Mean-variance portfolio selection under Volterra Heston model

# Mean-variance portfolio selection under Volterra Heston model

Bingyan Han Department of Statistics, The Chinese University of Hong Kong, Hong Kong, byhan@link.cuhk.edu.hk    Hoi Ying Wong Department of Statistics, The Chinese University of Hong Kong, Hong Kong, hywong@cuhk.edu.hk
###### Abstract

Motivated by empirical evidence for rough volatility models, this paper investigates the continuous-time mean-variance (MV) portfolio selection under Volterra Heston model. Due to the non-Markovian and non-semimartingale nature of the model, classic stochastic optimal control frameworks are not directly applicable to the associated optimization problem. By constructing an auxiliary stochastic process, we obtain the optimal investment strategy that depends on the solution to a Riccati-Volterra equation. The MV efficient frontier is shown to maintain a quadratic curve. Numerical studies show that both roughness and volatility of volatility affect the optimal strategy materially.
Keywords: Mean-variance portfolio, Volterra Heston model, Riccati–Volterra equations, rough volatility.
Mathematics Subject Classification: 93E20, 60G22, 49N90, 60H10.

## 1 Introduction

Recently, there is a growing interest in studying rough volatility models (Gatheral et al., 2018; El Euch and Rosenbaum, 2019; Guennoun et al., 2018). The terminology “rough” can be roughly thought of as the level of Hölder regularity. Typically, a rough diffusion model has trajectories that are rougher than the paths of a standard Brownian motion. Therefore, rough volatility models are stochastic volatility models that the volatility process is rough.

Rough volatility models offer a much better empirical fit to the stylized facts of financial data. An investigation in the time series of realized volatility111See, for example, Oxford-Man Institute’s realized library at https://realized.oxford-man.ox.ac.uk/data from high frequency data obtains the Hurst parameter close to , which is much smaller than for the standard Brownian motion. The Hurst parameter is used to reflect the memoryness of a time series and associated with the roughness of the limiting fractional Brownian motion. The smaller the , the rougher the time series model. Therefore, the empirical finding suggests a rougher realized path of volatility than that of the standard Brownian motion. Meanwhile, the simulated paths under small Hurst parameter look close to the realized ones.

Rough volatility models also better capture the term structure of implied volatility surface, especially for the explosion of at-the-money (ATM) skew when maturity goes to zero. More precisely, let be the implied volatility of an option where is log-moneyness and is time to expiration. The ATM skew at maturity is defined by

 (1.1)

Empirical evidence shows that ATM skew explodes when . However, conventional volatility models such as the Heston model generates a constant ATM skew for a small . Fukasawa (2011) shows that if the volatility is modeled by a fractional Brownian motion with Hurst parameter , then ATM skew has an approximation property,

 ϕ(τ)≈τH−1/2, when τ↓0. (1.2)

An application of rough volatility models fits the explosion remarkably well, with only one extra Hurst parameter.

Volatility has a long memory property. This stylized fact has received many attentions and debates. Gatheral et al. (2018) try to resolve this long-standing question from the view of rough volatility and provides an asymptotic behavior of the autocorrelation function of volatility. We refer readers to Gatheral et al. (2018) for the details of the analysis.

With the success in capturing empirical properties, many works provide elegant theoretical foundations. Fukasawa (2011) offers an expansion formula for implied volatility with martingale expansion theory. Fractional Brownian motion is considered as a special case in Fukasawa (2011, Section 3.3). El Euch and Rosenbaum (2019) introduce the rough Heston model, which is a scaling limit of proper Hawkes processes. The characteristic function is derived in closed form, up to the solution of a fractional Riccati equation. Along with this line of research, El Euch et al. (2018) show a similar link between Hawkes processes and (rough) Heston models. El Euch and Rosenbaum (2018) render a hedging strategy for options under rough Heston model. Abi Jaber et al. (2017) propose affine Volterra processes that embrace the rough Heston model as a special case. The explicit exponential-affine representations of Fourier-Laplace functional is derived using Riccati-Volterra equations. Keller-Ressel et al. (2018) present applications of affine Volterra processes in finance. Besides, Guennoun et al. (2018) coin an alternative definition of a rough version of the Heston model and derive some asymptotic results.

