Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps
We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have <i>q</i> players and are driven by a general-dimensional vector Lévy process. By es...
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2024-09-01
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| author | Wanyang Dai |
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| description | We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have <i>q</i> players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form It<i>o</i>-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>γ</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula> under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI. |
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| publishDate | 2024-09-01 |
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| series | Mathematics |
| spelling | doaj-art-88009a0fda43471980997317b74cedda2025-08-20T01:55:38ZengMDPI AGMathematics2227-73902024-09-011218289110.3390/math12182891Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy JumpsWanyang Dai0School of Mathematics, Nanjing University, Nanjing 210093, ChinaWe establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have <i>q</i> players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form It<i>o</i>-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>,</mo><msub><mi>γ</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>}</mo></mrow></semantics></math></inline-formula> under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI.https://www.mdpi.com/2227-7390/12/18/2891stochastic differential gamenon-zero-sum gamezero-sum gamenon-Gaussian noisestochastic partial differential equationdiscontinuous Lévy jump |
| spellingShingle | Wanyang Dai Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps Mathematics stochastic differential game non-zero-sum game zero-sum game non-Gaussian noise stochastic partial differential equation discontinuous Lévy jump |
| title | Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps |
| title_full | Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps |
| title_fullStr | Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps |
| title_full_unstemmed | Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps |
| title_short | Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps |
| title_sort | stochastic differential games and a unified forward backward coupled stochastic partial differential equation with levy jumps |
| topic | stochastic differential game non-zero-sum game zero-sum game non-Gaussian noise stochastic partial differential equation discontinuous Lévy jump |
| url | https://www.mdpi.com/2227-7390/12/18/2891 |
| work_keys_str_mv | AT wanyangdai stochasticdifferentialgamesandaunifiedforwardbackwardcoupledstochasticpartialdifferentialequationwithlevyjumps |