Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control

Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control Erhan Bayraktar University of Michigan joint work with Virginia R. Young, University of Michigan K αρλoβασi, ΣAMOΣ, June 2010

Probability of Lifetime Ruin with Stochastic Consumption

Probability of Lifetime Ruin with Stochastic Consumption Consumption rate follows a geometric Brownian motion given dc t = c t (a dt + b db c t ), c 0 = c > 0.

Probability of Lifetime Ruin with Stochastic Consumption Consumption rate follows a geometric Brownian motion given dc t = c t (a dt + b db c t ), c 0 = c > 0. The individual invests in a risky asset whose price at time t, S t, follows geometric Brownian motion given by ds t = S t (µ dt + σ db S t ), S 0 = S > 0.

Probability of Lifetime Ruin with Stochastic Consumption Consumption rate follows a geometric Brownian motion given dc t = c t (a dt + b db c t ), c 0 = c > 0. The individual invests in a risky asset whose price at time t, S t, follows geometric Brownian motion given by ds t = S t (µ dt + σ db S t ), S 0 = S > 0. Assume that B c and B S are correlated Brownian motions with correlation coefficient ρ [ 1, 1].

Probability of Lifetime Ruin with Stochastic Consumption Consumption rate follows a geometric Brownian motion given dc t = c t (a dt + b db c t ), c 0 = c > 0. The individual invests in a risky asset whose price at time t, S t, follows geometric Brownian motion given by ds t = S t (µ dt + σ db S t ), S 0 = S > 0. Assume that B c and B S are correlated Brownian motions with correlation coefficient ρ [ 1, 1]. The wealth dynamics dw t = (r W t + (µ r) π t c t ) dt + σ π t db t, W 0 = w > 0.

Probability of Lifetime Ruin with Stochastic Consumption

Probability of Lifetime Ruin with Stochastic Consumption Minimizing the probability of lifetime ruin is our objective ψ(w, c) = inf π A Pw,c (τ 0 < τ d )

Probability of Lifetime Ruin with Stochastic Consumption Minimizing the probability of lifetime ruin is our objective ψ(w, c) = inf π A Pw,c (τ 0 < τ d ) τ 0 = inf{t 0 : W t 0}.

Probability of Lifetime Ruin with Stochastic Consumption Minimizing the probability of lifetime ruin is our objective ψ(w, c) = inf π A Pw,c (τ 0 < τ d ) τ 0 = inf{t 0 : W t 0}. τ d is exponentially distributed with parameter λ (Time of death).

Our Goal ψ given is decreasing and convex with respect to w, increasing with respect to c and is the unique classical solution of the following HJB equation λ v = (rw c) v w + a c v c + 1 2 b2 c 2 v cc + min [(µ r) π v w + 12 ] σ2 π 2 v ww + σ π b c ρ v wc, π v(0, c) = 1 and v(w, 0) = 0. (1)

Our Goal ψ given is decreasing and convex with respect to w, increasing with respect to c and is the unique classical solution of the following HJB equation λ v = (rw c) v w + a c v c + 1 2 b2 c 2 v cc + min [(µ r) π v w + 12 ] σ2 π 2 v ww + σ π b c ρ v wc, π v(0, c) = 1 and v(w, 0) = 0. (1) The optimal investment strategy π is given in feedback form by πt = (µ r) ψ w (Wt, c t ) + σ b ρ c t ψ wc (Wt, c t ) σ 2 ψ ww (Wt,, c t ) in which W is the optimally controlled wealth process.

Outline of the Proof

Outline of the Proof Dimension Reduction. We reduce the dimension of the problem from two variables to one and obtain another problem which also is a ruin minimization problem.

Outline of the Proof Dimension Reduction. We reduce the dimension of the problem from two variables to one and obtain another problem which also is a ruin minimization problem. Approximation. We, then, construct a regular sequence of convex functions that converges uniformly to the value function that we obtain after the dimension reduction.

