDE). This fundamental discrepancy b etw een the forw ard evolution of the state and the bac kward evolution of the v alue pro cess naturally gives rise to the theory of forw ard-backw ard sto chastic differen tial equa- tions (FBSDEs), whic h provides a unified mathematical framework to reconcile these dual dynamical structures (see [ 1 , 2 , 3 , 4 ]). On the other hand, the in troduction of random time (or time c hange) in to stochastic differen tial equations has b ecome a vibrant and rapidly ev olving research fron tier (see [ 5 , 6 , 7 ]). By replacing the standard time incremen t dt and Bro wnian motion increment dB t with the incremen ts dE t and dB E t of a random, ∗ Corresponding author. ( chenj22@mails.tsinghua.edu.cn ) † Y au Mathematical Sciences Cen ter, Tsingh ua Universit y , Beijing, 100084, China. ‡ Department of Mathematical Sciences, Tsinghua Univ ersity , Beijing, 100084, China. § Department of Automation, Center for Brain-Inspired Computing Research, Tsinghua Universit y , Beijing, 100084, China. 1 non-decreasing op erational time dE t , this approach em b eds sto c hastic dynamics within a flexible, time- transformed framew ork. In financial applications, this time-c hanged structure can effectively mo del phenomena such as the stagnation of asset prices during trading halts or the intermitten t arriv al of mark et information, where the “effective” time scale of the system deviates from calendar time. In ph ysics, it pro vides a p ow erful tool to describ e the trapping effects of particles in sub diffusiv e pro cesses, where the mov emen t of particles is constrained by irregular, time-inhomogeneous environmen ts. The study of stochastic control for time-c hanged stochastic con trol systems represen ts an ev en more recen t and underdeveloped area. Nane and Ni [ 8 ] made a pioneering con tribution by establishing the sto c hastic maxim um principle for a sto c hastic con trol problem driven by time-changed L´ evy noise, ex- tending classical control theory to accommo date jumps and time-inogeneohom us volatilit y . Jin and Song [ 9 ] further adv anced the field b y deriving the sto c hastic maxim um principle for a class of mean field game problems with time-c hanged Brownian motion. Despite these adv ances, the interpla y betw een FBSDEs and time-changed systems remains largely unexplored. In this pap er, w e inv estigate the following system of time-c hanged forward-bac kw ard sto c hastic differen- tial equations with L ´ evy noise (TCFBSDEwLN)                      dX v t = f ( t, E t , X v t , v ( t )) dE t + σ ( t, E t , X v t , v ( t )) dB E t + R | z | 0 , t ≥ 0 , (2.1) where the Laplace exp onen t ψ : (0 , ∞ ) → (0 , ∞ ) is ψ ( ξ ) = R ∞ 0 (1 − e − ξy )Π( dz ) , ξ > 0 and the L ´ evy measure Π satisfies R ∞ 0 ( z ∧ 1)Π( dz ) < ∞ . This pap er fo cuses on the infinite L ´ evy measure case, i.e. 2 Π(0 , ∞ ) = ∞ . Let E = ( E t ) t ≥ 0 b e the inv erse of D , i.e. E t := inf { u > 0 : D u > t } , t ≥ 0 . (2.2) If D is a stable sub ordinator, then E has Mittag-Leffler distributions, see [ 10 ]. Let E B , E D and E denote the exp ectation under the probability measures P B , P D and P , resp ectively . Supp ose B and E are m utually indep enden t, then the pro duct measure satisfies P = P B × P D . In the conten t going forward, denote C as generic p ositiv e constants that ma y change from line to line. F or any given s ∈ [0 , T ], we introduce the following spaces. • L 2 (Ω , F s T ; R n ): the space of F s T -measurable R n -v alued squared integrable random v ariables ξ such that E h | ξ | 2 i < ∞ . • L 2 F ([ s, T ]; R n ): the space of F s T -adapted R n -v alued squared integrable processes φ ( t ) suc h that E R T s | φ ( t ) | 2 dt < ∞ . • L ∞ F ([ s, T ]; R n ): the space of F s T -adapted R n -v alued essentially b ounded pro cesses suc h that ∥ φ ( · ) ∥ ∞ := ess sup ( t,ω ) ∈ [ s,T ] × Ω | φ t ( ω ) | < ∞ . • L 2 F ,p ([ s, T ]; R n ): the space of F s T -predictable R n -v alued squared in tegrable pro cesses such that E R T s | ϕ ( t ) | 2 dt < ∞ . • F 2 p ([ s, T ]; R n ): the space of R n − v al ued F s T -predictable processes f ( · , · , · ) defined on Ω × [0 , T ] × E suc h that E R T 0 R E | f ( · , t, z ) | 2 Π( dz ) dE t < ∞ . Let U b e a nonempty conv ex subset of R k . W e define the admissible control set U ad =  v ( · ) ∈ L 2 F,p  [0 , T ]; R k  ; v ( t ) ∈ U , a.e.t ∈ [0 , T ] , P − a, s.  . F or any giv en admissible control v ( · ) ∈ U ad and in tial condition x 0 ∈ R n , we consider the time-changed forw ard-backw ard sto ch astic con trol system with L´ evy noise as shown in ( 1.1 ), or equiv alently in the in tegral form:            X v t = x 0 + R t 0 f ( s, E s , X v s , v ( s )) dE s + R t 0 σ ( s, E s , X v s , v ( s )) dB E s + R t 0 R | z | 0 and u, m : Ω × R + → R + b e the G t -me asur able functions which ar e inte gr able with r esp e ct to E t . L et n ( t ) b e a p ositive, monotonic, non-de cr e asing function. Then, the ine quality u ( t ) ≤ n ( t ) + Z t 0 m ( s ) u ( s ) dE s , t ≥ 0 (2.8) implies almost sur ely u ( t ) ≤ n ( t ) exp  Z t 0 m ( s ) dE s  , t ≥ 0 . (2.9) 5 2.2 Dualit y W e now introduce the following dual system to ( 1.1 ):                      dX v , ∗ t = f  D t , t, X v , ∗ t , v ( D t )  dt + σ  D t , t, X v , ∗ t , v ( D t )  dB t + R | z | 0, X v t is the cash flow of an agen t, v ( t ) is a control strategy of the agent and is regarded as the rate of capital injection or withdra wal, Y v t is the utility from v ( · ), ( A v t ) 2 is the volatilit y of utility . F or any v ( · ) ∈ U ad , ( 4.1 ) has a unique solution  X v · , Y v · , A v · , r v ( · , · )  . In tro duce a cost functional: J ( v ( · )) := E " 1 2 Z T 0 ( v ( t ) − κ ( t )) 2 dE t − y v 0 # . (4.2) where κ ( t ) is a deterministic and b ounded function with v alue in R , serving as a dynamic b enchmark. Then, the cash mangemen t problem with sto c hastic recursive utilit y is as follows. Problem 4.1. Find an optimal c ontr ol str ate gy u ( · ) ∈ U ad such that J ( u ( · )) = inf v ( · ) ∈U ad J ( v ( · )) . (4.3) subje ct to ( 4.1 ) . W e can chec k that Assumption 2.1 is satisfied. Then we can use the maxim um principle Theorem 3.1 to solv e the abov e problem. The Hamiltonian function and the adjoint equation are then reduced to H ( t 1 , t 2 , x, y , a, r ( · ) , v , p, q , k , R ( · )) = ⟨ q , − µ 1 x + β 1 v ⟩ + ⟨ k , σ v ⟩ − ⟨ p, ( − µ 1 y + µ 2 x + β 2 v ) ⟩ + Z | z |

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