Asset-Liability Management via Risk-Sensitive Control: Jump-Diffusion

Transcription

1 Asset-Liability Management via Risk-Sensitive Control: Jump-Diffusion Mark Davis 1 and Sébastien Lleo 2 ICSP 2013 University of Bergamo, July 8, Department of Mathematics, Imperial College London, London SW7 2A, England, mark.davis@imperial.ac.uk 2 Finance Department and Behavioral Sciences Research Centre, Reims Management School, 59 rue Pierre Taittinger, Reims, France, sebastien.lleo@reims-ms.fr

2 Outline Outline Asset and Liability Management Risk Sensitive Control: A Definition The Risk-Sensitive ALM Problem Equity and Leverage How to Solve a Stochastic Control Problem Existence of a C 1,2 Solution to the HJB PDE Verification Theorem Concluding Remarks

3 Asset and Liability Management Asset and Liability Management Asset and liability management (ALM) is critical for funded investors such as endowment funds and pension funds, but also for investors who have the ability to grow their asset base by borrowing: banks and hedge funds. In this talk we solve an ALM problem using Risk-Sensitive Control methods. Our work is related to the articles on surplus management by Rudolf and iemba [13] and Benk [2]. In our approach, the investor s objective is to jointly select an optimal amount of leverage and an optimal asset allocation to maximise the expected utility of the equity or surplus of his/her portfolio. Also, we allow securities prices and the value of the liability to be influenced by a number of underlying factors.

4 Asset and Liability Management Balance Sheet of a Funded Investor Assets at 'me t: Valued at V(t) Cons'tuted of a por5olio of m securi'es. Liabili'es at 'me t: Valued at L(t) Equity at 'me t: Values at E(t) E(t)=V(t)- L(t) Leverage ra'o: ρ(t)=v(t)/e(t)

5 Risk Sensitive Control: A Definition Risk Sensitive Control: A Definition 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 J(x, t, h; θ) := 1 θ ln E [e θf (t,x,h)] (1) where t is the time, x is the state variable, F is a given reward function, and the risk sensitivity θ ( 1, 0) (0, ) is an exogenous parameter representing the decision maker s degree of risk aversion.

6 Risk Sensitive Control: A Definition An Intuitive View of the Criterion A Taylor expansion of the previous expression around θ = 0 evidences the vital role played by the risk sensitivity parameter: J(x, t, h; θ) = E [F (t, x, h)] θ 2 Var [F (t, x, h)] + O(θ2 ) (2) θ 0, risk-null : corresponds to classical stochastic control; θ < 0: risk-seeking case corresponding to a maximization of the expectation of a convex decreasing function of F (t, x, h); θ > 0: risk-averse case corresponding to a minimization of the expectation of a convex increasing function of F (t, x, h).

7 Risk Sensitive Control: A Definition From Risk-Sensitive Control to Risk-Sensitive Asset Management If we chose F (t, x) = ln Wealth T, the Taylor expression is tantamount to a dynamic version of Markowitz mean-variance analysis, but with a built-in correction for higher moments. This leads us to the risk-sensitive asset management model: J(x, t, h; θ) := 1 ] [e θ ln E θ ln Wealth T = 1 [ ] θ ln E Wealth θ T = E [ln Wealth T ] θ 2 Var [ln Wealth T ] + O(θ 2 )

8 Risk Sensitive Control: A Definition Emergence of a Risk-Sensitive Asset Management (RSAM) Theory Jacobson [9], Whittle [14], Bensoussan [3] led the theoretical development of risk sensitive control. Risk-sensitive control first applied to finance by Lefebvre and Montulet [11] in a corporate finance context and by Fleming [6] in a portfolio selection context. Bielecki and Pliska [4]: first to apply continuous time risk-sensitive control as a practical tool to solve real world portfolio selection problems. Major contribution by Kuroda and Nagai [10]: elegant solution method based on a change of measure argument.

9 Risk Sensitive Control: A Definition Extensions to a Jump-Diffusion Setting The Risk-Sensitive Asset Management (RSAM) theory was developed based on diffusion models. In a jump-diffusion setting, Davis and Lleo [5] consider a finite time horizon problem with random jumps in the asset prices. They prove the existence of an optimal control and showed that the value function is a smooth (strong) solution of the Hamilton Jacobi Bellman Partial Differential Equation (HJB PDE). Davis and Lleo [6] consider a finite time horizon problem with random jumps in both asset prices and factors. Under standard control assumptions they prove the existence of an optimal control and showed that the value function is a smooth (strong) solution of the Hamilton Jacobi Bellman Partial Differential Equation (HJB PDE). Davis and Lleo [7] extend these results to a benchmarked asset management problem.

