観測に対して行動を対応させる規則（決定関数）のもとで起こる確率現象を解析するのが数理統計学の課題である．統計的決定理論は決定関数の総体における最適性，許容性の問題を扱うが，その基礎となるのが分布の計算である．確率過程のように分布の構造が複雑な場合は漸近的決定理論に頼ることになり，その適用のためには確率過程のクラス（マルコフ過程，ミキシング過程，セミマルチンゲール，弱および強従属過程，摂動モデル，．．．）に応じた極限定理の研究が必要となる．
　 高次の極限定理である漸近展開が重要になっている．これは分布の近似精度の改善のみならず，多変量解析，時系列解析，高次漸近決定理論，ブートストラップ法，情報量規準，統計的予測，情報幾何などの研究の基礎となる．セミマルチンゲールのような連続時間確率過程に対して（分布論的）漸近展開を与え，これを基礎に確率微分方程式の高次統計推測論が現在発展しており，統計量の有効性の証明，モデル選択における情報量規準の構成などに役立っている．展開公式の解析的正当性（validity）の証明には無限次元確率解析（マリアバン解析）が用いられる．
　 オプション価格の漸近展開の方法は，ファイナンスに現れる多くのモデルに適用されており，保険数理におけるＣＴＥなどのリスク尺度の近似計算，条件つき漸近展開および非線形フィルタへの応用も研究している．

1. Yoshida, N.: Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe, Probab. Theory Related Fields, 92, 275-311 (1992).
2. Yoshida, N.: Asymptotic expansion for statistics related to small diffusions. J. Japan Statist. Soc. 22, 139-159 (1992).
3. Yoshida, N.: Estimation for diffusion processes from discrete observation, J. Multivariate Analysis, 41, 220-242 (1992).
4. Yoshida, N.: Asymptotic expansion of Bayes estimators for small diffusions, Probab. Theory Related Fields, 95 , 429-450 (1993).
5. Yoshida, N.: Malliavin calculus and asymptotic expansion for martingales, Probab. Theory Related Fields, 109, 301-342 (1997).
6. Kusuoka, S., Yoshida, N.: Malliavin calculus, strong mixing, and expansion of diffusion functionals. Prob. Theory Related Fields 116, 457-484 (2000).
7. Sakamoto, Y., Yoshida, N.: Asymptotic expansion formulas for functionals of epsilon-Markov processes with a mixing property. Annals of the Institute of Statistical Mathematics, 56, 545-597 (2004).
8. Yoshida, N.: Partial mixing and conditional Edgeworth expansion for diffusions with jumps. Probab. Theory Related Fields 129, 559-624 (2004).
9. Hayashi, T., Yoshida, N.: On covariance estimation of nonsynchronously observed diffusion processes. Bernoulli, 11, 359-379 (2005).
10. Shimizu, Y., Yoshida, N.: Estimation of parameters for diffusion processes with jumps from discrete observations. Stat. Inferecne Stoch. Process., 9, 227-277 (2006).
11. Hayashi, T., Yoshida, N.: Asymptotic normality of a covariance estimator for nonsynchronously observed diffusion processes, Annals of the Institute of Statistical Mathematics 60, 367-406 (2008).
12. Sakamoto, Y., Yoshida, N.: Third-order asymptotic expansion of M-estimators for diffusion processes, Annals of the Institute of Statistical Mathematics 61, 629-661 (2009).
13. Yoshida, N. : Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations, Annals of the Institute of Statistical 63, 3, 431-479 (2011).
14. Hayashi, T., Yoshida, N.: Nonsynchronous covariation process and limit theorems, Stochastic Processes and their Applications, 121, 10, 2416-2454 (2011).
15. Ogihara, T., Yoshida, N.: Quasi-likelihood analysis for the stochastic differential equation with jumps, Statistical Inference for Stochastic Processes, 14, 3, 189-229 (2011).
16. Uchida, M., Yoshida, N.: Adaptive estimation of an ergodic diffusion process based on sampled data, Stochastic Processes and their Applications, 122, 8, 2885-2924 (2012).
17. Yoshida, N.: Martingale expansion in mixed normal limit, Stochastic Processes and their Applications, 123, 3, 887-933 (2013).
18. Uchida, M., Yoshida, N.: Quasi likelihood analysis of volatility and nondegeneracy of statistical random field, Stochastic Processes and their Applications, 123, 7, 2851-2876 (2013).
19. YUIMA Project Team: The YUIMA Project: A Computational framework for simulation and inference of stochastic differential equations, Journal of Statistical Software, 57, 4, 1-51 (2014).
20. Podolskij, M., Yoshida, N.: Edgeworth expansion for functionals of continuous diffusion processes, Annals of Applied Probability, 26, 6, 3415-3455 (2016).
21. Muni Toke, I., Yoshida, N.: Modelling intensities of order flows in a limit order book, Quantitative Finance, 17, 5, 683-701 (2017).
22. Podolskij, M., Veliyev, B., Yoshida, N.: Edgeworth expansion for the pre-averaging estimator, Stochastic Processes and their Applications, 127, 11, 3558-3595 (2017).
23. Tudor, C., Yoshida, N.: Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos, Stochastic Processes and their Applications, 129, 9, 3499-3526 (2019).
    
24. Nualart, D., Yoshida, N.: Asymptotic expansion of Skorohod integrals, Electron. J. Probab., 24, paper no. 119, 64 pp. (2019).

数理統計学．朝倉書店　2006
Iacus, S., Yoshida, N.: Simulation and Inference for Stochastic Processes with YUIMA: A Comprehensive R Framework for SDEs and Other Stochastic Processes, Springer 2018

日本数学会2006年度解析学賞,

第１回日本統計学会研究業績賞(2007),

第１４回日本統計学会賞(2009),

第８回藤原洋数理科学賞大賞(2019)

Bernoulli Society, Executive Committee,

統計数理研究所リスク解析戦略研究センター客員教授,

日本統計学会評議員，

日本アクチュアリー会評議員，

Statistical Inference for Stochastic Processes, editorial board,

8th World Congress in Probability and Statistics, Istanbul, Program Committee