Jump-Diffusion Risk-Sensitive Asset Management




                  Jump-Diffusion Risk-Sensitive Asset
                   ...
Jump-Diffusion Risk-Sensitive Asset Management
  Outline



     Outline

      1   Introduction

      2   The Risk-Sensit...
Jump-Diffusion Risk-Sensitive Asset Management
  Introduction



     Introduction

      Risk-sensitive control is a gener...
Jump-Diffusion Risk-Sensitive Asset Management
  Introduction




      Jacobson [?], Whittle [?], Bensoussan [?] led the t...
Jump-Diffusion Risk-Sensitive Asset Management
  The Risk-Sensitive Investment Problem



     The Risk-Sensitive Investmen...
Jump-Diffusion Risk-Sensitive Asset Management
  The Risk-Sensitive Investment Problem



      Note:
             W (t) is...
Jump-Diffusion Risk-Sensitive Asset Management
  The Risk-Sensitive Investment Problem




      The wealth, V (t) of the i...
Jump-Diffusion Risk-Sensitive Asset Management
  The Risk-Sensitive Investment Problem




      By Itˆ,
           o
     ...
Jump-Diffusion Risk-Sensitive Asset Management
  The Risk-Sensitive Investment Problem




      and the Dol´ans exponentia...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem



     Solving the Stochastic Cont...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Change of Measure


     Change...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Change of Measure



      As a...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    The HJB PDE


     The HJB PDEs...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    The HJB PDE


      We can addr...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Identifying a (Unique) Candidat...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Verification Theorem


     Veri...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Solving the Stochastic Control Problem
    Existence of a C 1,2 Solution t...
Jump-Diffusion Risk-Sensitive Asset Management
  Concluding Remarks



     Concluding Remarks

             We have seen t...
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Jump-Diffusion Risk-Sensitive Asset Management

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This presentation provides and overview of the paper "Jump-Diffusion Risk-Sensitive Asset Management." The paper proposes a solution to a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure,
with drifts that are functions of an auxiliary diffusion ‘factor’ process.

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Jump-Diffusion Risk-Sensitive Asset Management