10 The Risk-Sensitive ALM Problem The Risk-Sensitive ALM Problem - General Model Let (Ω, {F t}, F, P) be the underlying probability space. Take a market with 1. A money market asset S 0 with dynamics ds 0(t) S 0(t) = a0 (t, X (t)) dt, S0(0) = s0 (3) 2. m risky assets following jump-diffusion SDEs ds i (t) = [ a ( t, X (t ) )] S i (t ) dt + N Σ i ik (t, X (t))dw k (t) + γ i (t, z) N(dt, dz), k=1 S i (0) = s i, i = 1,..., m (4) 3. A liability given by the same type of jump-diffusion process: dl(t) L(t) = c(t, X (t ))dt+ς (t, X (t))dw (t)+ η(t, z) N(dt, dz), L(0) = l (5) 4. A n-dimensional vector of factors X (t) following dx (t) = b ( t, X (t ) ) dt + Λ(t, X (t))dw (t) + ξ ( t, X (t ), z ) N(dt, dz), X (0) = x 0 R n. (6)

11 The Risk-Sensitive ALM Problem Note: W (t) is a R m+n+1 -valued (F t)-brownian motion with components W k (t), k = 1,..., (m + n + 1). N p(dt, dz) is a Poisson random measure (see e.g. Ikeda and Watanabe [8]) defined as N p(dt, dz) = { Np(dt, dz) ν(dz)dt =: Ñ p(dt, dz) if z 0 N p(dt, dz) if z \ 0

12 The Risk-Sensitive ALM Problem Two Implementations We consider two different implementations of this general model: 1. Affine dynamics, with jumps in assets and liabilities only; 2. Standard control assumptions with jumps in assets, liabilities and factors.

13 The Risk-Sensitive ALM Problem Assumption 1: Affine dynamics with jumps in assets and liabilities 1. The money market asset S 0 has a dynamics ds 0(t) S 0(t) = ( a 0 + A 0X (t) ) dt, S 0(0) = s 0 (7) 2. m risky assets following jump-diffusion SDEs ds i (t) S i (t ) = (a + AX (t)) i dt + N k=1 Σ ik dw k (t) + γ i (z) N(dt, dz), S i (0) = s i, i = 1,..., m (8) 3. A liability given by the same type of jump-diffusion process: dl(t) L(t) = ( c + C X (t) ) dt+ς dw (t)+ η(z) N(dt, dz), L(0) = l (9) 4. A n-dimensional vector of factors X (t) following dx (t) = (b + BX (t)) dt + ΛdW (t), X (0) = x 0 R n. (10)

14 The Risk-Sensitive ALM Problem Assumption 2: Standard control assumptions Under standard control assumptions, our model has the same form as the general model: 1. A money market asset S 0 with dynamics ds 0(t) S 0(t) = a0 (t, X (t)) dt, S0(0) = s0 (11) 2. m risky assets following jump-diffusion SDEs ds i (t) = [ a ( t, X (t ) )] S i (t ) dt + N Σ i ik (t, X (t))dw k (t) + γ i (t, z) N(dt, dz), k=1 S i (0) = s i, i = 1,..., m (12) 3. A liability given by the same type of jump-diffusion process: dl(t) L(t) = c(t, X (t ))dt+ς (t, X (t))dw (t)+ η(t, z) N(dt, dz), L(0) = l (13) 4. A n-dimensional vector of factors X (t) following dx (t) = b ( t, X (t ) ) dt + Λ(t, X (t))dw (t) + ξ ( t, X (t ), z ) N(dt, dz), X (0) = x 0 R n. (14)

15 The Risk-Sensitive ALM Problem... The functions a 0, a, b, c, Σ = [σ ij ], ς, Λ are Lipschitz continuous, bounded with bounded derivatives in terms of the variables t and x. Ellipticity condition: ΣΣ > 0 (15) The jump intensities ξ(z) and γ(z) satisfies appropriate well-posedness conditions. Independence of systematic (factor-driven) and idiosyncratic (asset-driven and liability-driven) jump: (t, x, z) [0, T ] R n, γ(t, z)ξ (t, x, z) = η(t, z)ξ (t, x, z) = 0.