While the rough volatility literature focuses on option pricing, only a few of them considers portfolio optimization under rough volatility models. To the best of our knowledge, Fouque and Hu (2018a), Fouque and Hu (2018b), Bäuerle and Desmettre (2018) are the only related results in this direction so far. All of them focus on utility maximization. The remarkable success of rough volatility models motivates us to consider mean-variance (MV) portfolio selection under a rough stochastic environment. Mean-variance criterion in portfolio selection is pioneered by Markowitz’s seminal work. And there are extensive extensions to relax the model assumptions and consider more complicated stochastic financial markets. We do not try to give a complete literature review here. Zhou and Li (2000), Lim and Zhou (2002), Lim (2004), and Shen (2015) are works in continuous-time setting which are more relevant to the problem in this paper than others.

### 1.1 Major contributions

We formulate the mean-variance (MV) portfolio selection under Volterra Heston model in a reasonable and rigorous manner. As mentioned in Abi Jaber et al. (2017) and Keller-Ressel et al. (2018), Volterra Heston model (2.6)-(2.7) has a unique in law weak solution, but the pathwise uniqueness is still an open question. This enforces us to consider the MV problem under a general filtration which satisfies usual conditions but may not be the augmented filtration generated by the Brownian motion. A similar general setting is considered in Jeanblanc et al. (2012). Let us emphasize that we always fix the probability basis and Brownian motions for the problem in Section 3. Therefore, our formulation is still referred to as a strong formulation, because the filtered probability space and Brownian motions are not parts of the control.

Under such a problem formulation, we construct an auxiliary stochastic process to make the completing the square procedure in solving the MV portfolio selection possible in Section 4. We prove several properties of in Theorem 4.1, which is one of the main results in this paper. Like the case in El Euch and Rosenbaum (2019), El Euch and Rosenbaum (2018) and Abi Jaber et al. (2017), the main difficulty is the non-Markovian and non-semimartingale structure in Volterra Heston model (2.6)-(2.7). Inspired by the exponential-affine formulas in Abi Jaber et al. (2017) and El Euch and Rosenbaum (2019), the process is constructed upon the forward variance under a proper alternative measure. The explicit solution for the optimal investment strategy is obtained in Theorem 4.2.

Under the rough Heston model, we investigate the relationship between the Hurst parameter, or equivalently the in our notation, and the optimal investment strategy . Recently, Glasserman and He (2019) study a trading strategy that leverages the information of roughness. Their strategy earns excess returns, which are not explained by the standard factor models like the CAPM model and Fama-French model. They propose to long the roughest stocks and short the smoothest stocks. We examine this trading signal under the MV setting. Our result suggests that the effect of roughness on investment strategy is opposite under different volatility of volatility. We then discuss the effect of on the efficient frontier.

The rest of the paper is organized as follows. Section 2 presents the Volterra Heston model and some useful properties. We discuss a related Riccati-Volterra equation. We then formulate the MV portfolio selection problem in Section 3 and solve it explicitly in Section 4. Numerical illustrations are given in Section 5. Section 6 concludes. Existence and uniqueness of the solution to Riccati-Volterra equations are summarized in Appendix A.

## 2 The Volterra Heston model

We always work under a given complete probability space , with a filtration satisfying the usual conditions, supporting a two-dimensional Brownian motion . The is not necessarily the augmented filtration generated by and so it can be a strictly larger filtration. It is a crucial difference between this work and previous ones like Lim and Zhou (2002), Lim (2004), and Shen (2015). In the literature, Jeanblanc et al. (2012) also consider the MV hedging problem under a general filtration. The reason to consider such a mathematical setting is that the stochastic Volterra equation (2.6)-(2.7) only has a unique in law weak solution but its strong uniqueness is left as an open question. Recall that for stochastic differential equations, is referred to as a strong solution if it is adapted to the augmented filtration generated by , and a weak solution otherwise. For weak solutions, the driving Brownian motion is also a part of the solution. See Chapter IX in Revuz and Yor (1999) for more details. Therefore, cannot be simply chosen as the augmented filtration generated by . Extra information may be needed in constructing solution to (2.6)-(2.7).

To proceed, we introduce a kernel where , and assume the following standing assumption through out the paper, in line with Abi Jaber et al. (2017) and Keller-Ressel et al. (2018). A function is called completely monotone if it is infinitely differentiable and for all , and .

