Indifference Pricing in Finance/Insurance First introduced by Hodges and Neuberger (1989), as a pricing principle for contingent claims in an incomplete market. The value is within the interval of arbitrage prices � � Q E Q { X e − rT } , sup E Q { X e − rT } inf , Q where Q runs over the set of all EMMs. Existing works for similar problems Cvitani´ c et al.(’01), Delbaen et al.(’02)... (martingale, duality) Rouge & El Karoui(’00) (BSDEs) M. Davis (’00), M. Musiela & Zariphopoulou(’02); Young and Zariphopoulou(’02) (PDE solutions, power/exponential utility) Bielecki, Jeanblanc and Rutkowski (’05) (defaultable claims) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 11/ 63
A Universal Variable Life Insurance Problem The Universal Variable Life (UVL for short) is an insurance product that offers a separate cash account besides a death benefit various investment options different risk/return relationships (may include money market, bond, common stocks, or even non-tradable equities.) Main Features The changes in the policy’s cash values and death benefits will be related directly to the investment performance of its underlying assets. The death benefit will not fall below a minimum amount (usually the initial face amount) even if the invested assets depreciate in value by a substantial amount. Although there is no similar “floor” to protect the cash values. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 12/ 63
The Death Benefit Consider a term life insurance with expiration date T > 0 and death benefit b t = g ( S 1 t , · · · , S d t , Z t ) = g ( S t , Z t ) , (5) where g : R d +1 �→ (0 , ∞ ) is some measurable function. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 13/ 63
The Death Benefit Consider a term life insurance with expiration date T > 0 and death benefit b t = g ( S 1 t , · · · , S d t , Z t ) = g ( S t , Z t ) , (5) where g : R d +1 �→ (0 , ∞ ) is some measurable function. Example g ( S t , Z t ) = S i t ∨ s i , for some i , g ( S t , Z t ) = Z t ∨ z . If Z is the retirement fund, one can set g ( Z t ) = Z t ∨ e ¯ rt z , t ≥ 0, where ¯ r is a certain growth rate (such as the interest rate or any contractually pre-determined rate. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 13/ 63
The Death Benefit Consider a term life insurance with expiration date T > 0 and death benefit b t = g ( S 1 t , · · · , S d t , Z t ) = g ( S t , Z t ) , (5) where g : R d +1 �→ (0 , ∞ ) is some measurable function. Example g ( S t , Z t ) = S i t ∨ s i , for some i , g ( S t , Z t ) = Z t ∨ z . If Z is the retirement fund, one can set g ( Z t ) = Z t ∨ e ¯ rt z , t ≥ 0, where ¯ r is a certain growth rate (such as the interest rate or any contractually pre-determined rate. Note: In this case the loss process is Y t = g ( S T , Z T ) 1 { T ( x ) ≤ t } , t ≥ 0. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 13/ 63
Some Optimization Problems We denote � T 0 | π t | 2 dt < ∞} A = { π : E E t , w , s , z { · } = E { · | W t = w , S t = s , Z t = z } . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 14/ 63
Some Optimization Problems We denote � T 0 | π t | 2 dt < ∞} A = { π : E E t , w , s , z { · } = E { · | W t = w , S t = s , Z t = z } . △ = E t , w , s , z { u ( W π J ( t , w , s , z ; π ) T − Y T ) } , △ J 0 ( t , w ; π ) = E t , w { u ( W π T ) } . ( T ( x ) > T , = ⇒ Y T = 0.) △ � = E t , w , s { u ( W π T − g ( S T ) Y T ) } . ( g = g ( S T )) J ( t , w , s ; π ) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 14/ 63
Some Optimization Problems We denote � T 0 | π t | 2 dt < ∞} A = { π : E E t , w , s , z { · } = E { · | W t = w , S t = s , Z t = z } . △ = E t , w , s , z { u ( W π J ( t , w , s , z ; π ) T − Y T ) } , △ J 0 ( t , w ; π ) = E t , w { u ( W π T ) } . ( T ( x ) > T , = ⇒ Y T = 0.) △ � = E t , w , s { u ( W π T − g ( S T ) Y T ) } . ( g = g ( S T )) J ( t , w , s ; π ) The Value Functions V 0 ( t , w ) = sup π ∈ A J 0 ( t , w ; π ) V ( t , w , s ) = sup π ∈ A � J ( t , w , s ; π ) U ( t , w , s , z ) = sup π ∈ A J ( t , w , s , z ; π ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 14/ 63
Solution for g = g ( S T ) First recall the Bellman Principle: for any h > 0, E t , w , s { V ( t + h , W π V ( t , w , s ) = sup t + h , S t + h ) } . (6) π ∈ A Since g ( S T ) involves all tradeable assets, and the benefit is paid at a fixed terminal time T , one can consider g ( S T ) as a contingent claim, and determine its present value by c ( t , s ) = E Q { e − r ( T − t ) g ( S T ) | S t = s } . If the death occurs during [ t , t + h ], then one can set aside the amount of c ( t + h , S t + h ) at time t + h to hedge the potential claim lost g ( S T ), and consider the remaining optimization problem on [ t + h , T ] as if there were no insurance involved. Thus, E t , w , s { V ( t + h , W π t + h , S t + h ) } = E t , w , s { V 0 ( t + h , W π t + h − c ( t + h , S t + h )) } . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 15/ 63