16 The Risk-Sensitive ALM Problem... plus an extra condition: Assumption The vector valued function γ(t, z) satisfy: ξ(t, x, z) ν(dz) <, (t, x) [0, T ] R n (16) Note that the minimal condition on ξ under which the factor equation (14) is well posed is 0 ξ(t, x, z) 2 ν(dz) <, However, for this paper it is essential to impose the stronger condition (16) in order to connect the viscosity solution of HJB partial integro-differential equation (PIDE) with the viscosity solution of a related parabolic PDE.

17 The Risk-Sensitive ALM Problem Next Steps: Find the Dynamics of The Assets and Equity Assets at 'me t: Valued at V(t) Cons'tuted of a por1olio of m securi'es. Liabili'es at 'me t: Valued at L(t) Equity at 'me t: Values at E(t) E(t)=V(t)- L(t) Leverage ra'o: ρ(t)=v(t)/e(t)

18 The Risk-Sensitive ALM Problem Wealth Dynamics The wealth, V (t) of the investor in response to an investment strategy h(t) H, follows the dynamics dv (t) V (t ) = (a 0 (t, X (t))) dt + h (t)â (t, X (t)) dt + h (t)σ(t, X (t))dw t + h (t)γ(t, z) N p(dt, dz) (17) with initial endowment V (0) = v, where â := a a 01 and 1 R m denotes the m-element unit column vector.

19 Equity and Leverage Equity and Leverage The time t equity or surplus, E(t), is the wealth belonging directly to the investor. It is is defined as the difference between the value of the assets and of the liabilities, i.e. E(t) = V (t) L(t), E(0) := e 0 = v l > 0 WLOG, we assume that e 0 = 1. The dynamics of the equity is given in differential form by de(t) = dv (t) dl(t)

20 Equity and Leverage The time t leverage ratio, ρ(t), is defined as the ratio of asset value to equity value: ρ(t) = V (t) E(t) Thus, V (t) = ρ(t)e(t) L(t) = (ρ(t) 1)E(t) In our case, leverage is a control variable: the investor s objective is to choose both an optimal level of leverage and an optimal investment strategy.

21 Equity and Leverage The dynamics of the equity in response to an ALM policy (h(t), ρ(t)) can be expressed as de(t) = V (t ) [ (a 0 (t, X (t))) dt + h (t)â (t, X (t)) dt + h (t)σ(t, X (t))dw t ] + h (t)γ(t, z) N(dt, dz) [ ] L(t ) c(t, X (t ))dt + ς (t, X (t))dw (t) + η(t, z) N(dt, dz)

22 Equity and Leverage Rewriting in terms of equity and leverage only, we get de(t) E(t ) where = c(t, X (t))dt +ρ(t) [ h (t)â(t, X (t)) ĉ(t, X (t)) ] dt + ( ς (t, X (t)) + ρ(t)[h (t)σ(t, X (t)) ς (t, X ())] ) dw (t), { [ + η(t, z) + ρ(t) h (t)γ(t, z) η(t, z) ]} N(dt, dz) = α(t, X (t), h(t), ρ(t))dt + β(t, X (t), h(t), ρ(t))dw (t), + ζ((t, z, h(t), ρ(t))) N(dt, dz) (18) α(t, x, h, ρ) := c(t, x) + ρ [ h â(t, x) ĉ(t, x) ] β(t, x, h, ρ) := ς (t, x) + ρ(h Σ(t, x) ς (t, x)) ζ(t, z, h, ρ) := η(t, z) + ρ(t) [ h (t)γ(t, z) η(t, z) ] and ĉ := c a 0.

23 Equity and Leverage Investment and Leverage Constraints We also consider r N fixed investment constraints expressed in the form Υ h(t) υ (19) where Υ R m R r is a matrix and υ R r is a column vector. Assumption The system Υ y υ for the variable y R m admits at least two solutions. We also introduce the following constraints on leverage: K := { ρ R : < ρ ρ(t) ρ + < } (20) where ρ, ρ + are two real constants. These assumptions guarantee that there will be at least one ALM policy satisfying the constraints.

