On partially observed jump diffusions II: the filtering density

IF 1.4 3区数学 Q2 MATHEMATICS, APPLIED

Stochastics and Partial Differential Equations-Analysis and Computations Pub Date : 2023-10-03 DOI:10.1007/s40072-023-00311-y

Alexander Davie, Fabian Germ, István Gyöngy

{"title":"On partially observed jump diffusions II: the filtering density","authors":"Alexander Davie, Fabian Germ, István Gyöngy","doi":"10.1007/s40072-023-00311-y","DOIUrl":null,"url":null,"abstract":"Abstract A partially observed jump diffusion $$Z=(X_t,Y_t)_{t\\in [0,T]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $$X_t$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:math> given the observations $$(Y_s)_{s\\in [0,t]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:math> exists and belongs to $$L_p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> if the conditional density of $$X_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> given $$Y_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>Y</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> exists and belongs to $$L_p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> .","PeriodicalId":48569,"journal":{"name":"Stochastics and Partial Differential Equations-Analysis and Computations","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastics and Partial Differential Equations-Analysis and Computations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s40072-023-00311-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

Abstract A partially observed jump diffusion $$Z=(X_t,Y_t)_{t\in [0,T]}$$ Z = ( X t , Y t ) t ∈ [ 0 , T ] given by a stochastic differential equation driven by Wiener processes and Poisson martingale measures is considered when the coefficients of the equation satisfy appropriate Lipschitz and growth conditions. Under general conditions it is shown that the conditional density of the unobserved component $$X_t$$ X t given the observations $$(Y_s)_{s\in [0,t]}$$ ( Y s ) s ∈ [ 0 , t ] exists and belongs to $$L_p$$ L p if the conditional density of $$X_0$$ X 0 given $$Y_0$$ Y 0 exists and belongs to $$L_p$$ L p .

查看原文本刊更多论文

部分观测到的跳跃扩散II:过滤密度

考虑由Wiener过程和泊松鞅测度驱动的随机微分方程给出的部分观测跳跃扩散$$Z=(X_t,Y_t)_{t\in [0,T]}$$ Z = (X t, Y t) t∈[0,t]，当方程的系数满足适当的Lipschitz条件和生长条件时。在一般条件下，如果$$X_0$$ X 0在$$Y_0$$ Y 0下的条件密度存在并属于$$L_p$$ L p，则在观测值$$(Y_s)_{s\in [0,t]}$$ (Y s) s∈[0,t]下的未观测分量$$X_t$$ X t的条件密度存在且属于$$L_p$$ L p。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Stochastics and Partial Differential Equations-Analysis and Computations Mathematics-Modeling and Simulation

CiteScore

2.70

自引率

13.30%

发文量

期刊介绍： Stochastics and Partial Differential Equations: Analysis and Computations publishes the highest quality articles presenting significantly new and important developments in the SPDE theory and applications. SPDE is an active interdisciplinary area at the crossroads of stochastic anaylsis, partial differential equations and scientific computing. Statistical physics, fluid dynamics, financial modeling, nonlinear filtering, super-processes, continuum physics and, recently, uncertainty quantification are important contributors to and major users of the theory and practice of SPDEs. The journal is promoting synergetic activities between the SPDE theory, applications, and related large scale computations. The journal also welcomes high quality articles in fields strongly connected to SPDE such as stochastic differential equations in infinite-dimensional state spaces or probabilistic approaches to solving deterministic PDEs.