Outline of the Proof Dimension Reduction. We reduce the dimension of the problem from two variables to one and obtain another problem which also is a ruin minimization problem. Approximation. We, then, construct a regular sequence of convex functions that converges uniformly to the value function that we obtain after the dimension reduction. Convex Duality. We construct this sequence by taking the Legendre transform of a controller-and-stopper problem of Karatzas.

Outline of the Proof Dimension Reduction. We reduce the dimension of the problem from two variables to one and obtain another problem which also is a ruin minimization problem. Approximation. We, then, construct a regular sequence of convex functions that converges uniformly to the value function that we obtain after the dimension reduction. Convex Duality. We construct this sequence by taking the Legendre transform of a controller-and-stopper problem of Karatzas. Analysis of the Controller and Stopper Problem.

Dimension Reduction

Dimension Reduction It turns out that ψ(w, c) = φ(w/c)

Dimension Reduction It turns out that ψ(w, c) = φ(w/c) in which φ is the unique classical solution of the following HJB equation on R + : λ f = ( rz 1) f + 1 2 b2 (1 ρ 2 ) z 2 f + f (0) = 1 and min π [(µ r σbρ) π f + 12 σ2 π 2 f ], lim f (z) = 0, z (2) in which r = r a + b 2 + (µ r σbρ)ρb/σ.

Interpretation of the Reduced Problem Consider two (risky) assets with prices S (1) and S (2) following the diffusions ( d S (1) t = S (1) t r dt + b ) 1 ρ 2 d B (1) t, and (2) d S t = in which µ = µ r + σbρ + r. ( (2) S t µ dt + b 2 (1 ρ 2 ) + σ 2 d ) (2) B t,

Interpretation of the Reduced Problem Consider two (risky) assets with prices S (1) and S (2) following the diffusions ( d S (1) t = S (1) t r dt + b ) 1 ρ 2 d B (1) t, and (2) d S t = ( (2) S t µ dt + b 2 (1 ρ 2 ) + σ 2 d ) (2) B t, in which µ = µ r + σbρ + r. Also, B (1) and B (2) are correlated standard Brownian motions with correlation coefficient ρ = b 1 ρ 2 b 2 (1 ρ 2 ) + σ 2.

Interpretation of the Reduced Problem

Interpretation of the Reduced Problem Suppose an individual has wealth Z t at time t, consumes at the constant rate of 1, and wishes to invest in these two assets in order to minimize her probability of lifetime ruin.

Interpretation of the Reduced Problem Suppose an individual has wealth Z t at time t, consumes at the constant rate of 1, and wishes to invest in these two assets in order to minimize her probability of lifetime ruin. With a slight abuse of notation, let π t be the dollar amount that the individual invests in the second asset at time t; then, Z t π t is the amount invested in the first asset at time t.

Interpretation of the Reduced Problem Suppose an individual has wealth Z t at time t, consumes at the constant rate of 1, and wishes to invest in these two assets in order to minimize her probability of lifetime ruin. With a slight abuse of notation, let π t be the dollar amount that the individual invests in the second asset at time t; then, Z t π t is the amount invested in the first asset at time t. The function φ is again a minimum probability of lifetime ruin! φ(z) = inf P z ( τ 0 < τ d ) π Ã

An Approximating Sequence Consider the hitting time τ M defined by and τ M = inf{t 0 : Z t M}, for M > 0.

An Approximating Sequence Consider the hitting time τ M defined by and τ M = inf{t 0 : Z t M}, for M > 0. Let us define the auxiliary problem φ M (z) = inf P z ( τ 0 < ( τ M τ d )), π Ã

An Approximating Sequence The modified minimum probability of lifetime ruin φ M is continuous on R + and is decreasing, convex, and C 2 on (0, M).

