On the Optimality of Kelly Strategies

Transcription

1 On the Optimality of Kelly Strategies Mark Davis 1 and Sébastien Lleo 2 Workshop on Stochastic Models and Control Bad Herrenalb, April 1, Department of Mathematics, Imperial College London, London SW7 2AZ, England, mark.davis@imperial.ac.uk 2 Finance Department, Reims Management School, 59 rue Pierre Taittinger, Reims, France, sebastien.lleo@reims-ms.fr

2 Outline Outline Where to invest? Kelly Strategies Back to Basics: Kelly Strategies in the Merton Model Insights from the Merton Model: Change of Measure and Duality Further Insights from the Merton Model... Getting Kelly Strategies to Work for Factor Models Beyond Factor Models: Random Coefficients Conclusion and Next Steps

3 Where to invest? Where to invest? The central question for investment management practitioners is: where should I put my money? Ideas such as the utility of wealth, and even expected portfolio returns (and risks) take a backseat. Yet, in finance theory, the optimal asset allocation appears as less important than the utility of wealth. This is particularly true in the duality/martingale approach, where the optimal asset allocation is often obtained last.

4 Kelly Strategies Kelly Strategies As a result of the differences between investment management theories and practice, a number of techniques and tools have been created to speed up the derivation of an optimal or near-optimal asset allocation. Fractional Kelly strategies are one such technique.

5 Kelly Strategies The Kelly (criterion) portfolio invest in the growth maximizing portfolio. In continuous time, this represents an investment in the optimal portfolio under the logarithmic utility function. The Kelly criterion comes from the signal processing/gambling literature (see Kelly [7]). See MacLean, Thorpe and Ziemba [11] for a thorough view of the Kelly criterion. A number of great investors from Keynes to Buffet can be viewed as Kelly investors. Others, such as Bill Gross probably are. The Kelly criterion portfolio has a number of interesting properties but it is inherently risky.

6 Kelly Strategies To reduce the risk while keeping some of the nice properties, MacLean and Ziemba proposed the concept of fractional Kelly investment: Invest a fraction k of the wealth in the Kelly portfolio; invest a fraction 1 k in the risk-free asset. Fractional Kelly strategies are optimal in the Merton world of lognormal asset prices and with a globally risk-free asset; not optimal when the lognormality assumption is removed.

7 Kelly Strategies Three questions: Why are fractional Kelly strategies not optimal outside of the Merton model? How is an optimal strategy constructed? Can we improve the definition of fractional Kelly strategies to guarantee optimality?

8 Kelly Strategies In this talk, we will use the term Kelly strategies to refer to: The Kelly portfolio. Fractional Kelly investment.

9 Back to Basics: Kelly Strategies in the Merton Model Back to Basics: Kelly Strategies in the Merton Model Key ingredients: Probability space (Ω, {F t }, F, P); R m -valued (F t )-Brownian motion W (t); Price at time t of the ith security is S i (t), i = 0,..., m; The dynamics of the money market account and of the m risky securities are respectively given by: ds i (t) S i (t) = µ idt + ds 0 (t) S 0 (t) = rdt, S 0(0) = s 0 (1) N σ ik dw k (t), S i (0) = s i, i = 1,..., m k=1 where r, µ R m and Σ := [σ ij ] is a m m matrix. (2)

10 Back to Basics: Kelly Strategies in the Merton Model Assumption The matrix Σ is positive definite. Let G t := σ(s(s), 0 s t) be the sigma-field generated by the security process up to time t. An investment strategy or control process is an R m -valued process with the interpretation that h i (t) is the fraction of current portfolio value invested in the ith asset, i = 1,..., m. Fraction invested in the money market account is h 0 (t) = 1 m i=1 h i(t).;

11 Back to Basics: Kelly Strategies in the Merton Model Definition An R m -valued control process h(t) is in class A(T ) if the following conditions are satisfied: 1. h(t) is progressively measurable with respect to {B([0, t]) G t } t 0 and is càdlàg; ( ) T 2. P 0 h(s) 2 ds < + = 1, T > 0; 3. the Doléans exponential χ h t, given by χ h t := { t exp γ h(s) ΣdW s 1 t } 0 2 γ2 h(s) ΣΣ h(s)ds 0 (3) is an exponential martingale, i.e. E [ χ h T ] = 1 We say that a control process h(t) is admissible if h(t) A(T ).

