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differential equation in Itˆo calculus, the Stratonovich Fokker Planck equation is nothing
else than the Itˆo Fokker Planck equation of the stochastic differential equation:
dx =
$
f (x, t) − C(x)
%
dt + B(x) dw,
1
2Ci(x) =
>m
j=1
>n
l=1
(Bi,j (x)
(xl
Bi,l (x) (3.36)
Thus, the Stratonovich Fokker Planck equation has the form:
(p(x|y, t)
(t
= −'y ·
$$
f (y) − C(y)
%
p(x|y, t)
%
+
1
2
tr
$
'yy
$
B(y, t)B(y, t)T p(x|y, t)
%%
(3.37)
The difference between the Stratonovich and Itˆo Fokker Planck PDEs is in the extra
term C(x). In the question, which calculus to use, the answer depends on the appli-cation
and the goal of the underlying derivation. It is generally accepted (Chirikjian
2009),(Øksendal 2003) that the Itˆo calculus is used for the cases where expectation op-erations
have to be evaluated while the Stratonovich calculus has similar properties with
the usual calculus. In this section we have derived the connection between the two cal-culi
and therefore, one could take advantage of both by transforming the Stratonovich
interpreted solution of a stochastic differential equation in to its Itˆo version and then
apply Itˆo calculus. Besides these conceptual differences between the two calculi, there
are additional characteristics of Itˆo integration which we do not find in the Stratonovich
calculus and vice versa. More detailed discussion on the properties of the Itˆo integration
can been found in (Øksendal 2003), (Karatzas & Shreve 1991), (Chirikjian 2009) and
(Gardiner 2004).
80
Object Description
| Title | Iterative path integral stochastic optimal control: theory and applications to motor control |
| Author | Theodorou, Evangelos A. |
| Author email | etheodor@usc.edu; theo0027@umn.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2011-01-11 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-04-29 |
| Advisor (committee chair) | Schaal, Stefan |
| Advisor (committee member) |
Valero-Cuevas, Francisco Sukhatme, Gaurav S. Todorov, Emo Schweighofer, Nicolas |
| Abstract | Motivated by the limitations of current optimal control and reinforcement learning methods in terms of their efficiency and scalability, this thesis proposes an iterative stochastic optimal control approach based on the generalized path integral formalism. More precisely, we suggest the use of the framework of stochastic optimal control with path integrals to derive a novel approach to RL with parameterized policies. While solidly grounded in value function estimation and optimal control based on the stochastic Hamilton Jacobi Bellman (HJB) equation, policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise. The resulting algorithm can be conceived of as model-based, semi-model-based, or even model free, depending on how the learning problem is structured. The new algorithm, Policy Improvement with Path Integrals (PI2), demonstrates interesting similarities with previous RL research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well. Applications to high dimensional robotic systems are presented for a variety of tasks that require optimal planning and gain scheduling.; In addition to the work on generalized path integral stochastic optimal control, in this thesis we extend model based iterative optimal control algorithms to the stochastic setting. More precisely we derive the Differential Dynamic Programming algorithm for stochastic systems with state and control multiplicative noise. Finally, in the last part of this thesis, model based iterative optimal control methods are applied to bio-mechanical models of the index finger with the goal to find the underlying tendon forces applied for the movements of, tapping and flexing. |
| Keyword | stochastic optimal control; reinforcement learning,; robotics |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3804 |
| Contributing entity | University of Southern California |
| Rights | Theodorou, Evangelos A. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Theodorou-4581 |
| Archival file | uscthesesreloadpub_Volume14/etd-Theodorou-4581.pdf |
Description
| Title | Page 94 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | differential equation in Itˆo calculus, the Stratonovich Fokker Planck equation is nothing else than the Itˆo Fokker Planck equation of the stochastic differential equation: dx = $ f (x, t) − C(x) % dt + B(x) dw, 1 2Ci(x) = >m j=1 >n l=1 (Bi,j (x) (xl Bi,l (x) (3.36) Thus, the Stratonovich Fokker Planck equation has the form: (p(x|y, t) (t = −'y · $$ f (y) − C(y) % p(x|y, t) % + 1 2 tr $ 'yy $ B(y, t)B(y, t)T p(x|y, t) %% (3.37) The difference between the Stratonovich and Itˆo Fokker Planck PDEs is in the extra term C(x). In the question, which calculus to use, the answer depends on the appli-cation and the goal of the underlying derivation. It is generally accepted (Chirikjian 2009),(Øksendal 2003) that the Itˆo calculus is used for the cases where expectation op-erations have to be evaluated while the Stratonovich calculus has similar properties with the usual calculus. In this section we have derived the connection between the two cal-culi and therefore, one could take advantage of both by transforming the Stratonovich interpreted solution of a stochastic differential equation in to its Itˆo version and then apply Itˆo calculus. Besides these conceptual differences between the two calculi, there are additional characteristics of Itˆo integration which we do not find in the Stratonovich calculus and vice versa. More detailed discussion on the properties of the Itˆo integration can been found in (Øksendal 2003), (Karatzas & Shreve 1991), (Chirikjian 2009) and (Gardiner 2004). 80 |