Solution for g = g ( S T ) Now for any π on [ t , t + h ], V ( t , w , s ) ≥ E t , w , s { V ( t + h , W π t + h , S t + h ) } h p x + t + E t , w , s { V 0 ( t + h , W π t + h − c ( t + h , S t + h )) } h q x + t . Assume that c ( · , · ) ∈ C 1 , 2 and satisfies the Black-Scholes PDE, we can apply Itˆ o to both V ( W t , t , S t ) and V 0 ( W t − c ( t , S t ) , t ) from t to t + h , and then take conditional expectations and rearrange terms to obtain V ( w , t , s ) h q x + t ≥ V 0 ( w − c ( t , s ) , t ) h q x + t h h � 1 � � t + h � � + E { V t + L [ V ]( u , W u , S u ) � W t = w h p x + t h t � 1 � t + h � � � { V 0 t + L [ V 0 ]( r , W u , S u ) + E � W t = w h q x + t . h t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 16/ 63
Solution for g = g ( S T ) Letting h → 0, noting that h → 0 h q x + t / h = λ x ( t ) , lim lim h → 0 h p x + t = 1 , lim h → 0 h q x + t = 0 , and using the fact that c satisfies the Black-Scholes PDE, we obtain the HJB Equation for V : π { ( µ − r ) π V w + 1 2 σ 2 π 2 V ww + s σ 2 π V ws } + rwV w 0= V t +max + s µ V s + 1 2 σ 2 s 2 V ss + λ x ( t )( V 0 ( w − c , t ) − V ( w , t , s )) , V ( T , w , s ) = u ( w ) . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 17/ 63
Solution for g = g ( S T ) Letting h → 0, noting that h → 0 h q x + t / h = λ x ( t ) , lim lim h → 0 h p x + t = 1 , lim h → 0 h q x + t = 0 , and using the fact that c satisfies the Black-Scholes PDE, we obtain the HJB Equation for V : π { ( µ − r ) π V w + 1 2 σ 2 π 2 V ww + s σ 2 π V ws } + rwV w 0= V t +max + s µ V s + 1 2 σ 2 s 2 V ss + λ x ( t )( V 0 ( w − c , t ) − V ( w , t , s )) , V ( T , w , s ) = u ( w ) . Note: In the Black-Scholes world, the HJB equation for V 0 is � 1 � V 0 2 | σπ | 2 V 0 ww + � π, µ − r � V 0 + rwV 0 t + max w = 0 , w (7) π ∈ R + V 0 ( T , w ) = u ( w ) . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 17/ 63
The Case of Exponential Utility Consider now the case of exponential utility. I.e., u ( w ) = − 1 α e − α w . V 0 has the close form solution: α exp {− α we r ( T − t ) − ( µ − r ) 2 V 0 ( t , w ) = − 1 ( T − t ) } (8) 2 σ 2 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 18/ 63
The Case of Exponential Utility Consider now the case of exponential utility. I.e., u ( w ) = − 1 α e − α w . V 0 has the close form solution: α exp {− α we r ( T − t ) − ( µ − r ) 2 V 0 ( t , w ) = − 1 ( T − t ) } (8) 2 σ 2 Assume V ( t , w , s ) = V 0 ( t , w )Φ( t , s ), then Φ t + rS Φ s + σ 2 s 2 Φ ss − s 2 σ 2 Φ 2 + λ x ( e { c α e r ( T − t ) } − Φ) = 0 s 2 2Φ Φ( T , s ) = 1 . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 18/ 63
The Case of Exponential Utility Consider now the case of exponential utility. I.e., u ( w ) = − 1 α e − α w . V 0 has the close form solution: α exp {− α we r ( T − t ) − ( µ − r ) 2 V 0 ( t , w ) = − 1 ( T − t ) } (8) 2 σ 2 Assume V ( t , w , s ) = V 0 ( t , w )Φ( t , s ), then Φ t + rS Φ s + σ 2 s 2 Φ ss − s 2 σ 2 Φ 2 + λ x ( e { c α e r ( T − t ) } − Φ) = 0 s 2 2Φ Φ( T , s ) = 1 . Define h ( t , s ) = c ( t , s ) α e r ( T − t ) − ln Φ. Then one shows that � h t + srh s + 1 2 σ 2 s 2 h ss − λ x ( t )( e h − 1) = 0 (9) h ( T , s ) = α g ( s ) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 18/ 63
The Case of Exponential Utility If we change the variable: v = log s , τ = T − t , (9) becomes: � h τ = ( r − 1 2 σ 2 ) h v + 1 2 σ 2 h vv − λ x ( T − τ )( e h − 1) (10) h (0 , v ) = α g ( e v ) Note: The reaction-diffusion PDE (10) has a exponential growth, and we must show that it does not blow-up in finite time! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 19/ 63
The Case of Exponential Utility If we change the variable: v = log s , τ = T − t , (9) becomes: � h τ = ( r − 1 2 σ 2 ) h v + 1 2 σ 2 h vv − λ x ( T − τ )( e h − 1) (10) h (0 , v ) = α g ( e v ) Note: The reaction-diffusion PDE (10) has a exponential growth, and we must show that it does not blow-up in finite time! Now consider the Initial-Boundary value version of (10) with h ( t , ± N ) = α g ( ± N ) . h (0 , x ) = α g ( x ) , and denote its solution by h N ( t , x ). Define ˜ K = | α |� g � ∞ , and let � T △ = − log(1 − (1 − e − ˜ K ) e 0 λ ( u ) du ) . K Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 19/ 63
The Case of Exponential Utility Consider the function � t △ = − log { 1 − (1 − e − K ) e − 0 λ ( u ) du } , t ≥ 0 . β K ( t ) Since β K ( t ) is decreasing in t , we have ˜ K = β K ( T ) ≤ β K ( t ) ≤ β K (0) = K , ∀ t ∈ [0 , T ] . △ It can be easily checked that h ( t , x ) = β K ( t ), solves (10) with the Initial-Boundary value: h (0 , x ) = K , h ( t , ± N ) = β K ( t ) . (11) Thus by Comparison Theorem of PDE h N ( · , · ) is bounded by β ˜ K ( · ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 20/ 63
The Case of Exponential Utility Similarly, denote v N ( τ, x ) = ∂ x h N ( τ, x ), and apply the Comparison Theorem to v N one sees that v N ( · , · ) is bounded � T λ ( t ) dt , with K ′ = | α |� g ′ � ∞ . v ( t , x ) = K ′ e by the function ˜ t We can now apply the Arzela-Ascoli Theorem to obtain a uniformly bounded solution of the Cauchy problem by letting N → ∞ ! The indifference price of the UVL insurance is given by p = c (0 , s ) − h (0 , s ) e − rT , α Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 21/ 63