12 Back to Basics: Kelly Strategies in the Merton Model Taking the budget equation into consideration, the wealth, V (t), of the asset in response to an investment strategy h H follows the dynamics dv (t) V (t) = rdt + h (t) (µ r1) dt + h (t)σdw t (4) with initial endowment V (0) = v and where 1 R m is the m-element unit column vector.

13 Back to Basics: Kelly Strategies in the Merton Model The objective of an investor with a fixed time horizon T, no consumption and a power utility function is to maximize the expected utility of terminal wealth: [ V γ ] [ T e γ ln V T ] J(t, h; T, γ) = E [U(V T )] = E = E γ γ with risk aversion coefficient γ (, 0) (0, 1).

14 Back to Basics: Kelly Strategies in the Merton Model Define the value function Φ corresponding to the maximization of the auxiliary criterion function J(t, x, h; θ; T ; v) as By Itô s lemma, where e γ ln V (t) = v γ exp Φ(t) = sup J(t, h; T, γ) (5) h A { t } γ g(h(s); γ)ds χ h t (6) 0 g(h; γ) = 1 2 (1 γ) h ΣΣ h + h (µ r1) + r and the Doléans exponential χ h t is defined in (3)

15 Back to Basics: Kelly Strategies in the Merton Model We can solve the stochastic control problem associated with (5) by a change of measure argument (see exercise 8.18 in [8]). Define a measure P h on (Ω, F T ) via the Radon-Nikodým derivative For h A(T ), W h t is a standard P h -Brownian motion. dp h dp := χh T (7) t = W t γ Σ h(s)ds 0 The control criterion under this new measure is I (t, h; T, γ) = v γ [ { T }] exp γ g(h(s); θ)ds γ Eh t where E h t [ ] denotes the expectation taken with respect to the measure P h at an initial time t. t (8)

16 Back to Basics: Kelly Strategies in the Merton Model Under the measure P h, the control problem can be solved through a pointwise maximisation of the auxiliary criterion function I (v, x; h; t, T ). The optimal control h is simply the maximizer of the function g(x; h; t, T ) given by h = 1 1 γ (ΣΣ ) 1 (µ r1) which represents a position of 1 1 γ in the Kelly criterion portfolio.

17 Back to Basics: Kelly Strategies in the Merton Model The value function Φ(t), or optimal utility of wealth, equals Φ(t) = v γ { [ ] } γ exp 1 γ r + 2(1 γ) (µ r1) (ΣΣ ) 1 (µ r1) (T t) Substituting (9) into (3), we obtain an exact form for the Doléans exponential χ t associated with the control h : { γ χ t := exp 1 γ (µ r1) Σ 1 W (t) 1 ( ) γ 2 (µ r1) (ΣΣ ) 1 (µ r1) t} 2 1 γ (9) We can easily check that χ t is indeed an exponential martingale. Therefore h is an admissible control.

18 Back to Basics: Kelly Strategies in the Merton Model In the Merton model, fractional Kelly strategies appear naturally as a result of a classical Fund Separation Theorem: Theorem (Fund Separation Theorem) Any portfolio can be expressed as a linear combination of investments in the Kelly (log-utility) portfolio h K (t) = (ΣΣ ) 1 (µ r) (10) and the risk-free rate. Moreover, if an investor has a risk aversion γ, then the proportion of the Kelly portfolio will equal 1 1 γ.

19 Insights from the Merton Model: Change of Measure and Duality Insights from the Merton Model: Change of Measure and Duality The measure P h defined in (7) is also used in the martingale/duality approach to dynamic portfolio selection (see for example [14] and references therein). In complete market, the change of measure technique is equivalent to the martingale approach: the change of measure approach relies on the optimal asset allocation to identify the equivalent martingale measure, the martingale approach works in the opposite direction.

20 Further Insights from the Merton Model... Further Insights from the Merton Model... The level of risk aversion γ dictates the choice of measure P h. Two special cases are worth mentioning. Case 1: The Physical Measure The measures P h is the physical measure P in the limit as γ 0, that is in the log utility or Kelly criterion case. This observation forms the basis for the benchmark approach to finance (see [13]).

21 Further Insights from the Merton Model... Case 2: The Dangers of Overbetting A well established and somewhat surprising folk theorem holds that when an agent overbets by investing in twice the Kelly portfolio, his/her expected return will be equal to the risk-free rate. (see for instance [15] and [11]). Why is that???

22 Further Insights from the Merton Model... The risk aversion of an agent who invest into twice the Kelly portfolio must be γ = 1 2. Hence, the measures P h coincides with the equivalent martingale measure Q! Under this measure, the portfolio value discounted at the risk-free rate is a martingale.

