Completeness and semi-flows for stochastic differential equations with monotone drift Michael Scheutzow Technische Universit¨ at Berlin Bielefeld, October 4th, 2012 Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Stochastic Differential Equations An SDE d X t = b ( X t ) d t + P m j = 1 � j ( X t ) d W j ( t ) , X 0 = x 2 R d . Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Stochastic Differential Equations An SDE d X t = b ( X t ) d t + P m j = 1 � j ( X t ) d W j ( t ) , X 0 = x 2 R d . Well-known Existence and uniqueness of solutions, continuous dependence on initial condition and existence of solution flow of homeomorphisms if b , � globally Lipschitz. Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Stochastic Differential Equations An SDE d X t = b ( X t ) d t + P m j = 1 � j ( X t ) d W j ( t ) , X 0 = x 2 R d . Well-known Existence and uniqueness of solutions, continuous dependence on initial condition and existence of solution flow of homeomorphisms if b , � globally Lipschitz. Are these properties still true (at least locally) in case infinite number of driving BM (or Kunita-type sdes) local one-sided Lipschitz condition? Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Existence and uniqueness of local solutions Consider the sde d X t = b ( X t ) d t + M ( d t , X t ) , X 0 = x 2 R d , where b continuous, M cont. martingale field s.t. a ( x , y ) := d d t [ M ( t , x ) , M ( t , y )] is cont. and determ., a ( y , y )(= d A ( x , y ) := a ( x , x ) � a ( x , y ) � a ( y , x ) + d t [ M ( t , x ) � M ( t , y )]) and One-sided local Lipschitz condition 2 h b ( x ) � b ( y ) , x � y i + Tr A ( x , y ) K N | x � y | 2 , | x | , | y | N . Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Existence and uniqueness of local solutions Consider the sde d X t = b ( X t ) d t + M ( d t , X t ) , X 0 = x 2 R d , where b continuous, M cont. martingale field s.t. a ( x , y ) := d d t [ M ( t , x ) , M ( t , y )] is cont. and determ., a ( y , y )(= d A ( x , y ) := a ( x , x ) � a ( x , y ) � a ( y , x ) + d t [ M ( t , x ) � M ( t , y )]) and One-sided local Lipschitz condition 2 h b ( x ) � b ( y ) , x � y i + Tr A ( x , y ) K N | x � y | 2 , | x | , | y | N . Theorem The sde has a unique local solution. Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Existence and uniqueness of local solutions Consider the sde d X t = b ( X t ) d t + M ( d t , X t ) , X 0 = x 2 R d , where b continuous, M cont. martingale field s.t. a ( x , y ) := d d t [ M ( t , x ) , M ( t , y )] is cont. and determ., a ( y , y )(= d A ( x , y ) := a ( x , x ) � a ( x , y ) � a ( y , x ) + d t [ M ( t , x ) � M ( t , y )]) and One-sided local Lipschitz condition 2 h b ( x ) � b ( y ) , x � y i + Tr A ( x , y ) K N | x � y | 2 , | x | , | y | N . Theorem The sde has a unique local solution. In the “classical” case M ( t , x ) = P m j = 1 � j ( x ) W j ( t ) A ( x , y ) = ( � ( x ) � � ( y ))( � ( x ) � � ( y )) t , Tr A ( x , y ) = k � ( x ) � � ( y ) k 2 Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Idea of proof (Krylov, Pr´ evˆ ot/R¨ ockner): Euler approx. 0 := x and for t 2 ( k n , k + 1 For n 2 N let � n n ] : R t R t � n t := � n k / n b ( � n k / n M ( d s , � n k / n + k / n ) d s + k / n ) . Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Idea of proof (Krylov, Pr´ evˆ ot/R¨ ockner): Euler approx. 0 := x and for t 2 ( k n , k + 1 For n 2 N let � n n ] : R t R t � n t := � n k / n b ( � n k / n M ( d s , � n k / n + k / n ) d s + k / n ) . So, Up to appropriate stopping time: R t R t t | 2 ... 2 K R s | 2 d s + | � n t � � m 0 | � n s � � m 0 sth. small d s + N t Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Idea of proof (Krylov, Pr´ evˆ ot/R¨ ockner): Euler approx. 0 := x and for t 2 ( k n , k + 1 For n 2 N let � n n ] : R t R t � n t := � n k / n b ( � n k / n M ( d s , � n k / n + k / n ) d s + k / n ) . So, Up to appropriate stopping time: R t R t t | 2 ... 