  1. 1. Jump-Diffusion Risk-Sensitive Asset Management Jump-Diffusion Risk-Sensitive Asset Management Mark Davis and S´bastien Lleo e Department of Mathematics Imperial College London Full paper available at http://arxiv.org/abs/0905.4740v1 Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  2. 2. Jump-Diffusion Risk-Sensitive Asset Management Outline Outline 1 Introduction 2 The Risk-Sensitive Investment Problem 3 Solving the Stochastic Control Problem Change of Measure The HJB PDE Identifying a (Unique) Candidate Optimal Control Verification Theorem Existence of a C 1,2 Solution to the HJB PDE 4 Concluding Remarks Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  3. 3. Jump-Diffusion Risk-Sensitive Asset Management Introduction Introduction Risk-sensitive control is a generalization of classical stochastic control in which the degree of risk aversion or risk tolerance of the optimizing agent is explicitly parameterized in the objective criterion and influences directly the outcome of the optimization. In risk-sensitive control, the decision maker’s objective is to select a control policy h(t) to maximize the criterion 1 J(x, t, h; θ) := − ln E e −θF (t,x,h) (1) θ where t is the time, x is the state variable, F is a given reward function, and the risk sensitivity θ ∈ (0, ∞) is an exogenous parameter representing the decision maker’s degree of risk aversion. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  4. 4. Jump-Diffusion Risk-Sensitive Asset Management Introduction Jacobson [?], Whittle [?], Bensoussan [?] led the theoretical development of risk sensitive control. Risk-sensitive control was first applied to solve financial problems by Lefebvre and Montulet [?] in a corporate finance context and by Fleming [?] in a portfolio selection context. However, Bielecki and Pliska [?] were the first to apply the continuous time risk-sensitive control as a practical tool that could be used to solve ‘real world’ portfolio selection problems. A major contribution was made by Kuroda and Nagai [?] who introduced an elegant solution method based on a change of measure argument which transforms the risk sensitive control problem in a linear exponential of quadratic regulator. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  5. 5. Jump-Diffusion Risk-Sensitive Asset Management The Risk-Sensitive Investment Problem The Risk-Sensitive Investment Problem Let (Ω, {Ft } , F, P) be the underlying probability space. Take a market with a money market asset S0 with dynamics dS0 (t) = a0 + A0 X (t) dt, S0 (0) = s0 (2) S0 (t) and m risky assets following jump-diffusion SDEs N dSi (t) ¯ = (a + AX (t))i dt + σik dWk (t) + γi (z)Np (dt, dz), Si (t − ) Z k=1 Si (0) = si , i = 1, . . . , m (3) X (t) is a n-dimensional vector of economic factors following dX (t) = (b + BX (t))dt + ΛdW (t), X (0) = x (4) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  6. 6. Jump-Diffusion Risk-Sensitive Asset Management The Risk-Sensitive Investment Problem Note: W (t) is a Rm+n -valued (Ft )-Brownian motion with components Wk (t), k = 1, . . . , (m + n). ¯ Np (dt, dz) is a Poisson random measure (see e.g. Ikeda and Watanabe [?]) defined as ¯ Np (dt, dz) ˜ Np (dt, dz) − ν(dz)dt =: Np (dt, dz) if z ∈ Z0 = Np (dt, dz) if z ∈ ZZ0 the jump intensity γ(z) satisfies appropriate well-posedness conditions. assume that ΣΣ > 0 (5) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  7. 7. Jump-Diffusion Risk-Sensitive Asset Management The Risk-Sensitive Investment Problem The wealth, V (t) of the investor in response to an investment strategy h(t) ∈ H, follows the dynamics dV (t) = a ˆ a0 + A0 X (t) dt + h (t) ˆ + AX (t) dt + h (t)ΣdWt V (t − ) + ¯ h (t)γ(z)Np (dt, dz) (6) Z with initial endowment V (0) = v , where ˆ := a − a0 1, a ˆ A := A − 1A0 and 1 ∈ Rm denotes the m-element unit column vector. The objective is to maximize a function of the log-return of wealth 1 1 J(x, t, h; θ) := − ln E e −θ ln V (t,x,h) = − ln E V −θ (t, x, h) θ θ (7) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  8. 8. Jump-Diffusion Risk-Sensitive Asset Management The Risk-Sensitive Investment Problem By Itˆ, o t e −θ ln V (t) = v −θ exp θ g (Xs , h(s); θ)ds χh t (8) 0 where 1 g (x, h; θ) = a ˆ (θ + 1) h ΣΣ h − a0 − A0 x − h (ˆ + Ax) 2 1 −θ + 1 + h γ(z) − 1 + h γ(z)1Z0 (z) ν(dz) Z θ (9) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  9. 9. Jump-Diffusion Risk-Sensitive Asset Management The Risk-Sensitive Investment Problem and the Dol´ans exponential χh is given by e t t t 1 χh := exp −θ t h(s) ΣdWs − θ2 h(s) ΣΣ h(s)ds 0 2 0 t + ˜ ln (1 − G (z, h(s); θ)) Np (ds, dz) 0 Z t + {ln (1 − G (z, h(s); θ)) + G (z, h(s); θ)} ν(dz)ds , 0 Z (10) with −θ G (z, h; θ) = 1 − 1 + h γ(z) (11) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  10. 10. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Solving the Stochastic Control Problem The process involves 1 change of measure; 2 deriving the HJB PDE; 3 identifying a (unique) candidate optimal control; 4 proving a verification theorem; 5 proving existence of a C 1,2 solution to the HJB PDE. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  11. 11. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Change of Measure Change of Measure This step is due to Kuroda and Nagai [?]. Let Pθ be the measure h on (Ω, FT ) defined via the Radon-Nikod´m derivative y dPθ h := χh T (12) dP For a change of measure to be possible, we must ensure that G (z, h(s); θ) < 1, which is satisfied iff h (s)γ(z) > −1 a.s. dν. t Wth = Wt + θ Σ h(s)ds 0 is a standard Brownian motion under the measure Pθ and X (t) h satisfies the SDE: dX (t) = b + BX (t) − θΛΣ h(t) dt + ΛdWth , t ∈ [0, T ] (13) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  12. 12. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Change of Measure As a result, introduce two auxiliary criterion functions under Pθ : h the risk-sensitive control problem: T 1 I (v , x; h; t, T ; θ) = − ln Eh,θ exp θ t,x g (Xs , h(s); θ)ds − θ ln v θ t (14) where Et,x [·] denotes the expectation taken with respect to the measure Pθ and with initial conditions (t, x). h the exponentially transformed criterion T ˜(v , x, h; t, T ; θ) := Eh,θ exp θ I t,x g (s, Xs , h(s); θ)ds − θ ln v t (15) Note that the optimal control problem has become a diffusion problem. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  13. 13. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem The HJB PDE The HJB PDEs The HJB PDE associated with the risk-sensitive control criterion (14) is ∂Φ (t, x) + sup Lh Φ(t, x) = 0, t (t, x) ∈ (0, T ) × Rn (16) ∂t h∈J where Lh Φ(t, x) = t b + Bx − θΛΣ h(s) DΦ 1 θ + tr ΛΛ D 2 Φ − (DΦ) ΛΛ DΦ − g (x, h; θ) 2 2 (17) and subject to terminal condition Φ(T , x) = ln v This is a quasi-linear PDE with two sources of non-linearity: the suph∈J ; the quadratic growth term (DΦ) ΛΛ DΦ; Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  14. 14. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem The HJB PDE We can address the second linearity by considering instead the semi-linear PDE associated with the exponentially-transformed problem (15): ˜ ∂Φ 1 ˜ ˜ ˜ (t, x) + tr ΛΛ D 2 Φ(t, x) + H(t, x, Φ, D Φ) = 0 (18) ∂t 2 ˜ subject to terminal condition Φ(T , x) = v −θ and where H(s, x, r , p) = inf b + Bx − θΛΣ h(s) p + θg (x, h; θ)r h∈J (19) for r ∈ R, p ∈ Rn . ˜ In particular Φ(t, x) = exp {−θΦ(t, x)}. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  15. 15. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Identifying a (Unique) Candidate Optimal Control Identifying a (Unique) Candidate Optimal Control The supremum in (16) can be expressed as sup Lh Φ t h∈J 1 θ = (b + Bx) DΦ + tr ΛΛ D 2 Φ − (DΦ) ΛΛ DΦ + a0 + A0 x 2 2 1 a ˆ + sup − (θ + 1) h ΣΣ h − θh ΣΛ DΦ + h (ˆ + Ax) h∈J 2 1 −θ − 1 + h γ(z) − 1 + θh γ(z)1Z0 (z) ν(dz) (20) θ Z Under Assumption 5 the supremum is concave in h ∀z ∈ Z a.s. dν. ˆ The supremum is reached for a unique maximizer h(t, x, p). ˆ can be taken as a Borel By measurable selection, h measurable function on [0, T ] × Rn × Rn . Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  16. 16. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Verification Theorem Verification Theorem Broadly speaking, the verification theorem states that if we have a C 1,2 ([0, T ] × Rn ) bounded function φ which satisfies the HJB PDE (16) and its terminal condition; the stochastic differential equation dX (t) = b + BX (t) − θΛΣ h(t) dt + ΛdWtθ defines a unique solution X (s) for each given initial data X (t) = x; and, there exists a Borel-measurable maximizer h∗ (t, Xt ) of h → Lh φ defined in (17); then Φ is the value function and h∗ (t, Xt ) is the optimal Markov control process. ˜ . . . and similarly for Φ and the exponentially-transformed problem. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  17. 17. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Existence of a C 1,2 Solution to the HJB PDE ˜ To show that there exists a unique C 1,2 solution Φ to the HJB PDE (18) for the exponentially transformed problem, we follow similar arguments to those developed by Fleming and Rishel [?] (Theorem 6.2 and Appendix E). Namely, we use an approximation in policy space alongside functional analysis-related results on linear parabolic partial differential equations. The approximation in policy space algorithm was originally proposed by Bellman in the 1950s (see Bellman [?] for details) as a numerical method to compute the value function. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  18. 18. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Our approach has two steps. First, we use the approximation in policy space algorithm to show existence of a classical solution in a bounded region. Next, we extend our argument to unbounded state space. To derive this second result we follow a different argument than Fleming and Rishel [?] which makes more use of the actual structure of the control problem. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  19. 19. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Zero Beta Policy: by reference to the definition of the function g ˇ in equation (9), a ‘zero beta’ (0β) control policy h(t) is an admissible control policy for which the function g is independent from the state variable x (see for instance Black [?]). A zero beta policy exists as long as the coefficient matrix A has full rank. Without loss of generality, in the following we will fix a 0β control ˇ h as a constant function of time so that ˇ g (x, h; θ) = g ˇ where g is a constant. ˇ Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  20. 20. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Functional analysis notation: denote by Lη (K ) the space of η-th power integrable functions on K ⊂ Q; · η η,K the norm in L (K ); L η (Q), 1 < η < ∞ the space of all functions ψ such that for ∂ψ ∂ 2 ψ(t, x) and all its generalized partial derivatives ∂ψ , ∂xi , ∂xiψj , ∂t x i, j = 1, . . . , n are in Lη (K ); (2) ψ η,K the Sobolev-type norm associated with L η (Q), 1 < η < ∞ and defined as n n (2) ∂ψ ∂ψ ∂2ψ ψ η,K := ψ η,K + + + ∂t η,K ∂xi η,K ∂xi xj η,K i=1 i,j=1 Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  21. 21. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Step 1: Approximation in policy space - bounded space Consider the following auxiliary problem: fix R > 0 and let BR be the open n-dimensional ball of radius R > 0 centered at 0 defined as BR := {x ∈ Rn : |x| < R}. We construct an investment portfolio by solving the optimal risk-sensitive asset allocation problem as long as X (t) ∈ BR for R > 0. Then, as soon as X (t) ∈ BR , we switch all of the wealth / ˇ into the 0β policy h from the exit time t until the end of the investment horizon at time T . Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  22. 22. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE The HJB PDE for this auxiliary problem can be expressed as ˜ ∂Φ 1 ˜ ˜ ˜ + tr ΛΛ (t)D 2 Φ + H(t, x, Φ, D Φ) = 0 ∂t 2 ∀(t, x) ∈ QR := (0, T ) × BR subject to boundary conditions ˜ Φ(t, x) = Ψ(t, x) ∀(t, x) ∈ ∂ ∗ QR := ((0, T ) × ∂BR ) ∪ ({T } × BR ) and where Ψ(T , x) = e −θ ln v ∀x ∈ BR ; Ψ(t, x) := ψ(t) := e θˇ (T −t) ∀(t, x) ∈ (0, T ) × ∂BR and g ˇ where h is a fixed arbitrary 0β policy. ψ is obviously of class C 1,2 (Q ) and the Sobolev-type norm R (2) (2) Ψ η,∂ ∗ QR = ψ η,QR (21) is finite. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  23. 23. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ ˜ ˜ Define a sequence of functions Φ1 , Φ2 ,... Φk ,... on QR = [0, T ] × BR and of bounded measurable feedback control laws h0 , h1 ,... hk ,... where h0 is an arbitrary control. Assuming hk ˜ is defined, Φk+1 solves the boundary value problem: ˜ ∂ Φk+1 1 + tr ΛΛ (t)D 2 Φk+1˜ ∂t 2 ˜ ˜ +f (t, x, hk ) D Φk+1 + θg (t, x, hk )Φk+1 = 0 (22) subject to boundary conditions ˜ Φk+1 (t, x) = Ψ(t, x) ∀(t, x) ∈ ∂ ∗ QR := ((0, T ) × ∂BR ) ∪ ({T } × BR ) Based on standard results on parabolic Partial Differential Equations (Appendix E in Fleming and Rishel [?], Chapter IV in Ladyzhenskaya, Solonnikov and Uralceva [?]), the boundary value problem (22) admits a unique solution in L η (QR ). Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  24. 24. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Moreover, for almost all (t, x) ∈ QR , k = 1, 2, . . ., we define hk+1 by the prescription hk+1 = Argminh∈J ˜ ˜ f (t, x, h) D Φk+1 + θg (t, x, h)Φk+1 (23) so that ˜ ˜ f (t, x, hk+1 ) D Φk+1 + θg (t, x, hk+1 )Φk+1 = inf ˜ ˜ f (t, x, h) D Φk+1 + θg (t, x, h)Φk+1 h∈J ˜ ˜ = H(t, x, Φk+1 , D Φk+1 ) (24) Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  25. 25. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ Observe that the sequence Φk is globally bounded: k∈N bounded from below by 0 (by Feynman-Kac). bounded from above (optimality principle and ‘zero beta’ (0β) control policy) These bounds do not depend on the radius R and are therefore valid over the entire space (0, T ) × Rn . Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  26. 26. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Step 2: Convergence Inside the Cylinder (0, T ) × BR It can be shown using a control argument that the sequence ˜ Φk ˜ is non increasing and as a result converges to a limit Φ k∈N ˜ (2) as k → ∞. Since the Sobolev-type norm Φk+1 is bounded η,QR for 1 < η < ∞, we can show that the H¨lder-type norm |Φk |1+µ is o ˜ QR also bounded by apply the following estimate given by equation (E.9) in Appendix E of Fleming and Rishel (2) |Φk |1+µ ≤ MR Φk ˜ QR ˜ η,QR (25) for some constant MR (depending on R) and where n+2 µ = 1− η n |Φk |1+µ = |Φk |µ R + ˜ QR ˜ Q |Φki |µ R ˜x Q i=1 Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  27. 27. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ ˜ |Φk (t, x) − Φk (t, y )| |Φk |µ R ˜ Q = ˜ sup |Φk (t, x)| + sup (t,x)∈QR |x − y |µ (x, y ) ∈ G 0≤t≤T ˜ ˜ |Φk (s, x) − Φk (t, x)| + sup |s − t|µ/2 x ∈G 0 ≤ s, t ≤ T As k → ∞, ˜ ˜ D Φk converges to D Φ uniformly in Lη (QR ) ; ˜ ˜ D 2 Φk converges to D 2 Φ weakly in Lη (QR ) ; and ˜ ∂ Φk ˜ ∂Φ ∂t converges to ∂t weakly in Lη (QR ). ˜ We can then prove that Φ ∈ C 1,2 (QR ). Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  28. 28. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Step 3: Convergence from the Cylinder [0, T ) × BR to the State Space [0, T ) × Rn Let {Ri }i∈N > 0 be a non decreasing sequence with limi→∞ Ri → ∞ and let {τi }i∈N be the sequence of stopping times defined as τi := inf {t : X (t) ∈ BRi } ∧ T / Note that {τi }i∈N is non decreasing and limi→∞ τi = T . ˜ Denote by Φ(i) the limit of the sequence Φk˜ on k∈N (0, T ) × BRi , i.e. ˜ ˜ Φ(i) (t, x) = lim Φk (t, x) ∀(t, x) ∈ (0, T ) × BRi (26) k→∞ Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  29. 29. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ Figure: Convergence of the Sequence Φ(i) i∈N Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  30. 30. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ The sequence (Φ(i) )i∈N is bounded and non increasing: it ˜ converges to a limit Φ. This limit satisfies the boundary condition. ˜ We now apply Ascoli’s theorem to show that Φ is C 1,2 and satisfies the HJB PDE. These statements are local properties so we can restrict ourselves to a finite ball QR . Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  31. 31. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Using the following estimate given by equation (E8) in Appendix E of Fleming and Rishel, we deduce that ˜ (2) (2) Φ(i) η,QR ≤M Ψ η,∂ ∗ QR (27) for some constant M. ˜ (2) Combineing (27) with assumption (21) implies that Φ(i) η,QR is bounded for η > 1. Critically, the bound M does not depend on k. ˜ ˜ Moreover, by Step 2 Φ(i) and D Φ(i) are uniformly bounded on any ˜ (2) compact subset of Q0 . By equation (27) we know that Φ η,QR is bounded for any bounded set QR ⊂ Q0 . Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  32. 32. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ On QR , Φ(i) also satisfies the H¨lder estimate o (2) |Φ(i) |1+µ ≤ M1 Φ(i) ˜ QR ˜ η,QR for some constant M1 depending on QR and η. ˜ (i) 2 ˜ (i) We find, that ∂ Φ and ∂ Φxj also satisfy a uniform H¨lder ∂t ∂xi o condition on any compact subset of Q. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  33. 33. Jump-Diffusion Risk-Sensitive Asset Management Solving the Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE ˜ By Ascoli’s theorem, we can find a subsequence Φl of l∈N ˜ l ˜ Φ(i) ˜ such that Φl , ∂Φ ˜ , D Φl and i∈N l∈N ∂t l∈N l∈N ˜ ˜ ˜ ∂Φ ˜ ˜ D 2 Φl tends to respective limits Φ, ∂t D Φ and D 2 Φ l∈N uniformly on each compact subset of [0, T ] × Rn . ˜ Finally, the function Φ is the desired solution of equation (18) with ˜ terminal condition Φ(T , x) = e −θ ln v Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management
  34. 34. Jump-Diffusion Risk-Sensitive Asset Management Concluding Remarks Concluding Remarks We have seen that risk-sensitive asset management can be extended to include the possibility of infinite activity jumps in asset prices. In this case a unique optimal admissible control policy and a unique classical C 1,2 ((0, T ) × Rn ) solution exists. This approach extends naturally and with similar results to a jump-diffusion version of the risk-sensitive benchmarked asset management problem (see Davis and Lleo [?] for the original paper on benchmarks in a diffusion setting). We want to extend this approach to cover credit risk, for which we needed asset price processes with jumps. We are also working on extending this approach to include jumps in the factor processes as well as holding constraints. Mark Davis and Sebastien Lleo Jump-Diffusion Risk-Sensitive Asset Management

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