The General Case: g = g ( S T , Z T ) Note: Since Z is non-tradable, this is an “incomplete market” case and the arbitrage free price for the payoff g ( S T , Z T ) cannot be determined as in the previous case. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 22/ 63
The General Case: g = g ( S T , Z T ) Note: Since Z is non-tradable, this is an “incomplete market” case and the arbitrage free price for the payoff g ( S T , Z T ) cannot be determined as in the previous case. A Dynamic Strategy We consider the following more aggressive (or adventurous) strategy: Assuming that the death of the insured occurs before t + h Instead of putting aside a certain amount of money at the t + h to hedge the future claim, the insurer simply continue to invest all of his current wealth freely, but knowing that he is liable to pay g ( S T , Z T ) at time T . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 22/ 63
The General Case: g = g ( S T , Z T ) Consider an auxiliary control problem assuming death happens before T △ ˜ = E t , x , s , z { u ( X π J ( t , x , s , z ; π ) T ) − g ( S T , Z T ) } , with the corresponding value function ˜ U ( t , x , s , z ). Then U satisfies a HJB equation: (assuming µ = r ) � 1 � 2 σ 2 π 2 U ww + ( U ws S σ 2 + U wz Z σ Z σ ) π 0 = U t + max π + rwU w + U s S µ + U z Z µ Z + 1 2 σ 2 U ss S 2 +1 σ 2 + σ Z 2 ) + U sz SZ σσ Z + λ x ( t )(˜ 2 U zz Z 2 (˜ U − U ) , U ( w , T , s , z ) = u ( w ) , where ˜ U satisfies a similar HJB equation with λ x ≡ 0. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 23/ 63
The General Case: g = g ( S T , Z T ) Using the similar techniques as before, modulo the technicalities of showing the no blow-ups, we can derive the indifference price in this case: The premium p ( t , s , z ) = 1 α e − r ( T − t ) h ( T − t , log s , log z ), h is a bounded, classical solution to the PDE σ 2 + σ z 2 ) h y 2 y 2 − σσ z h y 1 y 2 h τ − 1 y 2 − 1 2 σ 2 h y 1 y 1 − 1 σ 2 h 2 2 ˜ 2 (˜ � 2 σ 2 � � � µ z − µ − r σ σ z − ˜ σ 2 + σ z 2 r − 1 − h y 1 − h y 2 2 − λ x ( T − τ )( e ˜ h − h − 1) = 0; h (0 , y 1 , y 2 ) = 0 , and ˜ h is a bounded, classical solution to a similar PDE as above, with λ x ≡ 0, and ˜ h (0 , y 1 , y 2 ) = α g ( e y 1 , e y 2 ). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 24/ 63
Multiple-decrement Case Main Features Allowing “multiple decrement”: such as short/long term disabilities, withdrawl, retirement, death, etc. benefit payable at a random time, e.g., “moment of death”. the payments may depend on the different status as well as the transitions between them. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 25/ 63
Multiple-decrement Case Main Features Allowing “multiple decrement”: such as short/long term disabilities, withdrawl, retirement, death, etc. benefit payable at a random time, e.g., “moment of death”. the payments may depend on the different status as well as the transitions between them. The State/Status Process { X t } t ≥ 0 A Markov chain with finite state space { 0 , 1 , ..., m } , representing the numerical code of the “status”. i = 1 to be the “cemetary state” (death), and X 0 = 0 denote I i t = 1 { X t = i } to be the “status indicator” and define the counting process △ N ij = # { transitions of X from state i to j during [0 , t ] } . t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 25/ 63
Multiple-decrement Case Some Important Quantities for each t , denote τ t = inf { s ≥ t : X s � = X t } ; and for i = 0 , ..., m , define τ i t = τ t , if X τ t = i and ∞ otherwise. △ p i t ¯ = P { τ s > t | X s = i } ; s △ q ij = P { τ j s = τ s ≤ t | X s = i } , s ≤ t , i , j ∈ { 0 , ..., m } . t ¯ s q 1 j p 1 Clearly, t ¯ s = 1; t ¯ s = 0, for all j � = 1; and � p i q ij t ¯ s + t ¯ s = 1 , ∀ i = 0 , 1 , · · · , m , 0 ≤ s < t . (12) j � = i “ force of decrement of status i due to cause j ” as q ij t + h ¯ △ λ ij ¯ t = lim , i , j = 0 , 1 , · · · m . (13) t h h → 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 26/ 63
Some Remarks If m = 1, then the state process X becomes the one as in the simple life model, and τ 1 0 = T ( x ). In that case we should have p 0 t q 01 t ¯ s = t − s p x + s , = t − s q x + s . s Being a Markov chain, the process X has its transition probability and the corresponding transition intensity t + h q ij △ λ ij t t q ij s = P { X t = j | X s = i } ; i � = j . = lim , t h h ↓ 0 There are natural links between p ij ’s and ¯ p ij ’s. For example: ¯ λ ij t = λ ij t , for all t ≥ 0, i , j = 0 , 1 , · · · , m ; � t + h � t + h � s ds } ; t + h p ij p i λ ij p i t λ ij t + h ¯ t = exp {− t = τ ¯ τ d τ , t t j � = i ∀ h > 0, i , j = 0 , · · · , m . ... ... ... ... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 27/ 63
The Payment Process A t : Two types of payments will be considered: “life-annuity” and “life-insurance”. Since the non-tradability of the asset Z will not make significant difference in the optimization problem, we will not distinguish Z from S . The cumulative payment process is defined by � t � � I i u a i ( u , S u ) du + a ij ( u , S u ) dN ij A t = u , t ≥ 0 , 0 i i � = j (14) — an F -adapted, c` adl` ag , non-decreasing process in which a i ( t , s ) — rate of payments of annuity at state i , given S t = s ; a ij ( t , s ) — rate of payments of insurance when transit from state i to j , given S t = s . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 28/ 63