23 Further Insights from the Merton Model... In the setting of the Merton model and its lognormally distributed asset prices, the definition of Fractional Kelly allocations guarantees optimality of the strategy. However, Fractional Kelly strategies are no longer optimal as soon as the lognormality assumption is removed (see Thorpe in [11]). This situation suggests that the definition of Fractional Kelly strategies could be broadened in order to guarantee optimality. We can take a first step in this direction by revisiting the ICAPM (see Merton[12]) in which the drift rate of the asset prices depend on a number of Normally-distributed factors.

24 Getting Kelly Strategies to Work for Factor Models Getting Kelly Strategies to Work for Factor Models Key ingredients: Probability space (Ω, {F t }, F, P); R N -valued (F t )-Brownian motion W (t), N := n + m; S i (t) denotes the price at time t of the ith security, with i = 0,..., m; X j (t) denotes the level at time t of the jth factor, with j = 1,..., n. For the time being, we assume that the factors are observable.

25 Getting Kelly Strategies to Work for Factor Models The dynamics of the money market account is given by: ds 0 (t) S 0 (t) = ( a 0 + A 0X (t) ) dt, S 0 (0) = s 0 (11) The dynamics of the m risky securities and n factors are and ds(t) = D (S(t)) (a + AX (t))dt + D (S(t)) ΣdW (t), S i (0) = s i, i = 1,..., m (12) dx (t) = (b + BX (t))dt + ΛdW (t), X (0) = x (13) where X (t) is the R n -valued factor process with components X j (t) and the market parameters a, A, b, B, Σ := [σ ij ], Λ := [Λ ij ] are vectors and matrices of appropriate dimensions.

26 Getting Kelly Strategies to Work for Factor Models Assumption The matrices ΣΣ and ΛΛ are positive definite. The objective of an investor remains the maximization of criterion (15) at a fixed time horizon T ; The wealth V (t) of the portfolio in response to an investment strategy h A(T ) is now factor-dependent with dynamics: dv (t) V (t) = ( a 0 + A 0X (t) ) dt + h (t)(â + ÂX (t))dt + h (t)σdw t (14) with â := a a 0 1, Â := A 1A 0, and initial endowment V (0) = v.

27 Getting Kelly Strategies to Work for Factor Models The expected utility of terminal wealth J(t, x, h; T, γ) si factor dependent: [ V γ ] [ T e γ ln V T ] J(t, x, h; T, γ) = E [U(V T )] = E = E γ γ By Itô s lemma, where e γ ln V (t) = v γ exp { t } γ g(x s, h(s); θ)ds χ h t (15) 0 g(x, h; γ) = 1 2 (1 γ) h ΣΣ h + h (â + Âx) + a 0 + A 0x(16) and the exponential martingale χ h t is still given by (3).

28 Getting Kelly Strategies to Work for Factor Models Applying the change of measure argument, we obtain the control criterion under the measure P h I (t, x, h; T, γ) = v γ [ { T }] γ Eh t,x exp γ g(x s, h(s); θ)ds (17) t where E h t,x [ ] denotes the expectation taken with respect to the measure P h and with initial conditions (t, x). The P h -dynamics of the state variable X (t) is dx (t) = ( b + BX (t) + γλσ h(t) ) dt + ΛdW h t, t [0, T ] (18)

29 Getting Kelly Strategies to Work for Factor Models Then value function Φ associated with the auxiliary criterion function I (t, x; h; T, γ) is defined as Φ(t, x) = sup I (t, x; h; T, γ) (19) h A(T ) We use the Feynman-Kač formula to write down the HJB PDE: Φ t (t, x) tr ( ΛΛ D 2 Φ(t, x) ) + H(t, x, Φ, DΦ) = 0 (20) subject to terminal condition Φ(T, x) = v γ γ (21) and where { (b H(s, x, r, p) = sup + Bx + γλσ h ) } p γg(x, h; γ)r (22) h R for r R and p R n.

30 Getting Kelly Strategies to Work for Factor Models To obtain the optimal control in a more convenient format and derive the value function Φ(t, x) more easily, we find it convenient to consider the logarithmically transformed value function Φ(t, x) := 1 γ ln γφ(t, x) with associated HJB PDE where Φ (t, x) + inf t h R Lh m t (t, x, DΦ, D 2 Φ) = 0 (23) L h t (s, x, p, M) = ( b + Bx + γλσ h(s) ) p tr ( ΛΛ M ) + γ 2 p ΛΛ p g(x, h; γ) (24) for r R and p R n and subject to terminal condition Φ(T, x) = ln v (25) This is in fact a risk-sensitive asset management problem (see [1], [9], [6]).