An Approximating Sequence The modified minimum probability of lifetime ruin φ M is continuous on R + and is decreasing, convex, and C 2 on (0, M). Additionally, φ M is the unique solution of the following HJB equation on [0, M] :

An Approximating Sequence The modified minimum probability of lifetime ruin φ M is continuous on R + and is decreasing, convex, and C 2 on (0, M). Additionally, φ M is the unique solution of the following HJB equation on [0, M] : λ f = ( rz 1) f + 1 2 b2 (1 ρ 2 ) z 2 f + min π f (0) = 1, f (M) = 0. [(µ r σbρ) π f + 12 σ2 π 2 f ],

An Approximating Sequence The modified minimum probability of lifetime ruin φ M is continuous on R + and is decreasing, convex, and C 2 on (0, M). Additionally, φ M is the unique solution of the following HJB equation on [0, M] : λ f = ( rz 1) f + 1 2 b2 (1 ρ 2 ) z 2 f + min π f (0) = 1, f (M) = 0. Furthermore, on R +, we have [(µ r σbρ) π f + 12 σ2 π 2 f ], lim φ M(z) = φ(z). M

A Controller and Stopper Problem

A Controller and Stopper Problem Define a controlled stochastic process Y α by dy α t [ = Yt α (λ r) dt + µ r σbρ d σ [ + α t ] (1) ˆB t b 1 ρ 2 dt + d ˆB (2) t ].

A Controller and Stopper Problem Define a controlled stochastic process Y α by dy α t [ = Yt α (λ r) dt + µ r σbρ d σ [ + α t ] (1) ˆB t b 1 ρ 2 dt + d ˆB (2) Admissible strategies, A(y): (α t ) t 0 that satisfy the integrability condition E[ t 0 α2 s ds] <, and Yt α 0 almost surely, for all t 0. t ].

A Controller and Stopper Problem

A Controller and Stopper Problem The controller-and-stopper problem ˆφ M (y) = [ τ sup inf Ê y α A(y) τ 0 ] e λt Yt α dt + e λτ u M (Yτ α ),

A Controller and Stopper Problem The controller-and-stopper problem ˆφ M (y) = [ τ sup inf Ê y α A(y) τ 0 ] e λt Yt α dt + e λτ u M (Yτ α ), Here payoff function u M for y 0 is given by u M (y) := min(my, 1).

Continuation Region

Continuation Region D = {y R + : ˆφ M (y) < u M (y)},

Continuation Region D = {y R + : ˆφ M (y) < u M (y)}, There exist 0 y M 1/M y 0 such that D = (y M, y 0 )

Continuation Region D = {y R + : ˆφ M (y) < u M (y)}, There exist 0 y M 1/M y 0 such that D = (y M, y 0 ) Suppose that y1 > 0 is such that ˆφ M (y 1 ) = u M (y 1 ). First, suppose that y 1 1/M; then, because ˆφ M (0) = 0 and because ˆφ M is non-decreasing, concave, and bounded above by the line My it must be that ˆφ M (y) = My for all 0 y y 1. Thus, if y 1 1/M is not in D, then the same is true for y [0, y 1 ].

Continuation Region D = {y R + : ˆφ M (y) < u M (y)}, There exist 0 y M 1/M y 0 such that D = (y M, y 0 ) Suppose that y1 > 0 is such that ˆφ M (y 1 ) = u M (y 1 ). First, suppose that y 1 1/M; then, because ˆφ M (0) = 0 and because ˆφ M is non-decreasing, concave, and bounded above by the line My it must be that ˆφ M (y) = My for all 0 y y 1. Thus, if y 1 1/M is not in D, then the same is true for y [0, y 1 ]. Finally, suppose that y1 1/M; then, because ˆφ M is non-decreasing, concave, and bounded above by the horizontal line 1 it must be that ˆφ M (y) = 1 for all y y 1. Thus, if y 1 1/M is not in D, then the same is true for y [y 1, ).

Viscosity Solutions g C 0 (R + ) is a viscosity supersolution (respectively, subsolution) if [ max λg(y 1 ) y 1 (λ r)y 1 f (y 1 ) my1 2 f (y 1 ) [ max b 1 ρ 2 αf (y 1 ) + 1 ] α 2 α2 f (y 1 ), ] g(y 1 ) u M (y 1 ) 0 (respectively, 0) whenever f C 2 (R + ) and g f has a global minimum (respectively, maximum) at y = y 1 0. (ii) g is a viscosity solution of if it is both a viscosity super- and subsolution.