Dynamics of General Reserve Dynamics of general reserve d ˆ t = [ r t ˆ W π W π t + π t ( µ t − r t )] dt + π t σ t dB t − dA t , where � � I i ( t ) a i ( t , S t ) dt + a ij ( t , S t ) dN ij dA t = t i i � = j △ t = 1 { X t = i } , N ij I i = # { jumps of X from i to j during [0 , t ] } t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 29/ 63
Dynamics of General Reserve Dynamics of general reserve d ˆ t = [ r t ˆ W π W π t + π t ( µ t − r t )] dt + π t σ t dB t − dA t , where � � I i ( t ) a i ( t , S t ) dt + a ij ( t , S t ) dN ij dA t = t i i � = j △ t = 1 { X t = i } , N ij I i = # { jumps of X from i to j during [0 , t ] } t Hamiltonian = 1 H k △ 2 | σ t π | 2 ψ + [ � π, µ t − r t 1 � + r t w − a k ( t , s )] ϕ + � π, σ t σ T t tr D [ s ] p � , k = 0 , 1 , · · · , m , △ H k ( t , w , s , ϕ, ψ, p ) = sup π H k ( t , w , s , ϕ, ψ, p ; π ) . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 29/ 63
The HJB Equation Theorem (Yu, ’07; M.-Yu, ’10) Under suitable conditions, the value function U = ( U 0 , U 1 , ..., U m ) is the unique viscosity solution to the system of PDDE’s: � U k t + F k ( t , w , s , DU k , D 2 U k ) + ( H k U ) = 0 , (15) U k ( T , w , s ) = u ( w ) , k = 0 , · · · , m , where � � w + 1 π ( µ t − r t ) U k 2 | σ t π | 2 U k ww + πσ 2 t sU k F k ( · · · ) = sup ws π ∈ Π s + 1 + µ t sU k 2 σ 2 t s 2 U k ss + ( r t w − a k ( t , s )) U k w � λ kj t ( U j ( t , w − a kj ( t , s ) , s ) − U k ( t , w , s )) . ( H k U ) = j � = k Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 30/ 63
Viscosity Solution for System of PDDEs Main Difficulties Definition of viscosity solution for the system of PDDE. Uniqueness Different from Ishii et al.’s results: Parabolic PDDE vs. Elliptic PDEs Different from Pardoux et al.’s results: Fully Nonlinear System vs. Semilinear System Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 31/ 63
Viscosity Solution for System of PDDEs Main Difficulties Definition of viscosity solution for the system of PDDE. Uniqueness Different from Ishii et al.’s results: Parabolic PDDE vs. Elliptic PDEs Different from Pardoux et al.’s results: Fully Nonlinear System vs. Semilinear System Main idea: Taking the index vector of the value function as an additional “spatial” variable with values in a finite set: the system of PDDEs becomes a single PDDE! The abstract framework of viscosity solutions (e.g., Fleming & Soner book) applies! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 31/ 63
Abstract Dynamic Programming Principle Revisited Recall Fleming-Soner (II.3) Σ — a closed subset of a Banach space C — a collection of functions on Σ T tr , 0 ≤ t ≤ r ≤ T — a family of operators on C , s.t., (i) T tt ϕ = ϕ ; (iia) T tr ϕ ≤ T ts ψ , if ϕ ≤ ( T rs ψ ), ∀ 0 ≤ t ≤ r ≤ s ; (iib) T tr ϕ ≥ T ts ψ , if ϕ ≥ ( T rs ψ ), ∀ 0 ≤ t ≤ r ≤ s . Note r = s in (ii) = ⇒ monotonicity : T tr ϕ ≤ T tr ψ , if ϕ ≤ ψ , (iia) ⊕ (iib) = ⇒ semigroup property : T ts ϕ = T tr ( T rs ϕ ) , t ≤ r ≤ s ≤ T , if T tr ϕ ∈ C , ∀ ϕ ∈ C . Of course, the fact that T tr ϕ ∈ C must be verified! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 32/ 63
Abstract Bellman (Dynamic Programming) Principle R n , and C = M (Σ), Σ ⊆ O , where O is an open set in I �� r � △ T t , r ; u ψ ( x ) = J ( t , r ; u ) = E t , x L ( s , X s , u s ) ds + ψ ( X r ) . t △ T t , r ψ ( x ) = inf u ∈ U ad T t , r ; u ψ ( x ) (Thus, T t , T ψ ( x ) = V ( t , x )!). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 33/ 63
Abstract Bellman (Dynamic Programming) Principle R n , and C = M (Σ), Σ ⊆ O , where O is an open set in I �� r � △ T t , r ; u ψ ( x ) = J ( t , r ; u ) = E t , x L ( s , X s , u s ) ds + ψ ( X r ) . t △ T t , r ψ ( x ) = inf u ∈ U ad T t , r ; u ψ ( x ) (Thus, T t , T ψ ( x ) = V ( t , x )!). Note Semigroup Property = (Abstract) Bellman Principle(!) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 33/ 63
Abstract Bellman (Dynamic Programming) Principle R n , and C = M (Σ), Σ ⊆ O , where O is an open set in I �� r � △ T t , r ; u ψ ( x ) = J ( t , r ; u ) = E t , x L ( s , X s , u s ) ds + ψ ( X r ) . t △ T t , r ψ ( x ) = inf u ∈ U ad T t , r ; u ψ ( x ) (Thus, T t , T ψ ( x ) = V ( t , x )!). Note Semigroup Property = (Abstract) Bellman Principle(!) Let { G t } t ≥ 0 be the “ infinitesimal generator ” of the semigroup T , that is, for all ϕ ∈ D , y ∈ Σ, 1 h { ( T tt + h ϕ ( t + h , · ))( y ) − ϕ ( t , y ) } = [ ∂ lim ∂ t + G t ] ϕ ( t , y ) , h ↓ 0 where D ⊂ C ([0 , T ) × Σ) is the set of “ test functions ” [i.e., ∂ ∀ ϕ ∈ D , ∂ t ϕ ( t , y ) and ( G t ϕ ( t , · ))( y ) are continuous.] Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 33/ 63
Abstract form of HJB Equation Assume V ∈ C 1 , 2 ⊂ D . Then use the semigroup property one derives the HJB equation: 1 0 = lim h { ( T tt + h V ( t + h , · ))( y ) − V ( t , y ) } h ↓ 0 = [ ∂ (16) ∂ t + G t ] V ( t , y ) , ∀ y ∈ Σ , V ( T , y ) = ψ ( y ) . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 34/ 63
