One Stage Deviation Principle in Continuous Stochastic Games

Abstract

In this paper, we consider McKean-Vlasov stochastic differential equations (MVSDEs) driven by Lévy noise. By identifying the right equations satisfied by the solutions of the MVSDEs with shifted driving Lévy noise, we build up a framework to fully apply the weak convergence method to establish large and moderate deviation principles for MVSDEs. In the case of ordinary SDEs, the rate function is calculated by using the solutions of the corresponding skeleton equations simply replacing the noise by the elements of the Cameron-Martin space. It turns out that the correct rate function for MVSDEs is defined through the solutions of skeleton equations replacing the noise by smooth functions and replacing the distributions involved in the equation by the distribution of the solution of the corresponding deterministic equation (without the noise). This is somehow surprising. With this approach, we obtain large and moderate deviation principles for much wider classes of MVSDEs in comparison with the existing literature see Dos Reis et al. (Ann. Appl. Probab. 29, 1487–1540, 2019).

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Authors and Affiliations

1. School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, 430072, People's Republic of China

   Wei Liu

2. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, Hubei, 430072, People's Republic of China

   Wei Liu

3. Department of Mathematics, Nanjing University, Nanjing, 210093, People's Republic of China

   Yulin Song

4. School of Mathematical Science, University of Science and Technology of China, Hefei, 230026, People's Republic of China

   Jianliang Zhai & Tusheng Zhang

Corresponding author

Correspondence to Jianliang Zhai.

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Tusheng Zhang and Jianliang Zhai's research is supported by NSFC (No. 11971456, 12131019, 11721101), School Start-up Fund (USTC) KY0010000036, the Fundamental Research Funds for the Central Universities (No. WK3470000016).

Yulin Song is supported by NSFC (No. 11971227, 11790272).

Wei Liu is supported by NSFC (No. 12071361, 12131019, 11731009), the Fundamental Research Funds for the Central Universities (No. 2042020kf0031, 2042020kf0217).

Appendix

Appendix

The Proof of Theorem 3.6

For the fixed J ∈ P r(D([0,T],H)), because of the existence of the strong solution, by the Yamada-Watanabe theorem (see [53] for the Wiener case and [63] for the PRM case), there exists a unique map \({\Gamma }_{J}:C([0,T],K)\times M_{FC}([0,T]\times Z)\rightarrow \mathbb {D}\) such that for any \((\bar {\Omega },\bar {\mathcal {F}},\bar {P},\{\bar {\mathcal {F}}_{t},\ t\in [0,T]\},\bar {W},\eta )\) satisfying that

• \((\bar {\Omega },\bar {\mathcal {F}},\bar {P})\) is a complete probability space;

• \(\{\bar {\mathcal {F}}_{t},\ t\in [0,T]\}\) is a right continuous filtration on \(\{\bar {\Omega },\bar {\mathcal {F}}\}\) augmented by the \(\bar {P}\)-zero sets;

• on the stochastic basis \((\bar {\Omega },\bar {\mathcal {F}},\bar {P},\{\bar {\mathcal {F}}_{t},\ t\in [0,T]\})\), \(\bar {W}=\{\bar {W}(t),t\in [0,T]\}\) is a cylindrical Brownian motion taking values in K, η is a PRM with intensity measure L e b _T ⊗ ν;

• \(\bar {W}\) and η are independent;

the following properties hold:

• (A0) \(\{{\Gamma }_{J}(\bar {W},\eta )(t),t\in [0,T]\}\) is an \(\{\bar {\mathcal {F}}_{t},\ t\in [0,T]\}\)-adapted process with paths in \(\mathbb {D}\);

• (A1)

  $$ \begin{array}{@{}rcl@{}}{{\int}_{0}^{T}}\|b(t,{\Gamma}_{J}(\bar{W},\eta),J)\|_{E}\mathrm{d} t + {{\int}_{0}^{T}}\|\sigma(t,{\Gamma}_{J}(\bar{W},\eta),J)\|^{2}_{\mathcal{L}_{2}}\mathrm{d} t + & {{\int}_{0}^{T}}{\int}_{Z}\|G(t,{\Gamma}_{J}(\bar{W},\eta),J,z)\|^{2}_{H}\nu(\mathrm{d} z)\mathrm{d} t\\ &<\infty,\ \bar{P}\text{-a.s.;}\end{array} $$

• (A2) as a stochastic equation on E one has

  $$ \begin{array}{@{}rcl@{}} {\Gamma}_{J}(\bar{W},\eta)(t) &=& h+{{\int}_{0}^{t}}b(s,{\Gamma}_{J}(\bar{W},\eta),J)\mathrm{d} s + {{\int}_{0}^{t}}\sigma(s,{\Gamma}_{J}(\bar{W},\eta),J)\mathrm{d} \bar{W}(s) \\ &&+ {{\int}_{0}^{t}}{\int}_{Z}G(s,{\Gamma}_{J}(\bar{W},\eta),J,z)\widetilde{\eta}(\mathrm{d} z,\mathrm{d} s),\ t\in[0,T],\ \bar{P}\text{-a.s.} \end{array} $$

  where \(\widetilde {\eta }\) is the corresponding compensated PRM with respect to η.

Therefore we have Y = Γ _J (W,N ¹), since (Y,J) is a solution of (3.1) and pathwise uniqueness holds with the fixed J.