Back to the Continuation Region

Back to the Continuation Region If M > 1/λ, then D = (y M, y 0 ) is non-empty. In particular, y M < 1/M < λ y 0.

Back to the Continuation Region If M > 1/λ, then D = (y M, y 0 ) is non-empty. In particular, y M < 1/M < λ y 0. Suppose M > 1/λ, and suppose that D is empty. Then, for all y 0, we have ˆφ M (y) = u M (y) = min(my, 1). ˆφ M = u M is a viscosity solution. Because M > 1/λ, there exists y 1 (1/M, λ). The value function is identically 1 in a neighborhood of y 1, the QVI evaluated at y = y 1 becomes max[λ y 1, 0] = 0, which contradicts y 1 < λ. Thus, the region D is non-empty.

Smooth Fit

Smooth Fit Assume that M > 1/λ. Let y 0 <. The function ˆφ M satisfies the smooth pasting condition, that is,

Smooth Fit Assume that M > 1/λ. Let y 0 <. The function ˆφ M satisfies the smooth pasting condition, that is, D ˆφM (y 0 ) = 0, and D + ˆφM (y M ) = M.

Smooth Fit Assume that M > 1/λ. Let y 0 <. The function ˆφ M satisfies the smooth pasting condition, that is, Assume that Let. Then the function D ˆφM (y 0 ) = 0, and D + ˆφM (y M ) = M. D + ˆφ M (y 0 ) < D ˆφ M (y 0 ). δ (D + (y 0 ) ˆφ M, D ˆφ M (y 0 )). ψ ε (y) = 1 + δ(y y 0 ) (y y 0) 2, 2ε dominates ˆφ M locally at y 0. Since ˆφ M is a viscosity subsolution of we have that λ y 0 (λ r)λδ + mλ2 ε + 1 2 b2 (1 ρ 2 ) δ2 ε 0.

Regularity of the Controller-and-Stopper Problem

Regularity of the Controller-and-Stopper Problem ˆφM is the unique classical solution of the following free-boundary problem: λg = y + (λ r)yg + my 2 g + max [b 1 ρ 2 αg + 12 ] α α2 g g(y M ) = My M and g(y 0 ) = 1. on D,

Regularity of the Controller-and-Stopper Problem ˆφM is the unique classical solution of the following free-boundary problem: λg = y + (λ r)yg + my 2 g + max [b 1 ρ 2 αg + 12 ] α α2 g g(y M ) = My M and g(y 0 ) = 1. on D, The value function for this problem, namely ˆφ M, is non-decreasing (strictly increasing on D ), concave (strictly concave on D), and C 2 on R + (except for possibly at y M where it is C 1 ).

Fenchel-Legendre Duality

Fenchel-Legendre Duality Define the convex dual Φ M (z) = max y 0 [ ˆφ M (y) zy] ( ).

Fenchel-Legendre Duality Define the convex dual Φ M (z) = max y 0 [ ˆφ M (y) zy] ( ). We have two cases to consider: (1) z M and (2) z < M.

Fenchel-Legendre Duality Define the convex dual Φ M (z) = max y 0 [ ˆφ M (y) zy] ( ). We have two cases to consider: (1) z M and (2) z < M. If z M, then Φ M (z) = 0 because ˆφ M (y) u M (y) My zy, from which it follows that the maximum on the right-hand side of (**) is achieved at y = y M.

Fenchel-Legendre Duality Define the convex dual Φ M (z) = max y 0 [ ˆφ M (y) zy] ( ). We have two cases to consider: (1) z M and (2) z < M. If z M, then Φ M (z) = 0 because ˆφ M (y) u M (y) My zy, from which it follows that the maximum on the right-hand side of (**) is achieved at y = y M. When z < M, y = I M (z) maximizes (**), in which I M is the inverse of ˆφ M on (y M, y 0 ].

Fenchel-Legendre Duality

Fenchel-Legendre Duality For z < M we have Φ M (z) = ˆφ M [I M (z)] zi M (z).

