Mean reflected SDE Paul-Eric Chaudru de Raynal Universit Savoie - PowerPoint PPT Presentation

Mean reflected SDE Paul-Eric Chaudru de Raynal Université Savoie Mont Blanc, LAMA 3rd Young researchers Meeting in Probability Numerics and Finance Joint work with P. Briand, A. Guillin and C. Labart 30 juin 2016

Motivations : measure at risk ‚ In finance : § The risk measure of a position is the (minimal) amount of own found needed by a company to hold the position § Given a risk measure, we can define a set of acceptable positions : the set of positions that do not require any own found to be hold § Given a set of acceptable positions for a company, we can define a risk measure : for any position the risk measure is the minimal amount of cash that makes the position acceptable ‚ Mathematical modelization : § p Ω , F q § X : Ω Q ω ÞÑ X p ω q P R value of the position § A Ă L 2 p Ω q set of acceptable positions § ρ A “ inf t m P R : m ` X P A u risk measure associated to the acceptable positions set § increasing § “translating invariant” : ρ p X ` m q “ ρ p X q ` m , m P R

Motivations : context § We are interesting in the positions X satisfying : E h p X q ě 0 where h : R Ñ R is an increasing function

Motivations : context § We are interesting in the positions X satisfying : E h p X q ě 0 where h : R Ñ R is an increasing function § The set of acceptable positions is A “ t X P L 2 p Ω q : E h p X q ě 0 u

Motivations : context § We are interesting in the positions X satisfying : E h p X q ě 0 where h : R Ñ R is an increasing function § The set of acceptable positions is A “ t X P L 2 p Ω q : E h p X q ě 0 u § The associated risk measure is ρ A “ inf t m P R : m ` X P A u

Motivations : context § We are interesting in the positions X satisfying : E h p X q ě 0 where h : R Ñ R is an increasing function § The set of acceptable positions is A “ t X P L 2 p Ω q : E h p X q ě 0 u § The associated risk measure is ρ A “ inf t m P R : m ` X P A u § Example : the “Value at Risk” at level α : VAR α p X q “ inf t m P R : P p m ` X q ď α u i.e. h : x ÞÑ 1 x ě 0 ´ p 1 ´ α q , 0 ă α ă 1

Motivations Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t X 0 “ x 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q § Stochastic dynamic for the value of a portfolio X through the time X t until a given date T ą 0

Motivations Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 @ t P r 0 , T s : E h p X t q ě 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q § Stochastic dynamic for the value of a portfolio X through the time X t until a given date T ą 0 § constrained to remain acceptable

Motivations Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 @ t P r 0 , T s : E h p X t q ě 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q § Stochastic dynamic for the value of a portfolio X through the time X t until a given date T ą 0 § constrained to remain acceptable § @ h ą 0 : K t ` h ´ K t amount of cash added in the portfolio between time t and t ` h to keep the position acceptable

Motivations Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q § Stochastic dynamic for the value of a portfolio X through the time X t until a given date T ą 0 § constrained to remain acceptable § @ h ą 0 : K t ` h ´ K t amount of cash added in the portfolio between time t and t ` h to keep the position acceptable § K t is the minimal amount of cash needed up to time t (increases)

Motivations Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q § Stochastic dynamic for the value of a portfolio X through the time X t until a given date T ą 0 § constrained to remain acceptable § @ h ą 0 : K t ` h ´ K t amount of cash added in the portfolio between time t and t ` h to keep the position acceptable § K t is the minimal amount of cash needed up to time t (increases) § Reflected stochastic differential equation

Motivations Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q § Stochastic dynamic for the value of a portfolio X through the time X t until a given date T ą 0 § constrained to remain acceptable § @ h ą 0 : K t ` h ´ K t amount of cash added in the portfolio between time t and t ` h to keep the position acceptable § K t is the minimal amount of cash needed up to time t (increases) § Reflected stochastic differential equation Ñ but the reflection acts on the law ë Mean reflected stochastic differential equation

Mean Reflected SDE : the constraint (heuristic) Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q ‚ How the process K looks like ?

Mean Reflected SDE : the constraint (heuristic) Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q ‚ How the process K looks like ? § Let Y be the solution on r 0 , T s of d Y t “ b p X t q d t ` σ p X t q d B t , Y 0 “ x 0 ë X t “ Y t ` K t ù ñ E r h p X t qs “ E r h p Y t ` K t qs § Set H t p K t q : “ E r h p Y t ` K t qs

Mean Reflected SDE : the constraint (heuristic) Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q ‚ How the process K looks like ? § Let Y be the solution on r 0 , T s of d Y t “ b p X t q d t ` σ p X t q d B t , Y 0 “ x 0 ë X t “ Y t ` K t ù ñ E r h p X t qs “ E r h p Y t ` K t qs § Set H t p K t q : “ E r h p Y t ` K t qs ż t § We want : E r h p X t qs ě 0 , d K t ě 0 , K 0 “ 0 , E r h p X s qs d K s “ 0 0 ‚ K t ?

Mean Reflected SDE : the constraint (heuristic) Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q ‚ How the process K looks like ? § Let Y be the solution on r 0 , T s of d Y t “ b p X t q d t ` σ p X t q d B t , Y 0 “ x 0 ë X t “ Y t ` K t ù ñ E r h p X t qs “ E r h p Y t ` K t qs § Set H t p K t q : “ E r h p Y t ` K t qs ż t § We want : E r h p X t qs ě 0 , d K t ě 0 , K 0 “ 0 , E r h p X s qs d K s “ 0 0 K t ě H ´ 1 ‚ p 0 q t

Mean Reflected SDE : the constraint (heuristic) Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q ‚ How the process K looks like ? § Let Y be the solution on r 0 , T s of d Y t “ b p X t q d t ` σ p X t q d B t , Y 0 “ x 0 ë X t “ Y t ` K t ù ñ E r h p X t qs “ E r h p Y t ` K t qs § Set H t p K t q : “ E r h p Y t ` K t qs ż t § We want : E r h p X t qs ě 0 , d K t ě 0 , K 0 “ 0 , E r h p X s qs d K s “ 0 0 K t ě p H ´ 1 q ` p 0 q ‚ t

Mean Reflected SDE : the constraint (heuristic) Consider on r 0 , T s , T ą 0 the system : d X t “ b p X t q d t ` σ p X t q d B t ` d K t , X 0 “ x 0 ş t @ t P r 0 , T s : E h p X t q ě 0 0 E r h p X s qs d K s “ 0 where B is a Brownian motion define on some filtered probability space p Ω , F , p F t q t ě 0 , P q ‚ How the process K looks like ? § Let Y be the solution on r 0 , T s of d Y t “ b p X t q d t ` σ p X t q d B t , Y 0 “ x 0 ë X t “ Y t ` K t ù ñ E r h p X t qs “ E r h p Y t ` K t qs § Set H t p K t q : “ E r h p Y t ` K t qs ż t § We want : E r h p X t qs ě 0 , d K t ě 0 , K 0 “ 0 , E r h p X s qs d K s “ 0 0 K t ě sup s ď t p H ´ 1 q ` p 0 q ‚ s