Abstract form of HJB Equation Assume V ∈ C 1 , 2 ⊂ D . Then use the semigroup property one derives the HJB equation: 1 0 = lim h { ( T tt + h V ( t + h , · ))( y ) − V ( t , y ) } h ↓ 0 = [ ∂ (16) ∂ t + G t ] V ( t , y ) , ∀ y ∈ Σ , V ( T , y ) = ψ ( y ) . Theorem (Fleming-Soner, Theorem II.5.1) If the value function of a control problem V ∈ C [0 , T ] × Σ), then V is a viscosity solution to the (abstract) HJB equation (16). Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 34/ 63
Abstract form of HJB Equation Assume V ∈ C 1 , 2 ⊂ D . Then use the semigroup property one derives the HJB equation: 1 0 = lim h { ( T tt + h V ( t + h , · ))( y ) − V ( t , y ) } h ↓ 0 = [ ∂ (16) ∂ t + G t ] V ( t , y ) , ∀ y ∈ Σ , V ( T , y ) = ψ ( y ) . Theorem (Fleming-Soner, Theorem II.5.1) If the value function of a control problem V ∈ C [0 , T ] × Σ), then V is a viscosity solution to the (abstract) HJB equation (16). Question: What are G , D ,..., etc. in our case? Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 34/ 63
Back to UVL Model Σ = { ( w , s , k ) : w , s ∈ R , k ∈ { 0 , 1 , ..., m }} , C = C (Σ). △ E w , s , k { ϕ ( ˆ W π ( T tr ϕ )( w , s , k ) = sup r , S r , X r ) } , t ≥ r π ∈ A ( T tT u )( w , s , k ) = U k ( t , w , s ), ∀ ( t , w , s ) and k Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 35/ 63
Back to UVL Model Σ = { ( w , s , k ) : w , s ∈ R , k ∈ { 0 , 1 , ..., m }} , C = C (Σ). △ E w , s , k { ϕ ( ˆ W π ( T tr ϕ )( w , s , k ) = sup r , S r , X r ) } , t ≥ r π ∈ A ( T tT u )( w , s , k ) = U k ( t , w , s ), ∀ ( t , w , s ) and k Note It is easy to check that the family { T tr } satisfies (i), (ii). Since U k ( t , w , s )’s are all continuous, the function ( t , w , s , k ) �→ U k ( t , w , s ) (on Σ) should satisfy an abstract HJB equation! Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 35/ 63
Back to UVL Model Σ = { ( w , s , k ) : w , s ∈ R , k ∈ { 0 , 1 , ..., m }} , C = C (Σ). △ E w , s , k { ϕ ( ˆ W π ( T tr ϕ )( w , s , k ) = sup r , S r , X r ) } , t ≥ r π ∈ A ( T tT u )( w , s , k ) = U k ( t , w , s ), ∀ ( t , w , s ) and k Note It is easy to check that the family { T tr } satisfies (i), (ii). Since U k ( t , w , s )’s are all continuous, the function ( t , w , s , k ) �→ U k ( t , w , s ) (on Σ) should satisfy an abstract HJB equation! Problems: Identify the infinitesimal generator of the semigroup T . Define the “viscosity solutions” to the corresponding abstract HJB equation (vs. the system of the HJB equations!) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 35/ 63
Abstract HJB Equation vs. System of PDDEs Denote U ( t , w , s , k ) = U k ( t , w , s ), and recall the PDDEs (15): ∂ ∂ t U k + F k ( t , w , s , DU k , D 2 U k ) + ( H k U )( t , w , s ) = 0 , (17) U k ( T , w , s ) = u ( w ) , k = 0 , · · · , m . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 36/ 63
Abstract HJB Equation vs. System of PDDEs Denote U ( t , w , s , k ) = U k ( t , w , s ), and recall the PDDEs (15): ∂ ∂ t U k + F k ( t , w , s , DU k , D 2 U k ) + ( H k U )( t , w , s ) = 0 , (17) U k ( T , w , s ) = u ( w ) , k = 0 , · · · , m . Theorem The viscosity solutions of the abstract HJB equation (16) with respect to the operator T and that of the system of PDDEs (17) are equivalent if and only if ( G t ϕ ( t , · ))( w , s , k ) = [ F k ( · , · , · , D ϕ, D 2 ϕ ) + ( H k ϕ )]( t , w , s ) . (18) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 36/ 63
Abstract HJB Equation vs. System of PDDEs Denote U ( t , w , s , k ) = U k ( t , w , s ), and recall the PDDEs (15): ∂ ∂ t U k + F k ( t , w , s , DU k , D 2 U k ) + ( H k U )( t , w , s ) = 0 , (17) U k ( T , w , s ) = u ( w ) , k = 0 , · · · , m . Theorem The viscosity solutions of the abstract HJB equation (16) with respect to the operator T and that of the system of PDDEs (17) are equivalent if and only if ( G t ϕ ( t , · ))( w , s , k ) = [ F k ( · , · , · , D ϕ, D 2 ϕ ) + ( H k ϕ )]( t , w , s ) . (18) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 36/ 63
The Case of Bereaved Partner Main Rationales The usual “Multi-Life Contingency” (e.g., pension plans) assumes independent mortality, even for married couples Empirical evidence of the bereaved spouse (Hu-Goldman (’90) Mariikainen-Valkonen (’96), and Valkonen et al. (’04)) indicated the possible correlated mortality. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 37/ 63
The Case of Bereaved Partner Main Rationales The usual “Multi-Life Contingency” (e.g., pension plans) assumes independent mortality, even for married couples Empirical evidence of the bereaved spouse (Hu-Goldman (’90) Mariikainen-Valkonen (’96), and Valkonen et al. (’04)) indicated the possible correlated mortality. T x 1 , T x 2 , · · · , T x n — future life time random variables, △ = min { T x 1 , · · · , T x n } — (Joint-life) T m = T x 1 , ··· , x n △ T M = T x 1 , ··· , x n = max { T x 1 , · · · , T x n } — (Last-survivor) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 37/ 63