31 Getting Kelly Strategies to Work for Factor Models Solving the optimization problem in (24) gives the optimal investment policy h (t) h (t) = 1 ( ΣΣ ) [ 1 â + ÂX (t) + γσλ D Φ(t, X (t))] 1 γ (26) The solution to HJB PDE (23) is Φ(t, x) = 1 2 x Q(t)x + x q(t) + k(t) where Q(t) satisfies a matrix Riccati equation, q(t) satisfies a vector linear ODE and k(t) is an integral. As a result, h (t) = 1 1 γ ( ΣΣ ) 1 [ â + ÂX (t) + γσλ (Q(t)X (t) + q(t))] (27)

32 Getting Kelly Strategies to Work for Factor Models Substituting (26) into (3), we obtain an exact form for the Doléans exponential χ t associated with the control h : χ t := { γ t [ ] exp â + ÂX (s) + γσλ (Q(s)X (s) + q(s)) 1 γ 0 (ΣΣ ) 1 ΣdW (s) 1 ( ) γ 2 t ( ) â + ÂX (s) + γσλ (Q(s)X (s) + q(s)) 2 1 γ 0 ) } (ΣΣ ) 1 â + ÂX (s) + γσλ (Q(s)X (s) + q(s)) ds (28)

33 Getting Kelly Strategies to Work for Factor Models We can interpret the Girsanov kernel γ [ ] 1 γ Σ (ΣΣ ) 1 â + ÂX (s) + γσλ (Q(s)X (s) + q(s)) as the projection of the factor-dependent returns â + ÂX (s) + γσλ (Q(s)X (s) + q(s)) on the subspace spanned by the asset volatilities Σ, scaled by γ 1 γ. This observation also implies that the appropriate Girsanov kernel in a complete (factor-free) setting is with a similar interpretation. γ 1 γ Σ (ΣΣ ) 1 (µ r1)

34 Getting Kelly Strategies to Work for Factor Models In the ICAPM, a new view of Fractional Kelly investing emerges: Theorem (ICAPM Fund Separation Theorem) Any portfolio can be expressed as a linear combination of investments into two funds with respective risky asset allocations: h K (t) = (ΣΣ ) 1 ( â + ÂX (t) ) h I (t) = (ΣΣ ) 1 ΣΛ (Q(t)X (t) + q(t)) (29) and respective allocation to the money market account given by ( ) h0 K (t) = 1 1 (ΣΣ ) 1 â + ÂX (t) h I 0(t) = 1 1 (ΣΣ ) 1 ΣΛ (Q(t)X (t) + q(t)) Moreover, if an investor has a risk aversion γ, then the respective 1 weights of each mutual fund in the investor s portfolio equal 1 γ and γ, respectively.

35 Getting Kelly Strategies to Work for Factor Models In the factor-based ICAPM, ( ΣΣ ) 1 [ â + ÂX (t) ] (30) represents the Kelly (log utility) portfolio and ( ΣΣ ) 1 ΣΛ (Q(t)X (t) + q(t)) (31) is the intertemporal hedging porfolio identified by Merton. This mutual fund theorem raises some questions as to the practicality of the intertemporal hedging portfolio as an investment option.

36 Beyond Factor Models: Random Coefficients Beyond Factor Models: Random Coefficients The change of measure approach still works well in a partial observation case where we would need to estimate the coefficients of the factor process through a Kalman filter. However, this presupposes that we know the form of the factor process. A more general approach would be to assume that the coefficients of the asset price dynamics are random (See Bjørk, Davis and Landén [14] and references therein).

37 Beyond Factor Models: Random Coefficients Key ingredients: The full underlying probability space (Ω, {F t }, F, P); The filtration Ft W := σ(w (s)), 0 s t) generated by an m-dimensional Brownian motions driving the asset returns (augmented by the P-null sets); The filtration Ft S := σ(s(s)), 0 s t) generated by the m asset price (also augmented by the P-null sets); The dynamics of the money market account is given by: ds 0 (t) S 0 (t) = r(t)dt, S 0(0) = s 0 (32) where r R + is a bounded Ft S -adapted process. We will denote by Z(t, T ) the discount factor: { T } Z(t, T ) = exp r(s)ds (33) t

38 Beyond Factor Models: Random Coefficients The dynamics of the m risky securities and n factors can be expressed as: ds i (t) S i (t) = µ i(t)dt+ m σ ik (t)dw k (t), S i (0) = s i, i = 1,..., m k=1 (34) where µ(t) = (µ 1 (t),..., µ m (t)) is an F t -adapted process and the volatility Σ(t) := [σ ij (t)], i = 1,..., m, j = 1,..., m is an F S t -adapted process. More synthetically, ds(t) = D(S(t))µ(t)dt + D(S(t))Σ(t)dW (t) (35) Note that no Markovian structure is either assumed or required. Assumption We assume that the matrix Σ(t) is positive definite t.