24 Equity and Leverage Problem Formulation The investor s objective is to maximise the risk-sensitive criterion J(h, ρ) J(h, ρ; θ) := 1 ] [e θ ln E θ ln E T = 1 [ ] θ ln E ln E θ T (21) where ln E T (h) can be interpreted as the log return on equity.

25 Equity and Leverage From (17) and the general Itô formula we find that the term e θ ln E(T ) can be expressed as { T } e θ ln E(T ) = exp θ g(t, X t, h(t))dt χ h (T ) (22) 0 where g(t, x, h) = 1 2 (θ + 1) ββ (t, x, h, ρ) α(t, x, h, ρ) { 1 [ ] } + (ζ(t, z, h, ρ)) θ 1 + ζ(t, z, h, ρ)1 0 (z) ν(dz) θ = 1 2 (θ + 1) [ ς (t, x) + ρ(h Σ(t, x) ς (t, x)) ] [ ς (t, x) + ρ(h Σ(t, x) ς (t, x)) ] c(t, x) ρ [ h â(t, x) ĉ(t, x) ] { 1 [ (η(t, [ + z) + ρ(t) h (t)γ(t, z) η(t, z) ]) θ 1] θ + ( η(t, z) + ρ(t) [ h (t)γ(t, z) η(t, z) ]) 1 0 (z) } ν(dz) (23)

26 Equity and Leverage and the Doléans exponential χ h t is given by { t χ h (t) := exp θ β(s, X (s), h(s), ρ(s))dw s with 0 1 t 2 θ2 β(s, X (s), h(s), ρ(s))β(s, X (s), h(s), ρ(s)) ds + + t 0 t 0 0 ln (1 G(s, z, h(s), ρ(s))) Ñ(ds, dz) } {ln (1 G(s, z, h(s), ρ(s))) + G(s, z, h(s), ρ(s))} ν(dz)ds, (24) G(t, z, h, ρ) = 1 (ζ(t, z, h, ρ)) θ = 1 ( η(t, z) + ρ(t) [ h (t)γ(t, z) η(t, z) ]) θ (25)

27 Equity and Leverage Change of Measure For h A, ρ R and θ > 0 let P h be the measure on (Ω, F T ) defined via the Radon-Nikodým derivative dp h dp = χh (T ), (26) and let E h denote the corresponding expectation. Then J(h, ρ) = 1 ( )] [exp θ ln Eh θ g(t, X t, h(t))dt. (27) Moreover, under P h, W h t = W t + θ = W t + θ t 0 t β(s, X (s), h(s), ρ(s))ds is a standard Brownian motion and ς (s, X (s)) + ρ(s)(h (s)σ(s, X (s)) ς (s, X (s)))ds

28 Equity and Leverage... the P h -compensated Poisson random measure is given by t Ñ h (ds, dz) = = = 0 t 0 t 0 t 0 0 N(ds, dz) 0 N(ds, dz) 0 N(ds, dz) 0 t 0 t 0 t 0 {1 G(s, z, h(s))} ν(dz)ds 0 { (ζ(s, z, h(s), ρ(s))) θ} ν(dz)ds 0 0 { (η(s, z) + ρ(s) [ h (s)γ(s, z) η(s, z) ]) θ } ν(dz)ds

29 Equity and Leverage As a result, under P h the factor process X (s), 0 s t satisfies the SDE: dx (s) = f (s, X (s), h(s))ds+λ(s, X (s))dws θ + ξ ( s, X (s ), z ) Ñ h (ds, dz), X (0) = x 0 where f (t, x, h) := b(t, x) θλβ(t, X (t), h, ρ) + [ ξ(t, x, z) (ζ(t, z, h, ρ)) θ] ν(dz) (28) = b(t, x) θλ [ ς (t, x) + ρ(h Σ(t, x) ς (t, x)) ] [ (η(t, [ + ξ(t, x, z) z) + ρ(t) h (t)γ(t, z) η(t, z) ]) ] θ 1 (z) 0 ν(dz) and b is the P-measure drift of the factor process. (29)

30 Equity and Leverage Following the change of measure we introduce two auxiliary criterion functions under P θ h: the risk-sensitive control problem: I (v, x; h; t, T ; θ) = 1 θ ln Eh,θ t,x [ { T }] exp θ g(x s, h(s); θ)ds t where E t,x [ ] denotes the expectation taken with respect to the measure P θ h and with initial conditions (t, x). the exponentially transformed criterion [ { Ĩ (v, x, h; t, T ; θ) := E h,θ t,x exp θ T t }] g(s, X s, h(s); θ)ds (30) (31)