The Case of Bereaved Partner Main Rationales The usual “Multi-Life Contingency” (e.g., pension plans) assumes independent mortality, even for married couples Empirical evidence of the bereaved spouse (Hu-Goldman (’90) Mariikainen-Valkonen (’96), and Valkonen et al. (’04)) indicated the possible correlated mortality. T x 1 , T x 2 , · · · , T x n — future life time random variables, △ = min { T x 1 , · · · , T x n } — (Joint-life) T m = T x 1 , ··· , x n △ T M = T x 1 , ··· , x n = max { T x 1 , · · · , T x n } — (Last-survivor) If n = 2, one has T M + T m = T x 1 + T x 2 , T M T m = T x 1 T x 2 . F M ( t ) + F m ( t ) = F T x 1 ( t ) + F T x 2 ( t ), t ≥ 0 where F T is the distribution function of T . If T x 1 ⊥ T x 2 , then F M ( t ) = F T x 1 ( t ) F T x 2 ( t )... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 37/ 63
The Case of Bereaved Partner Assume n = 2, and that the individual force of mortalities take the form: � µ x 1 ( t ) = λ x 1 ( t ) + 1 { T x 2 ≤ t } γ x 1 ( t − T x 2 ) t ≥ 0 , (19) µ x 2 ( t ) = λ x 2 ( t ) + 1 { T x 1 ≤ t } γ x 2 ( t − T x 1 ) , where λ x i ’s are the (marginal) force of mortality and n i γ x i ( t ) = r i e t + 1 , i = 1 , 2 , r 1 , r 2 , n 1 , n 2 > 0 . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 38/ 63
The Case of Bereaved Partner Assume n = 2, and that the individual force of mortalities take the form: � µ x 1 ( t ) = λ x 1 ( t ) + 1 { T x 2 ≤ t } γ x 1 ( t − T x 2 ) t ≥ 0 , (19) µ x 2 ( t ) = λ x 2 ( t ) + 1 { T x 1 ≤ t } γ x 2 ( t − T x 1 ) , where λ x i ’s are the (marginal) force of mortality and n i γ x i ( t ) = r i e t + 1 , i = 1 , 2 , r 1 , r 2 , n 1 , n 2 > 0 . Note: This essentially becomes a problem of “ Counter-Party Risk ”, a well-know topic in “Contagion Models” of correlated default! Existing literature include King-Wadhwani, Kodres-Pritsker, Collin-Dufresne, ... Jarrow-Yu, Yu (2001, counterparty, two firms) ......... Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 38/ 63
Basic Setup Let (Ω , F , { F t } , P ) be a given filtered probability space. P is risk neutral (in a default free bond market) ∃ a factor process X = { X t : t ≥ 0 } There are I firms, with default times τ i , i = 1 , · · · , I Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 39/ 63
Basic Setup Let (Ω , F , { F t } , P ) be a given filtered probability space. P is risk neutral (in a default free bond market) ∃ a factor process X = { X t : t ≥ 0 } There are I firms, with default times τ i , i = 1 , · · · , I Denote △ N i = 1 { τ i ≤ t } — default process with respect to τ i , t △ = F X t ∨ F 1 t ∨ ... ∨ F I t , where F i t = σ { N i s : 0 ≤ s ≤ t } , ∀ i F t t ∨ ... ∨ F i − 1 ∨ F i +1 H i t = F X t ∨ F 1 ∨ ... ∨ F I t , t t ⇒ F t = H i t ∨ F i = t . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 39/ 63
Basic Setup Define t = P { τ i > t | H i ⇒ S i is an H i -supermg) S i t } > 0 (= △ H i = − ln( S i t ), t ≥ 0 — Hazard Process t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 63
Basic Setup Define t = P { τ i > t | H i ⇒ S i is an H i -supermg) S i t } > 0 (= △ H i = − ln( S i t ), t ≥ 0 — Hazard Process t Note: t > 0 implies that τ i cannot be an H i -stopping time! S i Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 63
Basic Setup Define t = P { τ i > t | H i ⇒ S i is an H i -supermg) S i t } > 0 (= △ H i = − ln( S i t ), t ≥ 0 — Hazard Process t Note: t > 0 implies that τ i cannot be an H i -stopping time! S i � t If ∃ λ i t ∈ H i t , such that H i 0 λ i t = s ds , t ≥ 0, then � t � � t = P { τ i > t | H i S i λ i t } = exp − s ds . (20) 0 — λ i is called the (conditional) intensity process of τ i , and it holds that λ i t = − dS i t / S i t , t ≥ 0. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 40/ 63
A Useful Lemma Lemma For any F -measurable random variable Z we have, for any t ≥ 0 , E { 1 { τ i > t } Z | H i t } 1 { τ i > t } E { Z | F t } = 1 { τ i > t } (21) E { 1 { τ i > t } | H i t } Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 41/ 63
A Useful Lemma Lemma For any F -measurable random variable Z we have, for any t ≥ 0 , E { 1 { τ i > t } Z | H i t } 1 { τ i > t } E { Z | F t } = 1 { τ i > t } (21) E { 1 { τ i > t } | H i t } Idea: Define △ t , A ∩ { τ i > t } = B ∩ { τ i > t }} . F ∗ = { A ∈ F |∃ B ∈ H i t Then one can check that F t = F ∗ t , t ≥ 0. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 41/ 63
A Useful Lemma Lemma For any F -measurable random variable Z we have, for any t ≥ 0 , E { 1 { τ i > t } Z | H i t } 1 { τ i > t } E { Z | F t } = 1 { τ i > t } (21) E { 1 { τ i > t } | H i t } Idea: Define △ t , A ∩ { τ i > t } = B ∩ { τ i > t }} . F ∗ = { A ∈ F |∃ B ∈ H i t Then one can check that F t = F ∗ t , t ≥ 0. Applying “Monotone Class”, one shows that, ∀ Z ∈ F , ∃ X ∈ H i t , s.t. E { 1 { τ i > t } Z | F t } = 1 { τ i > t } E { Z | F t } = 1 { τ i > t } X . Taking E {· | H i t } on both sides and solve for X . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 41/ 63
The Conditional Survival Probability Note that P { τ i > T | F t } = 1 { τ i > t } E { 1 { τ i > T } | F t } . Applying Lemma we have E [ 1 { τ i > T } | H i t } P { τ i > T | F t } = 1 { τ i > t } t } . (22) E { 1 { τ i > t } | H i Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 63