2 K R s | 2 d s + | � n t � � m 0 | � n s � � m 0 sth. small d s + N t Now use Stochastic Gronwall Lemma (v. Renesse, S., 2010) Let Z � 0 , H be adapted cts., N cts. local mart, N 0 = 0 s.t. R t 0 Z ⇤ Z t K u d u + N t + H t , t � 0 . Then, for each 0 < p < 1 and ↵ > 1 + p 1 � p 9 c 1 , c 2 : T ) p c 1 exp { c 2 KT } ( E H ⇤ α T ) p / α . E ( Z ⇤ Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Existence of global solutions Theorem If, in addition, there exists a nondecr. ⇢ : [ 0 , 1 ) ! ( 0 , 1 ) s.t. R 1 0 1 / ⇢ ( u ) d u = 1 and 2 h b ( x ) , x i + Tr ( a ( x , x )) ⇢ ( | x | 2 ) , x 2 R d , then the local solution of the sde is global (( weakly ) complete ). Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Existence of global solutions Theorem If, in addition, there exists a nondecr. ⇢ : [ 0 , 1 ) ! ( 0 , 1 ) s.t. R 1 0 1 / ⇢ ( u ) d u = 1 and 2 h b ( x ) , x i + Tr ( a ( x , x )) ⇢ ( | x | 2 ) , x 2 R d , then the local solution of the sde is global (( weakly ) complete ). Itˆ o’s formula implies Z τ X 2 τ � X 2 0 = 2 h b ( X u ) , X u i + Tr ( a ( X u , X u )) d u + N τ 0 Z τ ⇢ ( | X u | 2 ) d u + N τ . 0 Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Existence of global solutions Theorem If, in addition, there exists a nondecr. ⇢ : [ 0 , 1 ) ! ( 0 , 1 ) s.t. R 1 0 1 / ⇢ ( u ) d u = 1 and 2 h b ( x ) , x i + Tr ( a ( x , x )) ⇢ ( | x | 2 ) , x 2 R d , then the local solution of the sde is global (( weakly ) complete ). Itˆ o’s formula implies Z τ X 2 τ � X 2 0 = 2 h b ( X u ) , X u i + Tr ( a ( X u , X u )) d u + N τ 0 Z τ ⇢ ( | X u | 2 ) d u + N τ . 0 Lemma (v. Renesse, S., 2010) Let Z � 0 be adapted cts. defined on [ 0 , � ) , N cts. local mart, N 0 = 0, C � 0 s.t. R t 0 ⇢ ( Z ⇤ Z t u ) d u + N t + C , t 2 [ 0 , � ) and lim t " σ Z ⇤ t = 1 on { � < 1 } . Then � = 1 almost surely. Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Continuous dependence on initial condition Question Are our conditions sufficient for Continuous dependence on initial conditions (or even the semi-flow property)? In particular: Do conditions for global existence of solutions ensure existence of a continuous map ' : [ 0 , 1 ) ⇥ R d ! R d which is a modification of the solution map ( strong completeness )? Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Continuous dependence on initial condition Question Are our conditions sufficient for Continuous dependence on initial conditions (or even the semi-flow property)? In particular: Do conditions for global existence of solutions ensure existence of a continuous map ' : [ 0 , 1 ) ⇥ R d ! R d which is a modification of the solution map ( strong completeness )? Answer: No! Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Continuous dependence on initial condition Question Are our conditions sufficient for Continuous dependence on initial conditions (or even the semi-flow property)? In particular: Do conditions for global existence of solutions ensure existence of a continuous map ' : [ 0 , 1 ) ⇥ R d ! R d which is a modification of the solution map ( strong completeness )? Answer: No! There exists a 2d sde without drift driven by a single BM with bounded and C 1 coefficient which is not strongly complete. Reference: Li, S.: Lack of strong completeness .. (Ann. Prob. 2011) Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
Continuous modification Lemma Assume that for some µ, K � 0, and all x , y 2 R d 2 h b ( x ) � b ( y ) , x � y i + Tr A ( x , y ) + µ k A ( x , y ) k K | x � y | 2 . Then the sde is weakly complete. Denote solutions by � t ( x ) . For each q 2 ( 0 , µ + 2 ) , there exist c 1 , c 2 s.t. E sup 0 s T | � s ( x ) � � s ( y ) | q c 1 | x � y | q exp { c 2 KT } holds for all x , y , T . Michael Scheutzow Technische Universit¨ at Berlin Completeness and semi-flows for stochastic differential equations
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