31 How to Solve a Stochastic Control Problem How to Solve a Stochastic Control Problem Our objective is to solve the control problem in a classical sense. The process involves 1. deriving the HJB P(I)DE; 2. identifying a (unique) candidate optimal control; 3. proving existence of a C 1,2 (classical) solution to the HJB P(I)DE. 4. proving a verification theorem;

32 How to Solve a Stochastic Control Problem The HJB P(I)DEs The HJB PIDE associated with the risk-sensitive control criterion (30) is where with Φ t + sup h J,ρ K ( ) L h t, x, Φ, DΦ, D 2 Φ = 0 (32) L h (t, x, u, p, M) = f (t, x, h) p tr ( ΛΛ (t, x)m ) θ 2 p ΛΛ (t, x)p I NL [t, x, u, p] = g(t, x, h) + I NL [t, x, u, p] (33) { 1 ( ) } e θ[u(t,x+ξ(t,x,z)) u(t,x)] 1 ξ(t, x, z) p ν(dz) (34) θ and subject to the terminal condition (recall our normalization e 0 = 1) Φ(T, x) = 0, x R n. (35) Condition (16) ensures that I NL is well defined, at least for bounded u.

33 How to Solve a Stochastic Control Problem To remove the quadratic growth term, we consider the PIDE associated with the exponentially-transformed problem (31): Φ t (t, x) + 1 ( ) 2 tr ΛΛ (t, x)d 2 Φ(t, x) + H(t, x, Φ, D Φ) + { Φ (t, x + ξ(t, x, z)) Φ(t, x) ξ(t, x, z) D Φ(t, x)} ν(dz) = 0(36) subject to terminal condition where for r R, p R n Φ(T, x) = 1 (37) H(s, x, r, p) = { inf f (s, x, h) p + θg(s, x, h)r } (38) h J In particular Φ(t, x) = exp { θφ(t, x)}.

34 How to Solve a Stochastic Control Problem Some Insights... The presence (or absence) of jumps plays a crucial role in our problem. If we think about our general model as a meta model for diffusion and jump-diffusion problems we observe that: 1. Pure diffusion: we have a pure diffusion problem and the hope of finding an analytical solution for the optimal policy pair (h, ρ ). The HJB equation is a parabolic PDE; 2. Jumps in asset and liabilities only: because of the jumps we will not generally be able to analytical solution for the optimal policy pair (h, ρ ). However, the HJB equation remains a parabolic PDE; 3. Jumps in factors only: we can potentially find an analytical solution for the optimal policy pair (h, ρ ), but now the jumps in the factor level have transformed the HJB equation is a parabolic PIDE; 4. Full jump diffusion: because of the jumps in assets and liabilities we will not generally be able to analytical solution for the optimal policy pair (h, ρ ). Moreover, the jumps in the factor level have transformed the HJB equation is a parabolic PIDE;

35 How to Solve a Stochastic Control Problem Identifying a (Unique) Candidate Optimal Control The supremum in (32) can be expressed as sup L h (t, x, u, p, M) h J,ρ K = b (t, x)p tr ( ΛΛ (t, x)m ) θ 2 p ΛΛ (t, x) p + c(t, x) + I NL [t, x, u, p] { 1 2 (θ + 1) [ ς (t, x) + ρ(h Σ(t, x) ς (t, x)) ] [ ς (t, x) + ρ(h Σ(t, x) sup h J,ρ K θλ [ ς (t, x) + ρ(h Σ(t, x) ς (t, x)) ] p + ρ [ h â(t, x) ĉ(t, x) ] 1 { (1 θξ(t, x, z) p ) [ ( [ η(t, z) + ρ(t) h (t)γ(t, z) η(t, z) ]) θ 1] θ + ( ρ(t) [ h (t)γ(t, z) η(t, z) ]) 1 0 (z) } ν(dz) }

36 How to Solve a Stochastic Control Problem This equation looks messy, but it is actually well structured: Because ΣΣ > 0 and because systematic jumps are independent from idiosyncratic jumps, this problem corresponds to the maximization of a concave function on a convex set of constraints. By the Lagrange Duality (see for example Theorem 1 in Section 8.6 in [12]), we conclude that the supremum in (22) admits a unique maximizing pair (ĥ, ˆρ)(t; x; p) for (t; x; p) [0; T ] R n R n. By measurable selection, (ĥ, ˆρ) can be taken as a Borel measurable function on [0; T ] R n R n.