The Conditional Survival Probability Note that P { τ i > T | F t } = 1 { τ i > t } E { 1 { τ i > T } | F t } . Applying Lemma we have E [ 1 { τ i > T } | H i t } P { τ i > T | F t } = 1 { τ i > t } t } . (22) E { 1 { τ i > t } | H i Since τ i > T | H i E { 1 { τ i > T } | H i T }| H i t } = E { P { t } = s ds � � � � T � 0 λ i e − � H i . E t � t 0 λ i E { 1 { τ i > t } | H i t } = e − s ds Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 63
The Conditional Survival Probability Note that P { τ i > T | F t } = 1 { τ i > t } E { 1 { τ i > T } | F t } . Applying Lemma we have E [ 1 { τ i > T } | H i t } P { τ i > T | F t } = 1 { τ i > t } t } . (22) E { 1 { τ i > t } | H i Since τ i > T | H i E { 1 { τ i > T } | H i T }| H i t } = E { P { t } = s ds � � � � T � 0 λ i e − � H i . E t � t 0 λ i E { 1 { τ i > t } | H i t } = e − s ds Consequently: � s ds � � � T � P { τ i > T | F t } = 1 { τ i > t } E λ i e − � H i . t t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 63
The Conditional Survival Probability Note that P { τ i > T | F t } = 1 { τ i > t } E { 1 { τ i > T } | F t } . Applying Lemma we have E [ 1 { τ i > T } | H i t } P { τ i > T | F t } = 1 { τ i > t } t } . (22) E { 1 { τ i > t } | H i Since τ i > T | H i E { 1 { τ i > T } | H i T }| H i t } = E { P { t } = s ds � � � � T � 0 λ i e − � H i . E t � t 0 λ i E { 1 { τ i > t } | H i t } = e − s ds Consequently: � s ds � � � T � P { τ i > T | F t } = 1 { τ i > t } E λ i e − � H i . t t � t △ M i = N i t − H i 0 1 { τ i > s } λ i t ∧ τ i = 1 { τ i ≤ t } − s ds , i = 1 , ..., I , are t { F t } -martingales. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 42/ 63
Standing Assumptions (H1) λ i t satisfy the following condition: � t � � I �� � λ i exp 2 s ds < ∞ , ∀ t < ∞ . E 0 i =1 (H2) For each i , P { τ i > 0 } = 1. Furthermore, there are no simultaneous defaults among the I firms. In other words, it holds that P { τ i � = τ j } = 1, whenever i � = j . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 43/ 63
Standing Assumptions (H1) λ i t satisfy the following condition: � t � � I �� � λ i exp 2 s ds < ∞ , ∀ t < ∞ . E 0 i =1 (H2) For each i , P { τ i > 0 } = 1. Furthermore, there are no simultaneous defaults among the I firms. In other words, it holds that P { τ i � = τ j } = 1, whenever i � = j . Main Task Find effective, tractable way to calculate the joint distribution (survival probability): P { τ 1 ≤ t 1 , · · · , τ I ≤ t I } , P { τ 1 > t 1 , · · · , τ I > t I } , and/or given the conditional intensities. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 43/ 63
Representation of Joint Survival Probability � t △ Define, for i = 1 , ..., I , Γ i 0 λ i = exp { s ds } , and t � � t � △ Z i = 1 { τ i > t } Γ i λ i t = 1 { τ i > t } exp s ds . (23) t 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 44/ 63
Representation of Joint Survival Probability � t △ Define, for i = 1 , ..., I , Γ i 0 λ i = exp { s ds } , and t � � t � △ Z i = 1 { τ i > t } Γ i λ i t = 1 { τ i > t } exp s ds . (23) t 0 Then Z i t ≥ 0; and Z i 0 = 1, ∀ i . Z i ’s are { F t } -adapted, and E { Z i t } = 1. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 44/ 63
Representation of Joint Survival Probability � t △ Define, for i = 1 , ..., I , Γ i 0 λ i = exp { s ds } , and t � � t � △ Z i = 1 { τ i > t } Γ i λ i t = 1 { τ i > t } exp s ds . (23) t 0 Then Z i t ≥ 0; and Z i 0 = 1, ∀ i . Z i ’s are { F t } -adapted, and E { Z i t } = 1. Proposition Assume (H1) and (H2). Then, for k = 1 , ..., I , the processes k k � � △ Z i 1 { τ i > t } Γ i = t , t ≥ 0 (24) t i =1 i =1 are all { F t } -martingales. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 44/ 63
Representation of Joint Survival Probability [Sketch of the proof.] (i) Z i t ’s are martingales. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 45/ 63
Representation of Joint Survival Probability [Sketch of the proof.] (i) Z i t ’s are martingales. E { Z i E { 1 { τ i > t } Γ i t | F s } = 1 { τ i > s } E { 1 { τ i > t } Γ i t | F s } = t | F s } E { 1 { τ i > t } Γ i t | H i t } = 1 { τ i > s } (Lemma) E { 1 { τ i > s } | H i s } E { 1 { τ i > t } Γ i t | H i s } = Z i s E { 1 { τ i > t } Γ i t | H i = 1 { τ i > s } s } (Γ i t ) − 1 = Z i s E { E { 1 { τ i > t } | H i t } Γ i t | H i s } = Z i s . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 45/ 63
Representation of Joint Survival Probability [Sketch of the proof.] (i) Z i t ’s are martingales. E { Z i E { 1 { τ i > t } Γ i t | F s } = 1 { τ i > s } E { 1 { τ i > t } Γ i t | F s } = t | F s } E { 1 { τ i > t } Γ i t | H i t } = 1 { τ i > s } (Lemma) E { 1 { τ i > s } | H i s } E { 1 { τ i > t } Γ i t | H i s } = Z i s E { 1 { τ i > t } Γ i t | H i = 1 { τ i > s } s } (Γ i t ) − 1 = Z i s E { E { 1 { τ i > t } | H i t } Γ i t | H i s } = Z i s . = � k t is an mg, then so is � k +1 △ (ii) If ˜ t = ˜ t Z k +1 Z k i =1 Z i i =1 Z i Z k . t t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 45/ 63
