Finance, Insurance, and Stochastic Control (IV) Jin Ma Spring School on “Stochastic Control in Finance” Roscoff, France, March 7-17, 2010 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 1/ 37
Outline Introduction 1 Normal martingales in a Wiener-Poisson space 2 The Stochastic Control Problem 3 The Dynamic Programming (Bellman) Principle 4 The HJB Equations 5 Uniqueness 6 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 2/ 37
A Different Look at the Reinsurance Problem Recall the general form of a reserve equation with reinsurance: � [ α f ]( t , x )˜ dX t = b ( X t , α t , π t ) dt + σ ( π t ) dB t − N ( dxdt ) , R + X 0 = x . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 3/ 37
A Different Look at the Reinsurance Problem Recall the general form of a reserve equation with reinsurance: � [ α f ]( t , x )˜ dX t = b ( X t , α t , π t ) dt + σ ( π t ) dB t − N ( dxdt ) , R + X 0 = x . Suppose that the random field α is such that there exists some predictable pair ( β , u ) so that the martingale � t � t � △ [ α f ]( s , x )˜ M u = β s dB s + N ( dxds ) t 0 0 R + satisfies the following property: d [ M u ] t = dt + u t dM u t , t ≥ 0 . (1) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 3/ 37
A Different Look at the Reinsurance Problem Recall the general form of a reserve equation with reinsurance: � [ α f ]( t , x )˜ dX t = b ( X t , α t , π t ) dt + σ ( π t ) dB t − N ( dxdt ) , R + X 0 = x . Suppose that the random field α is such that there exists some predictable pair ( β , u ) so that the martingale � t � t � △ [ α f ]( s , x )˜ M u = β s dB s + N ( dxds ) t 0 0 R + satisfies the following property: d [ M u ] t = dt + u t dM u t , t ≥ 0 . (1) t ) 2 = u t ∆ M u Note: Since ∆[ M u ] t = (∆ M u t , u exactly controls the jumps of the reserve, that is, the claim size! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 3/ 37
A Different Look at the Reinsurance Problem The equation (1) implies that � M � t = t . A martingale with such a property is called a “ normal martingale ” (Dellacherie (1989)) and the equation (1) is called the “ structure equation ” (Emery (1989)) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 4/ 37
A Different Look at the Reinsurance Problem The equation (1) implies that � M � t = t . A martingale with such a property is called a “ normal martingale ” (Dellacherie (1989)) and the equation (1) is called the “ structure equation ” (Emery (1989)) One can show (and will do) that for any bounded, predictable process u there are always such α and β , at least when the probability space is “nice”. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 4/ 37
A Different Look at the Reinsurance Problem The equation (1) implies that � M � t = t . A martingale with such a property is called a “ normal martingale ” (Dellacherie (1989)) and the equation (1) is called the “ structure equation ” (Emery (1989)) One can show (and will do) that for any bounded, predictable process u there are always such α and β , at least when the probability space is “nice”. Then, noting that the Brownian motion B itself satisfies (1) with u ≡ 0, one can rewrite (1) as � t � t σ ( π s ) dM u X t = x + b ( X s , u s , π s ) ds + ˜ s , t ≥ 0 , (2) 0 0 where M u is a (possibly multi-dimensional) martingale satisfying the Structure Equation. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 4/ 37
A Different Look at the Reinsurance Problem Note The process u “controls” exactly the jump sizes of M u (whence that of X ) π could be regarded as a “regular” control. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 5/ 37
A Different Look at the Reinsurance Problem Note The process u “controls” exactly the jump sizes of M u (whence that of X ) π could be regarded as a “regular” control. The system (2) provides a new model for stochastic control problems in which the control of the jump size is essential. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 5/ 37