37 Existence of a C 1,2 Solution to the HJB PDE Existence of a C 1,2 Solution to the HJB PDE Once we have cleared the hurdle of having two controls, we are back to PDE territory: The number and properties of the controls do not matter (much); We can rely on our earlier results (see [5, 6] Proving the existence of a strong, C 1,2, solution is the most difficult and intricate step in the process. However, this is a necessary step if we want to use the Verification Theorem to conclusively solve our optimal investment problem.

38 Verification Theorem Verification Theorem Broadly speaking, the verification theorem states that if we have a C 1,2 ([0, T ] R n ) bounded function φ which satisfies the HJB PDE (32) and its terminal condition; the stochastic differential equation dx (s) = f (s, X (s), h(s), ρ(s); θ)ds + Λ(s, X (s))dws θ + ξ ( s, X (s ), z ) Ñp θ (ds, dz) defines a unique solution X (s) for each given initial data X (t) = x; and, there exists a Borel-measurable maximizing pair (h (t, X t), ρ (t, X t)) of h L h φ defined in (33); then Φ is the value function and (h (t, X t), ρ (t, X t)) is the optimal pair of Markov control processes.... and similarly for Φ and the exponentially-transformed problem.

39 Concluding Remarks Concluding Remarks Factors give us a lot of flexibility in the way we setup and parametrize the problem. We could even consider behavioural factors in our analysis (see [1])! Even with liabilities, leverage and jumps, we still manage to get a smooth value function and a convex optimisation problem for the controls. This is promising from a numerical perspective. The final major question we face to implement the model is how to populate the set of parameters sensibly. Coming soon: Black-Litterman for ALM (diffusion and affine jump-diffusion).

40 Concluding Remarks Thank you! 3 Any question? 3 Adapted from W. Krawcewicz, University of Alberta

41 Concluding Remarks Thank you! XXXXXX Espresso 4 Any question? 4 Adapted from W. Krawcewicz, University of Alberta

42 Concluding Remarks G. Andruszkiewicz, M.H.A. Davis, and S. Lleo. Taming animal spirits: risk management with behavioural factors. Annals of Finance, 9: , Mareen Benk. Intertemporal surplus management with jump risk. In Horand I. Gassmann and William T. iemba, editors, Stochastic Programming: Applications in Finance, Energy, Planning and Logistics, volume 4 of World Scientific Series in Finance, pages World Scientific Publishing Company, A. Bensoussan. Stochastic Control of Partially Observable Systems. Cambridge University Press, T.R. Bielecki and S.R. Pliska. Risk-sensitive dynamic asset management. Applied Mathematics and Optimization, 39: , M.H.A. Davis and S. Lleo. Jump-diffusion risk-sensitive asset management I: Diffusion factor model. SIAM Journal on Financial Mathematics, 2:22 54, M.H.A. Davis and S. Lleo.

43 Concluding Remarks Jump-diffusion risk-sensitive asset management ii: Jump-diffusion factor model. SIAM Journal on Control and Optimization (forthcoming), M.H.A. Davis and S. Lleo. Jump-diffusion risk-sensitive benchmarked asset management. In H.I. Gassmann and W.T. iemba, editors, Stochastic Programming: Applications in Finance, Energy, Planning and Logistics, number 4 in World Scientific Series in Finance, pages World, N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland Publishing Company, D.H. Jacobson. Optimal stochastic linear systems with exponential criteria and their relation to deterministic differential games. IEEE Transactions on Automatic Control, 18(2): , K. Kuroda and H. Nagai. Risk-sensitive portfolio optimization on infinite time horizon. Stochastics and Stochastics Reports, 73: , M. Lefebvre and P. Montulet.

44 Concluding Remarks Risk-sensitive optimal investment policy. International Journal of Systems Science, 22: , D. Luenberger. Optimization by Vector Space Methods. John Wiley & Sons, Markus Rudolf and William T. iemba. Intertemporal surplus management. Journal of Economic Dynamics & Control, 28: , P. Whittle. Risk Sensitive Optimal Control. John Wiley & Sons, New York, 1990.