Representation of Joint Survival Probability [Sketch of the proof.] (i) Z i t ’s are martingales. E { Z i E { 1 { τ i > t } Γ i t | F s } = 1 { τ i > s } E { 1 { τ i > t } Γ i t | F s } = t | F s } E { 1 { τ i > t } Γ i t | H i t } = 1 { τ i > s } (Lemma) E { 1 { τ i > s } | H i s } E { 1 { τ i > t } Γ i t | H i s } = Z i s E { 1 { τ i > t } Γ i t | H i = 1 { τ i > s } s } (Γ i t ) − 1 = Z i s E { E { 1 { τ i > t } | H i t } Γ i t | H i s } = Z i s . = � k t is an mg, then so is � k +1 △ (ii) If ˜ t = ˜ t Z k +1 Z k i =1 Z i i =1 Z i Z k . t t � t � t ˜ t Z k +1 ˜ s − dZ k +1 0 + Z k +1 s − d ˜ s + [˜ Z k , Z k +1 ] t . Z k Z k Z k = + t s 0 + Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 45/ 63
Representation of Joint Survival Probability [Sketch of the proof.] (i) Z i t ’s are martingales. E { Z i E { 1 { τ i > t } Γ i t | F s } = 1 { τ i > s } E { 1 { τ i > t } Γ i t | F s } = t | F s } E { 1 { τ i > t } Γ i t | H i t } = 1 { τ i > s } (Lemma) E { 1 { τ i > s } | H i s } E { 1 { τ i > t } Γ i t | H i s } = Z i s E { 1 { τ i > t } Γ i t | H i = 1 { τ i > s } s } (Γ i t ) − 1 = Z i s E { E { 1 { τ i > t } | H i t } Γ i t | H i s } = Z i s . = � k t is an mg, then so is � k +1 △ (ii) If ˜ t = ˜ t Z k +1 Z k i =1 Z i i =1 Z i Z k . t t � t � t ˜ t Z k +1 ˜ s − dZ k +1 0 + Z k +1 s − d ˜ s + [˜ Z k , Z k +1 ] t . Z k Z k Z k = + t s 0 + Z k and Z k +1 are FV and quadratic pure jump, Since both ˜ � [˜ Z k , Z k +1 ] t = ˜ Z k 0 Z k +1 ∆˜ Z k s ∆ Z k +1 = ˜ Z k 0 Z k +1 + . 0 s 0 0 < s ≤ t Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 45/ 63
Representation of Joint Survival Probability Define � � k d P 1 , ··· , k � d P i � � △ △ = ˜ � = Z i � Z k Z i T ; T = T . (25) � � d P d P F T F T i =1 △ = E P 1 , ··· , k { X } = E { Z 1 and E 1 , ··· , k { X } T Z 2 T ... Z k T X } . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 46/ 63
Representation of Joint Survival Probability Define � � k d P 1 , ··· , k � d P i � � △ △ = ˜ � = Z i � Z k Z i T ; T = T . (25) � � d P d P F T F T i =1 △ = E P 1 , ··· , k { X } = E { Z 1 and E 1 , ··· , k { X } T Z 2 T ... Z k T X } . Then,for each k and A ∈ F t , it holds that E { 1 A ˜ Z k t E 1 , ··· , k { X | F t }} E { 1 A E { ˜ Z k T | F t } E 1 , ··· , k { X | F t }} = E 1 , ··· , k { 1 A E 1 , ··· , k { X | F t }} = E 1 , ··· , k { 1 A X } = E { 1 A E { ˜ Z k = T X | F t }} . Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 46/ 63
Representation of Joint Survival Probability Define � � k d P 1 , ··· , k � d P i � � △ △ = ˜ � = Z i � Z k Z i T ; T = T . (25) � � d P d P F T F T i =1 △ = E P 1 , ··· , k { X } = E { Z 1 and E 1 , ··· , k { X } T Z 2 T ... Z k T X } . Then,for each k and A ∈ F t , it holds that E { 1 A ˜ Z k t E 1 , ··· , k { X | F t }} E { 1 A E { ˜ Z k T | F t } E 1 , ··· , k { X | F t }} = E 1 , ··· , k { 1 A E 1 , ··· , k { X | F t }} = E 1 , ··· , k { 1 A X } = E { 1 A E { ˜ Z k = T X | F t }} . This leads to E { Z 1 T Z 2 T ... Z k T X | F t } = Z 1 t Z 2 t ... Z k t E 1 , ··· , k { X | F t } , P − a . s . (26) Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 46/ 63
Representation of Joint Survival Probability Assume I = 2, and t 1 ≤ t 2 . Apply (26) we get � � t 2 ) − 1 �� �� � P { τ 1 > t 1 , τ 2 > t 2 } Z 2 t 2 (Γ 2 = E 1 { τ 1 > t 1 } E � F t 1 t 2 ) − 1 �� � t 1 E P 2 � �� � 1 { τ 1 > t 1 } Z 2 (Γ 2 = E � F t 1 � t 1 E P 2 � t 2 ) − 1 �� �� � Z 1 t 1 Z 2 (Γ 1 t 1 ) − 1 (Γ 2 = � F t 1 E E 1 , 2 � E P 2 � t 2 ) − 1 �� �� � (Γ 1 t 1 ) − 1 (Γ 2 = � F t 1 . In particular, if t 1 = t 2 = t , then we have � t P { τ 1 > t , τ 2 > t } = E 1 , 2 � � �� ( λ 1 s + λ 2 exp − s ) ds . 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 47/ 63
Representation of Joint Survival Probability Theorem Assume (H1) and (H2). Then, (i) For any 0 ≤ t 1 ≤ t 2 ≤ ... ≤ t I < ∞ , it holds that P { τ 1 > t 1 , τ 2 > t 2 , ..., τ I > t I } E 1 , ··· , I � � E P I � t i ) − 1 �� � � � I � � � (Γ i = · · · � F t I − 1 · · · � F t 1 ; i =1 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 48/ 63
Representation of Joint Survival Probability Theorem Assume (H1) and (H2). Then, (i) For any 0 ≤ t 1 ≤ t 2 ≤ ... ≤ t I < ∞ , it holds that P { τ 1 > t 1 , τ 2 > t 2 , ..., τ I > t I } E 1 , ··· , I � � E P I � t i ) − 1 �� � � � I � � � (Γ i = · · · � F t I − 1 · · · � F t 1 ; i =1 (ii) Denote τ ∗ = min { τ 1 , · · · , τ I } , then for any 0 ≤ t ≤ T a) P { τ ∗ > t } = E 1 , ··· , I � s ds � � t � I i =1 λ i e − ; 0 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 48/ 63
Representation of Joint Survival Probability Theorem Assume (H1) and (H2). Then, (i) For any 0 ≤ t 1 ≤ t 2 ≤ ... ≤ t I < ∞ , it holds that P { τ 1 > t 1 , τ 2 > t 2 , ..., τ I > t I } E 1 , ··· , I � � E P I � t i ) − 1 �� � � � I � � � (Γ i = · · · � F t I − 1 · · · � F t 1 ; i =1 (ii) Denote τ ∗ = min { τ 1 , · · · , τ I } , then for any 0 ≤ t ≤ T a) P { τ ∗ > t } = E 1 , ··· , I � s ds � � t � I i =1 λ i e − ; 0 s ds � I 1 { τ i > t } E 1 , ··· , I � � � � T � b) P { τ ∗ > T | F t } = � I i =1 λ i e − � F t . t i =1 Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 48/ 63
Counter-Party Risk Models Two firm case: � λ A t = a 0 ( t ) + 1 { τ B ≤ t } a 1 ( t − τ B ) , (27) λ B t = b 0 ( t ) + 1 { τ A ≤ t } b 1 ( t − τ A ) , where a 0 , a 1 , b 0 , and b 1 are deterministic functions. Jin Ma (USC) Finance, Insurance, and Mathematics Roscoff 3/2010 49/ 63
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