39 Beyond Factor Models: Random Coefficients The critical step in this approach is to establish a projection onto the observable filtration. Once this is done, the partial observation problem related to the filtration F can be rewritten as an equivalent complete observation problem with respect to the filtration F S. This effectively changes a utility maximizaton problem set in an incomplete market into a standard utility maximization problem set in a complete market.

40 Beyond Factor Models: Random Coefficients Define the process Y (t) by: dy (t) = Σ 1 t D(S t ) 1 ds t = Σ 1 t µ t dt + Σ t dw (t) (36) For any F-adapted process X, define the filter estimate process ˆX as the optional projection of X onto the filtration F S : [ ] Ŷ t = E P Y t Ft S

41 Beyond Factor Models: Random Coefficients Next, define the innovation process U by du(t) = dz(t) (Σ 1 µ(t))dt = dz(t) Σ 1ˆµ(t)dt (37) where the second equality follows from the observability of Σ t. Note that the innovation process U(t) is a standard F S -Brownian motion (see for example Lipster and Shiryaev [10]). As a result, we can rewrite (35) using the filter estimate for α and the innovation process U obtained with respect to the filtration F S : ds(t) = D(S(t))ˆµ(t)dt + D(S(t))Σ(t)dU(t) (38) The remainder follows from a standard martingale argument.

42 Beyond Factor Models: Random Coefficients Definition The Girsanov kernel is the vector process ϕ t given by for all t. ϕ(t) = Σ 1 (t)(ˆµ(t) r1) (39) Note that he Girsanov kernel ϕ(t) and the market price of risk vector λ(t) are related by λ(t) = ϕ(t).

43 Beyond Factor Models: Random Coefficients Assumption We assume that the Girsanov kernel satisfies the Novikov condition [ E e 1 ] T 2 0 ϕ(t) 2 dt < (40) and the integrability condition [ E e 1 2 T 0 ( γ 1 γ ) 2 ϕ(t) 2 dt ] < (41)

44 Beyond Factor Models: Random Coefficients Definition The equivalent martingale measure Q is defined by the Radon-Nikodým derivative dq dp = L t := exp on F S t. { t ϕ (s)du(s) 1 t ϕ (s)ϕ(s)ds }, It follows from Assumption 4 that L t is a true martingale. Moreover, Q is unique: this is a direct consequence of the use of filtered estimates in (38).

45 Beyond Factor Models: Random Coefficients The objective is to maximize the expected utility of terminal wealth: [ V γ ] J(t, h; T, γ) = E P T γ subject to the budget constraint Next, form the Lagrangian E Q [K 0,T V T ] = v = E P [K 0,T L T V T ] (42) L(h, λ; T ) = 1 [ γ EP V γ T = Ω ] ( ) λ E P [K 0,T L T V T ] v 1 γ V γ T λ (K 0,T (ω)l T (ω)v T (ω) v) dp(ω)

46 Beyond Factor Models: Random Coefficients This separable problem can be maximized for each ω. The first order condition leads to V T = [λk 0,T L T ] 1 1 γ Substituting in the budget equation (42), we get [ ] λ 1 1 γ E P (K 0,T L T ) γ 1 γ = v and as a result, V T = v H 0 K γ 0,T L 1 γ T (43) with [ ] H 0 = E P (K 0,T L T ) γ 1 γ

47 Beyond Factor Models: Random Coefficients Therefore, the optimal expected utility is equal to: [ (V U 0 = E P T ) γ ] = H 1 γ v 0 γ γ (44) Observe that H 0 can be expressed as H 0 = E P [ L 0 T exp { T where { L 0 γ t := exp 1 γ t 0 0 ( r t + ϕ (s)du(s) 1 2 ) }] 1 2(1 γ) ϕ(t) 2 dt ( ) γ 2 } t ϕ (s)ϕ(s)ds (45) 1 γ 0 is a true martingale. This leads us to the following definition:

48 Beyond Factor Models: Random Coefficients Definition (i). The measure Q 0 is defined by the Radon-Nikodým derivative dq 0 dp = L 0 t, on F S t (ii). The process H is defined by { γ T ( ) } ] H t = E [exp 0 1 r s + 1 γ t 2(1 γ) ϕ(s) 2 ds F t (46) where the expectation E 0 [ ] is taken with respect to the Q 0 measure.