Fenchel-Legendre Duality For z < M we have Φ M (z) = ˆφ M [I M (z)] zi M (z). Which implies Φ M (z) = ˆφ M [I M(z)] I M (z) I M(z) zi M (z) = zi M (z) I M(z) zi M (z) = I M(z).

Fenchel-Legendre Duality For z < M we have Φ M (z) = ˆφ M [I M (z)] zi M (z). Which implies Φ M (z) = ˆφ M [I M(z)] I M (z) I M(z) zi M (z) = zi M (z) I M(z) zi M (z) = I M(z). Taking one more derivative Φ M (z) = I M (z) = 1/ ˆφ M [I M(z)].

Fenchel-Legendre Duality

Fenchel-Legendre Duality Letting y = I M (z) = Φ M (z) in the partial differential equation for ˆφ M we get

Fenchel-Legendre Duality Letting y = I M (z) = Φ M (z) in the partial differential equation for ˆφ M we get λ ˆφ M [I M (z)] = I M (z) + (λ r)i M (z) ˆφ M [I M(z)] + mim 2 (z) ˆφ M [I M(z)] ( 2 ˆφ M M(z)]) [I 1 2 b2 (1 ρ 2 ) ˆφ M [I M(z)].

Fenchel-Legendre Duality Letting y = I M (z) = Φ M (z) in the partial differential equation for ˆφ M we get λ ˆφ M [I M (z)] = I M (z) + (λ r)i M (z) ˆφ M [I M(z)] + mim 2 (z) ˆφ M [I M(z)] ( 2 ˆφ M M(z)]) [I 1 2 b2 (1 ρ 2 ) ˆφ M [I M(z)]. Rewrite this equation in terms of Φ M to get λφ M (z) = ( rz 1)Φ M (z) m (Φ M (z))2 Φ M (z) + 1 2 b2 (1 ρ 2 )z 2 Φ M (z).

Fenchel-Legendre Duality Letting y = I M (z) = Φ M (z) in the partial differential equation for ˆφ M we get λ ˆφ M [I M (z)] = I M (z) + (λ r)i M (z) ˆφ M [I M(z)] + mim 2 (z) ˆφ M [I M(z)] ( 2 ˆφ M M(z)]) [I 1 2 b2 (1 ρ 2 ) ˆφ M [I M(z)]. Rewrite this equation in terms of Φ M to get λφ M (z) = ( rz 1)Φ M (z) m (Φ M (z))2 Φ M (z) + 1 2 b2 (1 ρ 2 )z 2 Φ M (z). Also obtain the boundary conditions Φ M (M) = 0 and Φ M (0) = 1.

Fenchel-Legendre Duality Letting y = I M (z) = Φ M (z) in the partial differential equation for ˆφ M we get λ ˆφ M [I M (z)] = I M (z) + (λ r)i M (z) ˆφ M [I M(z)] + mim 2 (z) ˆφ M [I M(z)] ( 2 ˆφ M M(z)]) [I 1 2 b2 (1 ρ 2 ) ˆφ M [I M(z)]. Rewrite this equation in terms of Φ M to get λφ M (z) = ( rz 1)Φ M (z) m (Φ M (z))2 Φ M (z) + 1 2 b2 (1 ρ 2 )z 2 Φ M (z). Also obtain the boundary conditions Φ M (M) = 0 and Φ M (0) = 1. Thanks to a verification theorem Φ M = φ M.

The Scheme for the proofs

The Scheme for the proofs Show that ˆφ M is a viscosity solution of the quasi-variational inequality.

The Scheme for the proofs Show that ˆφ M is a viscosity solution of the quasi-variational inequality. Prove a comparison result for this quasi-variational inequality.

The Scheme for the proofs Show that ˆφ M is a viscosity solution of the quasi-variational inequality. Prove a comparison result for this quasi-variational inequality. Show that ˆφ M is C 2 and strictly concave in the continuation region.

The Scheme for the proofs Show that ˆφ M is a viscosity solution of the quasi-variational inequality. Prove a comparison result for this quasi-variational inequality. Show that ˆφ M is C 2 and strictly concave in the continuation region. Show that smooth pasting holds for the controller-and-stopper problem.