A Different Look at the Reinsurance Problem Note The process u “controls” exactly the jump sizes of M u (whence that of X ) π could be regarded as a “regular” control. The system (2) provides a new model for stochastic control problems in which the control of the jump size is essential. Some references: Ma-Protter-San Martin (1998) — Anticipating calculus and an Ocone-Haussmann-Clark type formula for normal martingales Dritschel-Protter (1999) — complete market with discontinuous security prices. Buckdahn-Ma-Rainer (2008) — Stochastic Control for Systems driven by normal mg. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 5/ 37
Normal Martingales and Structure Equation △ (Ω , F , P ; F )— a filtered probability space, and F = { F t } t ≥ 0 satisfies the “ usual hypotheses ”. 0 ( F , P ) — the space of all L 2 P -martingales s.t. X 0 = 0. M 2 X ∈ M 2 0 ( F , P ) is “normal” if � X � t = t , (i.e., [ X ] t = t + mg.) If a normal martingale also has the “Representation Property”, then there exists an F -predictable process u such that � t [ X ] t = t + u s dX s , ∀ t ≥ 0 . (3) 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 6/ 37
Normal Martingales and Structure Equation △ (Ω , F , P ; F )— a filtered probability space, and F = { F t } t ≥ 0 satisfies the “ usual hypotheses ”. 0 ( F , P ) — the space of all L 2 P -martingales s.t. X 0 = 0. M 2 X ∈ M 2 0 ( F , P ) is “normal” if � X � t = t , (i.e., [ X ] t = t + mg.) If a normal martingale also has the “Representation Property”, then there exists an F -predictable process u such that � t [ X ] t = t + u s dX s , ∀ t ≥ 0 . (3) 0 Warning A solution to the structure equation must be “normal” but the converse it not necessarily true! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 6/ 37
Some examples Example u ≡ 0 — Brownian motion △ u ≡ α ∈ R ∗ ( = R \ { 0 } ) — compensated Poisson process u t = − X t — Az´ ema’s martingale u t = − 2 X t — “ Parabolic martingale ” (Protter-Sharpe (1979)). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 7/ 37
Some examples Example u ≡ 0 — Brownian motion △ u ≡ α ∈ R ∗ ( = R \ { 0 } ) — compensated Poisson process u t = − X t — Az´ ema’s martingale u t = − 2 X t — “ Parabolic martingale ” (Protter-Sharpe (1979)). Characteristics of the solutions to structure equations Let X ∈ M 2 0 ( F , P ) be a solution to (3), and denote △ D X ( ω ) = { t > 0; ∆ X t ( ω ) � = 0 } , ω ∈ Ω. Then, ∆ X t = u t , for all t ∈ D X , P -a.s. Decomposing X = X c + X d , one has dX c t = 1 { u t =0 } dX t , and dX d t = 1 { u t � =0 } dX t , t ≥ 0. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 7/ 37
Normal martingales in a Wiener-Poisson space Note that in general the well-posedness of higher dimensional structure equation is not trivial. References include Meyer (1989), Kurtz-Protter (1991), and Phan (2001), ... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 8/ 37
Normal martingales in a Wiener-Poisson space Note that in general the well-posedness of higher dimensional structure equation is not trivial. References include Meyer (1989), Kurtz-Protter (1991), and Phan (2001), ... “Wiener-Poisson” space Assume that on (Ω , F , P ) there exist B — a d -dimensional standard Brownian motion µ — a Poisson random measure, such that B ⊥ ⊥ µ , and with △ the compensator � µ ( dtdx ) = ν ( dx ) dt , where ν is the L´ evy measure of µ . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 8/ 37
Normal martingales in a Wiener-Poisson space Note that in general the well-posedness of higher dimensional structure equation is not trivial. References include Meyer (1989), Kurtz-Protter (1991), and Phan (2001), ... “Wiener-Poisson” space Assume that on (Ω , F , P ) there exist B — a d -dimensional standard Brownian motion µ — a Poisson random measure, such that B ⊥ ⊥ µ , and with △ the compensator � µ ( dtdx ) = ν ( dx ) dt , where ν is the L´ evy measure of µ . Denote F B ,µ = { F B ,µ } t ≥ 0 to be the natural filtration generated by B t and µ , and let = F B ,µ P (augmentation) satisfies the usual hypothses. △ F Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 8/ 37
Normal martingales in a Wiener-Poisson space Martingale Representation Theorem (Jacod-Shiryaev (1987/2003)) For any X ∈ M 2 0 ( F , P ; R d ), There exists a unique pair ( α, β ) ∈ L 2 F ([0 , T ]; R d × d ) × L 2 F ([0 , T ] × R ∗ ; dt × d ν ; R d ), such that � t � X t = α s dB s + [0 , t ] × R ∗ β s ( x )˜ µ ( dsdx ) , t ≥ 0 . (4) 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 9/ 37
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