September  2018, 8(3&4): 501-508. doi: 10.3934/mcrf.2018020

Existence for nonlinear finite dimensional stochastic differential equations of subgradient type

Octav Mayer Institute of Mathematics of Romanian Academy, Iaşi, Romania

This paper is dedicated to Professor Jiongmin Yong on the occasion of his 60th birthday

Received  September 2017 Revised  January 2018 Published  September 2018

One proves via variational techniques the existence and uniqueness of a strong solution to the stochastic differential equation $dX+{\partial} {\varphi} (t,X)dt\ni \sum\limits^N_{i = 1}σ_i(X)d{β}_i,\ X(0) = x,$ where ${\partial}{\varphi} :{\mathbb{R}}^d\to2^{{\mathbb{R}}^d}$ is the subdifferential of a convex function ${\varphi}:{\mathbb{R}}^d\to{\mathbb{R}}$ and $σ_i∈ L({\mathbb{R}}^d,{\mathbb{R}}^d)$, $1≤ d<{∞}$.

Citation: Viorel Barbu. Existence for nonlinear finite dimensional stochastic differential equations of subgradient type. Mathematical Control & Related Fields, 2018, 8 (3&4) : 501-508. doi: 10.3934/mcrf.2018020
References:
[1]

V. Barbu, Nonlinear Differential Equations Of Monotone Type In Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5. Google Scholar

[2]

V. Barbu, Optimal Control of Variational Inequalities, Pitman Advanced Publishing Program, Boston. London. Melbourne, 1984. Google Scholar

[3]

V. Barbu, A variational approach to stochastic nonlinear parabolic problems, J. Math. Annal. Appl., 384 (2011), 2-15. doi: 10.1016/j.jmaa.2010.07.016. Google Scholar

[4]

V. Barbu, Optimal control approach to nonlinear diffusion equations driven by Wiener noise, J. Optim. Theory Appl., 153 (2012), 1-26. doi: 10.1007/s10957-011-9946-8. Google Scholar

[5]

V. Barbu, A variational approach to nonlinear stochastic differential equations with linear multiplicative noise, to appear.Google Scholar

[6]

V. BarbuG. Da Prato and M. Röckner, Existence of strong solutions for stochastic porous media equations under general monotonicity conditions, Ann. Probab., 37 (2009), 428-452. doi: 10.1214/08-AOP408. Google Scholar

[7]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, 2012. doi: 10.1007/978-94-007-2247-7. Google Scholar

[8]

V. Barbu and M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicataive noise, J. Eur. Math.Soc., 17 (2015), 1789-1815. doi: 10.4171/JEMS/545. Google Scholar

[9]

R. BuckdahnL. MaticiucE. Pardoux and A. Rǎşcanu, Stochastic variational inequalities on nonconvex domains, J. Diff. Equations, 259 (2015), 7332-7374. doi: 10.1016/j.jde.2015.08.023. Google Scholar

[10]

C. Prevot and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., 1905, Springer, 2007. Google Scholar

show all references

References:
[1]

V. Barbu, Nonlinear Differential Equations Of Monotone Type In Banach Spaces, Springer, 2010. doi: 10.1007/978-1-4419-5542-5. Google Scholar

[2]

V. Barbu, Optimal Control of Variational Inequalities, Pitman Advanced Publishing Program, Boston. London. Melbourne, 1984. Google Scholar

[3]

V. Barbu, A variational approach to stochastic nonlinear parabolic problems, J. Math. Annal. Appl., 384 (2011), 2-15. doi: 10.1016/j.jmaa.2010.07.016. Google Scholar

[4]

V. Barbu, Optimal control approach to nonlinear diffusion equations driven by Wiener noise, J. Optim. Theory Appl., 153 (2012), 1-26. doi: 10.1007/s10957-011-9946-8. Google Scholar

[5]

V. Barbu, A variational approach to nonlinear stochastic differential equations with linear multiplicative noise, to appear.Google Scholar

[6]

V. BarbuG. Da Prato and M. Röckner, Existence of strong solutions for stochastic porous media equations under general monotonicity conditions, Ann. Probab., 37 (2009), 428-452. doi: 10.1214/08-AOP408. Google Scholar

[7]

V. Barbu and T. Precupanu, Convexity and Optimization in Banach Spaces, Springer, 2012. doi: 10.1007/978-94-007-2247-7. Google Scholar

[8]

V. Barbu and M. Röckner, An operatorial approach to stochastic partial differential equations driven by linear multiplicataive noise, J. Eur. Math.Soc., 17 (2015), 1789-1815. doi: 10.4171/JEMS/545. Google Scholar

[9]

R. BuckdahnL. MaticiucE. Pardoux and A. Rǎşcanu, Stochastic variational inequalities on nonconvex domains, J. Diff. Equations, 259 (2015), 7332-7374. doi: 10.1016/j.jde.2015.08.023. Google Scholar

[10]

C. Prevot and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., 1905, Springer, 2007. Google Scholar

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