49 Beyond Factor Models: Random Coefficients Finally, we recall Proposition 4.1 in Bjørk, Davis and Landén [14]: Proposition The following hold: (i). The optimal wealth process V (t) is given by V (t) = H(t) ( ) 1 K0,t Lt 1 γ x H(0) (ii). The optimal weight vector h is given by h (t) = 1 1 γ (Σ tσ t) 1 (µ t r1) + σ H (t) ( Σ 1 ) t (47) where σ H is the volatility term of H, i.e. H has dynamics of the form H(t) = µ H (t)dt + σ H (t)du(t)

50 Beyond Factor Models: Random Coefficients In the limit as γ 0, the optimal wealth process V (t) is given by V (t) = ( K 0,t Lt ) 1 1 γ x and the optimal investment is the Kelly portfolio, h (t) = (Σ t Σ t) 1 (µ t r1) (48)

51 Beyond Factor Models: Random Coefficients With the choice γ = 1 2, and h (t) = 2(Σ t Σ t) 1 (µ t r1) + σ H (t) ( Σ 1 ) t (49) As a result, V (t) = H(t) { ( ) T } 2 K0,t L t x = x exp r(s)ds H(0) t as expected.

52 Beyond Factor Models: Random Coefficients However, there is still something unsatisfactory about this result: the optimal strategy depends on the volatility of the process H(t): { γ T ( ) } ] H t = E [exp 0 1 r s + 1 γ t 2(1 γ) ϕ(s) 2 ds F t As a result, getting some intuition on the intertemporal hedging term is difficult.

53 Conclusion and Next Steps Conclusion and Next Steps Get stronger statements of equivalence between the martingale/duality approach and the change of measure technique in incomplete markets; Study the practicality of intertemporal hedging portfolio as an investment option; Add jumps.

54 Conclusion and Next Steps Thank you! 3 Any question? 3 Copyright: W. Krawcewicz, University of Alberta

55 Conclusion and Next Steps T.R. Bielecki and S.R. Pliska. Risk-sensitive dynamic asset management. Applied Mathematics and Optimization, 39: , T.R. Bielecki and S.R. Pliska. Risk sensitive intertemporal CAPM. IEEE Transactions on Automatic Control, 49(3): , March M.H.A. Davis and S. Lleo. Risk-sensitive benchmarked asset management. Quantitative Finance, 8(4): , June M.H.A. Davis and S. Lleo. On the optimality of kelly strategies ii: Applications. working paper, M.H.A. Davis and S. Lleo.

56 Conclusion and Next Steps Jump-diffusion risk-sensitive asset management i: Diffusion factor model. SIAM Journal on Financial Mathematics (to appear), M.H.A. Davis and S. Lleo. The Kelly Capital Growth Investment Criterion: Theory and Practice, chapter Fractional Kelly Strategies for Benchmarked Asset Management. World Scientific, J. Kelly. A new interpretation of information rate. Bell System Technology, 35: , B. Øksendal. Stochastic Differential Equations. Universitext. Springer-Verlag, 6 edition, K. Kuroda and H. Nagai.

57 Conclusion and Next Steps Risk-sensitive portfolio optimization on infinite time horizon. Stochastics and Stochastics Reports, 73: , R. Lipster and A. Shiryaev. Statistics of Random Processes: I. General Theory. Probability and Its Applications. Springer-Verlag, 2 edition, L.C. MacLean, E. Throp, and W.T. Ziemba, editors. The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific, R.C. Merton. An intertemporal capital asset pricing model. Econometrica, 41( ), E. Platen and D. Heath. A Benchmark Approach to Quantitative Finance.

58 Conclusion and Next Steps Springer Finance. Springer-Verlag, T. Bjørk, M. Davis, and C. Landén. Optimal investment under partial information. Mathematical Methods of Operations Research, E. Thorp. Handbook of Asset and Liability Management, chapter The Kelly Criterion in Blackjack, Sports Betting and the Stock Market. Handbook in Finance. North Holland, 2006.