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- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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