Portfolio management under risk constraints Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie 12 septembre 2013 Journée des doctorants DIM 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers • Reflected backward stochastic differential equations with jumps and partial integro variational inequalities 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers • Reflected backward stochastic differential equations with jumps and partial integro variational inequalities • Double barrier reflected BSDEs with jumps and generalized Dynkin games 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers • Reflected backward stochastic differential equations with jumps and partial integro variational inequalities • Double barrier reflected BSDEs with jumps and generalized Dynkin games • Working papers 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers • Reflected backward stochastic differential equations with jumps and partial integro variational inequalities • Double barrier reflected BSDEs with jumps and generalized Dynkin games • Working papers • Portfolio management under amount constraints 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers • Reflected backward stochastic differential equations with jumps and partial integro variational inequalities • Double barrier reflected BSDEs with jumps and generalized Dynkin games • Working papers • Portfolio management under amount constraints • BSDEs with jumps and weak nonlinear terminal condition 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Plan • Key-tool: BSDEs • Submitted papers • Reflected backward stochastic differential equations with jumps and partial integro variational inequalities • Double barrier reflected BSDEs with jumps and generalized Dynkin games • Working papers • Portfolio management under amount constraints • BSDEs with jumps and weak nonlinear terminal condition • BSDEs with weak reflection 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
Key-tool: BSDEs Definition BSDEs A process ( X , Z ) is said to be a solution of the BSDE associated with driver f and terminal condition ξ if − dX t = f ( t , X t , Z t ) dt − Z t dW t X T = ξ T The interest for this kind of stochastic equations has increased steadily. This is due to the strong connections of these equations with mathematical finance and the fact that they gave a generalization of the well known Feynman-Kac formula to semi-linear partial differential equations and fully nonlinear equations (probabilistic representations). 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
RBSDEs with jumps and PIDVIs. Mathematical tools • Reflected BSDEs with jumps A process ( Y , Z , k ( . ) , A ) is said to be a solution of the reflected BSDE with jumps associated with driver g and obstacle ξ . if � R ∗ k t ( u ) ˜ − dY t = g ( t , Y t , Z t , k t ( · )) dt + dA t − Z t dW t − N ( dt , du ) Y T = ξ T (1) ξ t ≤ Y t 0 ≤ t ≤ T a.s. , � T 0 ( Y t − ξ t ) dA c t = 0 a.s. and ∆ A d τ = ∆ A d τ 1 { Y τ − = ξ τ − } 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
RBSDEs with jumps and PIDVIs. Mathematical tools • Reflected BSDEs with jumps in the Markovian case For each ( t , x ) ∈ [ 0 , T ] × R , let { X t , x s , t ≤ s ≤ T } be the unique R -valued solution of the SDE with jumps: � s � s � s � X t , x t b ( X t , x t σ ( X t , x R ∗ β ( X t , x r − , e ) ˜ = x + ) dr + ) dW r + N ( dr , de ) s r r t The obstacle ξ t , x and driver f are of the following form: ξ t , x := h ( s , X t , x s ) , s < T s ξ t , x := g ( X t , x T ) T f ( s , X t , x s ( ω ) , y , z , k ) := ϕ ( s , X t , x s ( ω ) , y , z , � R ∗ k ( e ) γ ( x , e ) ν ( de )) 1 s ≥ t 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
• Related obstacle problem for a PIDE min ( u ( t , x ) − h ( t , x ) , − ∂ u ∂ t ( t , x ) − Lu ( t , x ) − f ( t , x , u ( t , x ) , ( σ ∂ u ∂ x )( t , x ) , Bu ( t , x )) = 0 , ( t , x ) ∈ [ 0 , T ) × R u ( T , x ) = g ( x ) , x ∈ R (2) where • L := A + K 2 σ 2 ( x ) ∂ 2 φ • A φ ( x ) := 1 ∂ x 2 ( x )+ b ( x ) ∂φ ∂ x ( x ) , φ ∈ C 2 ( R ) � φ ( x + β ( x , e )) − φ ( x ) − ∂φ � • K φ ( x ) := � ∂ x ( x ) β ( x , e ) ν ( de ) R ∗ � • B φ ( x ) := R ∗ ( φ ( x + β ( x , e )) − φ ( x )) γ ( x , e ) ν ( de ) . 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
RBSDEs with jumps and PIDVIs. Literature/Contribution Literature Links between RBSDEs and obstacle problems for PDEs in the continuous case. Contribution We extend the previous results in the case when the dynamic of Y and the obstacle admit jumps. • In the general case: We establish new a priori estimates for RBSDEs with jumps. • In the Markovian case: We define: u ( t , x ) := Y t , x (3) . t • We introduce the related obstacle problem for a parabolic PIDE (2) • We establish an existence result. More precisely, we show that u defined by (3) is a solution of the PIDVI (2). The proof we propose is different and simpler than the one used in the previous literature in the continuous case. 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
RBSDEs with jumps and PIDVIs. Literature/Contribution • We prove properties of the function u defined by (3). • We give an uniqueness result in the class of continuous functions with polynomial growth. In order to obtain it, we establish a comparison theorem. The proof of this theorem : technical difficulties! 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
RBSDEs with jumps and PIDVIs. Financial application Financial application: optimal stopping problem for dynamic risk measures induced by BSDEs with jumps. In the framework of risk measures: the state process X = an index, an interest rate process, an economic factor, an indicator of the market, the value of a portfolio, which has an influence on the risk measure and the position. • Dynamic risk measure of the financial position ζ at time t: ρ t ( ζ , S ) := − X t ( ζ , S ) where X t ( ζ , S ) = X t denotes the solution of the following BSDE: � R ∗ l t ( u ) ˜ − d X t = f ( t , X t , π t , l t ( · )) dt − π t dW t − N ( dt , du ); X S = ζ • The minimal risk measure: v ( S ) := ess inf ρ S ( ξ τ , τ ) (4) τ ∈ T S 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
RBSDEs with jumps and PIDVIs. Financial application Proposition 1. The minimal risk measure at time S satisfies v ( S ) = − Y S = − u ( S , X S ) a . s . (5) where u is the unique viscosity solution of the PIDIV (2) . Moreover, the stopping time τ ∗ S defined by S := inf { t ≥ S , Y t = ξ t } = inf { t ≥ S , u ( t , X t ) = ¯ τ ∗ h ( t , X t ) } S , τ ∗ is optimal for (4), that is v ( S ) = ρ S ( ξ τ ∗ S ) a.s. 12 septembre 2013 Journée des doctorants DIM Roxana DUMITRESCU Advisors: Bruno Bouchard and Romuald Elie (Université Paris Dauphine) Portfolio management under risk constraints / 22
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