The Scheme for the proofs

The Scheme for the proofs Conclude that the convex dual, namely Φ M, of ˆφ M (via the Legendre transform) is a C 2 solution of φ M s HJB on [0, M] with Φ M (z) = 0 for z M.

The Scheme for the proofs Conclude that the convex dual, namely Φ M, of ˆφ M (via the Legendre transform) is a C 2 solution of φ M s HJB on [0, M] with Φ M (z) = 0 for z M. Show via a verification lemma that the minimum probability of ruin φ M defined in equals Φ M.

The Scheme for the proofs Conclude that the convex dual, namely Φ M, of ˆφ M (via the Legendre transform) is a C 2 solution of φ M s HJB on [0, M] with Φ M (z) = 0 for z M. Show via a verification lemma that the minimum probability of ruin φ M defined in equals Φ M. Show that lim M φ M is a viscosity solution of the HJB equation for φ.

The Scheme for the proofs Conclude that the convex dual, namely Φ M, of ˆφ M (via the Legendre transform) is a C 2 solution of φ M s HJB on [0, M] with Φ M (z) = 0 for z M. Show via a verification lemma that the minimum probability of ruin φ M defined in equals Φ M. Show that lim M φ M is a viscosity solution of the HJB equation for φ. Show that lim M φ M is smooth.

The Scheme for the proofs Conclude that the convex dual, namely Φ M, of ˆφ M (via the Legendre transform) is a C 2 solution of φ M s HJB on [0, M] with Φ M (z) = 0 for z M. Show via a verification lemma that the minimum probability of ruin φ M defined in equals Φ M. Show that lim M φ M is a viscosity solution of the HJB equation for φ. Show that lim M φ M is smooth. Show that lim M φ M = φ on R + and that φ is the unique smooth solution of the corresponding HJB.

The Scheme for the proofs Conclude that the convex dual, namely Φ M, of ˆφ M (via the Legendre transform) is a C 2 solution of φ M s HJB on [0, M] with Φ M (z) = 0 for z M. Show via a verification lemma that the minimum probability of ruin φ M defined in equals Φ M. Show that lim M φ M is a viscosity solution of the HJB equation for φ. Show that lim M φ M is smooth. Show that lim M φ M = φ on R + and that φ is the unique smooth solution of the corresponding HJB. A verification theorem shows that ψ(w, c) = φ(w/c).

References

References (1) Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control. Available on ArxiV.

References (1) Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control. Available on ArxiV. (2) Correspondence between Lifetime Minimum Wealth and Utility of Consumption, Finance and Stochastics, 2007, Volume 11 (2) 213-236.

References (1) Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control. Available on ArxiV. (2) Correspondence between Lifetime Minimum Wealth and Utility of Consumption, Finance and Stochastics, 2007, Volume 11 (2) 213-236. (3) Minimizing the Probability of Lifetime Ruin under Borrowing Constraints, Insurance Mathematics and Economics, 2007, 41: 196-221.

References (1) Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control. Available on ArxiV. (2) Correspondence between Lifetime Minimum Wealth and Utility of Consumption, Finance and Stochastics, 2007, Volume 11 (2) 213-236. (3) Minimizing the Probability of Lifetime Ruin under Borrowing Constraints, Insurance Mathematics and Economics, 2007, 41: 196-221. (4) Maximizing Utility of Consumption Subject to a Constraint on the Probability of Lifetime Ruin, Finance and Research Letters, (2008), 5 (4), 204-212.

References (1) Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control. Available on ArxiV. (2) Correspondence between Lifetime Minimum Wealth and Utility of Consumption, Finance and Stochastics, 2007, Volume 11 (2) 213-236. (3) Minimizing the Probability of Lifetime Ruin under Borrowing Constraints, Insurance Mathematics and Economics, 2007, 41: 196-221. (4) Maximizing Utility of Consumption Subject to a Constraint on the Probability of Lifetime Ruin, Finance and Research Letters, (2008), 5 (4), 204-212. (5) Optimal Investment Strategy to Minimize Occupation Time, Annals of Operations Research, 2010, 176 (1), 389-408.

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