For any given \(m\in (0,\infty )\) and \(u=(\phi ,\psi )\in \mathcal {S}^{m}_{1}\times \mathcal {S}^{m}_{2}\), ∀t ∈ [0,T], let

$$ \begin{array}{@{}rcl@{}} \mathcal{M}_{t}(\phi):=\exp\Big(-{{\int}_{0}^{t}}\langle\phi(s),\mathrm{d} W(s)\rangle_{K}-\frac{1}{2}{{\int}_{0}^{t}}\|\phi(s)\|^{2}_{K}\mathrm{d} s\Big) \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \mathcal{E}_{t}(\psi):=\exp\Big({{\int}_{0}^{t}}{\int}_{Z}{\int}_{[0,\psi]}\log \varphi(s,z)N(\mathrm{d} r,\mathrm{d} z,\mathrm{d} s)+{{\int}_{0}^{t}}{\int}_{Z}{\int}_{[0,\psi]}(-\varphi(s,z)+1)\mathrm{d} r\nu(\mathrm{d} z)\mathrm{d} s\Big), \end{array} $$

where \(\varphi :=\frac {1}{\psi }\). Then we have, on the probability space \(({\Omega },\mathcal {F},P)\)

• \(\{{\mathscr{M}}_{t}(\phi ),\ t\in [0,T]\}\) is a \(\mathbb {F}\)-martingale;

• \(\{\mathcal {E}_{t}(\psi ),\ t\in [0,T]\}\) is a \(\mathbb {F}\)-martingale by Theorem 6.1 in [10];

• moreover, \(\{{\mathscr{M}}_{t}(\phi )\mathcal {E}_{t}(\psi ),\ t\in [0,T]\}\) is a \(\mathbb {F}\)-martingale on \(({\Omega },\mathcal {F},P)\), thanks to the independence of W and N on \(({\Omega },\mathcal {F},P)\).

Let

$$ \begin{array}{@{}rcl@{}} Q(O):={\int}_{O}\mathcal{M}_{T}(\phi)\mathcal{E}_{T}(\psi)\mathrm{d} P,\ \forall O\in\mathcal{F}, \end{array} $$

((5.96))

then

• (Q1) Q is a probability measure on \(({\Omega },\mathcal {F})\);

• (Q2) the measures Q and P are equivalent;

• (Q3) By the Girsanov Theorem (see, e.g., [35, Theorem III.3.24], [16, Appendix A.1]), under the probability space \(({\Omega },\mathcal {F},\mathbb {F},Q)\), \((W(\cdot )+{\int \limits }_{0}^{\cdot }\phi (s)\mathrm {d} s,N^{\psi })\) has the same law as that of (W(⋅),N ¹) on \(({\Omega },\mathcal {F},\mathbb {F},P)\).

Let

$$ \begin{array}{@{}rcl@{}} Y^{u}:={\Gamma}_{J}(W(\cdot)+{\int}_{0}^{\cdot}\phi(s)\mathrm{d} s,N^{\psi}). \end{array} $$

((5.97))

By the property of Γ _J , it follows that (under the probability Q),

• (B0) Y ^u = {Y ^u (t),t ∈ [0,T]} is an \(\mathbb {F}\)-adapted process with paths in \(\mathbb {D}\);

• (B1) \({{\int \limits }_{0}^{T}}\|b(t,Y^{u}, J)\|_{E}\mathrm {d} t + {{\int \limits }_{0}^{T}}\|\sigma (t,Y^{u}, J)\|^{2}_{{\mathscr{L}}_{2}}\mathrm {d} t + {{\int \limits }_{0}^{T}}{\int \limits }_{Z}\|G(t,Y^{u},J,z)\|^{2}_{H}\nu (\mathrm {d} z)\mathrm {d} t <\infty ,\ Q\text {-a.s.;}\)

• (B2) as a stochastic equation on E one has

  $$ \begin{array}{@{}rcl@{}} Y^{u}(t) &=& h+{{\int}_{0}^{t}}b(s,Y^{u}, J)\mathrm{d} s + {{\int}_{0}^{t}}\sigma(s,Y^{u}, J)\mathrm{d} \Big(W(s)+ {{\int}_{0}^{s}}\phi(l)\mathrm{d} l\Big)\\ &&+ {{\int}_{0}^{t}}{\int}_{Z}G(s,Y^{u}, J,z)\Big(N^{\psi}(\mathrm{d} z,\mathrm{d} s)-\nu(\mathrm{d} z)\mathrm{d} s\Big),\ t\in[0,T],\ Q\text{-a.s.} \end{array} $$

  ((5.98))

Since stochastic integrals against semimartingales remain the same with respect to a class of equivalent probability measures and since Q and P are equivalent, we conclude that under the probability P, Y ^u fulfills the equation (5.98) as well. This completes the proof of Theorem 3.6.

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Liu, W., Song, Y., Zhai, J. et al. Large and Moderate Deviation Principles for McKean-Vlasov SDEs with Jumps. Potential Anal (2022). https://doi.org/10.1007/s11118-022-10005-0

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• Received: 08 April 2021

• Accepted: 09 April 2022

• Published: 10 May 2022

• DOI : https://doi.org/10.1007/s11118-022-10005-0

Keywords

• Large deviation
• Moderate deviation
• Weak convergence method
• McKean-Vlasov equation
• Lévy noise

Mathematics Subject Classification (2010)

• 60F10
• 60H10
• 60H15
• 60J75
• 37L55

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