###### Assumption 2.1.

is strictly positive and completely monotone. There is such that and for every .

The convolutions and for a measurable kernel on and a measure on of locally bounded variation are defined by

 (K∗L)(t)=∫[0,t]K(t−s)L(ds),(L∗K)(t)=∫[0,t]L(ds)K(t−s) (2.1)

for under proper conditions. The integral is extended to by right-continuity if possible. If is a function on , let

 (K∗F)(t)=∫t0K(t−s)F(s)ds. (2.2)

Let be a -dimensional continuous local martingale. The convolution between and is defined as

 (K∗dW)t=∫t0K(t−s)dWs. (2.3)

A measure on is called resolvent of the first kind to , if

 K∗L=L∗K≡id. (2.4)

Kernel is called the resolvent, or resolvent of the second kind, to if

 K∗R=R∗K=K−R. (2.5)

Further properties of these definitions can be found in Gripenberg et al. (1990) and Abi Jaber et al. (2017). Definitions in higher dimensions and in matrix form are possible. However, it suffices for us to consider the scalar case. Commonly used kernels are summarized in Table 1, which is also available at Abi Jaber et al. (2017).

The variance within the Volterra Heston model is defined as

 Vt=V0+κ∫t0K(t−s)(ϕ−Vs)ds+∫t0K(t−s)σ√VsdBs, (2.6)

where and are positive constants. The correlation between stock price and variance is also constant. Gatheral et al. (2018) documented that the general overall shape of the implied volatility surface does not change significantly. This indicates that it is still acceptable to consider a variance process whose parameters are independent of stock price and time. Rough Heston model in El Euch and Rosenbaum (2019) and El Euch and Rosenbaum (2018) is a special case of (2.6) with . Other definition of Heston model in rough version exists, see Guennoun et al. (2018). Bäuerle and Desmettre (2018) consider the power utility maximization under the model in Guennoun et al. (2018) with correlation .

Following Abi Jaber et al. (2017) as well as Kraft (2005), Zeng and Taksar (2013), and Shen and Zeng (2015), we assume the risky asset (stock) price follows

 dSt=St(rt+θVt)dt+St√VtdW1t,S0>0, (2.7)

with a deterministic bounded risk-free rate and constant . Then the market price of risk, or risk premium, is given by . The risk-free rate is the rate of return of a risk-free asset available in the market.

We need the existence and uniqueness result from Theorem 7.1 in Abi Jaber et al. (2017) and restate it as follows.

###### Theorem 2.2.

(Theorem 7.1 in Abi Jaber et al. (2017) ) Under Assumption 2.1, the stochastic Volterra equation (2.6)-(2.7) has a unique in law -valued continuous weak solution for any initial condition .

###### Remark 2.3.

For weak solutions, it is free to construct the Brownian motion as needed. It is also unknown whether pathwise uniqueness holds for (2.6)-(2.7). However, the MV objective only depends on the mathematical expectation on the distribution of the processes. In the sequel, we will only work with a version of the solution to (2.6)-(2.7) and fix the solution , since other solutions have the same law.

The following standing Assumption enables us to apply the Girsanov theorem and verify the admissibility of optimal strategy.

###### Assumption 2.4.

for a large enough constant .

To verify Assumption 2.4 holds under reasonable conditions, we consider a Riccati-Volterra equation (2.8) for as follows,

 g(a,t)=∫t0K(t−s)[a−κg(a,s)+σ22g2(a,s)]ds,g(a,0)=0. (2.8)

Existence and uniqueness of the solution to (2.8) are given in Lemmas A.2 and A.3.

###### Theorem 2.5.

Suppose the Riccati-Volterra equation (2.8) has a unique continuous solution on , then

 E[exp(a∫T0Vsds)]=exp[κϕ∫T0g(a,s)ds+V0∫T0[a−κg(a,s)+σ22g2(a,s)]ds]<∞. (2.9)

Moreover, if the resolvent of the first kind to exists and denote it as , then

 E[exp(a∫T0Vsds)]=exp[κϕ∫T0g(a,s)ds+V0∫T0g(a,T−s)L(ds)]. (2.10)
###### Proof.

By Theorem 4.3 in Abi Jaber et al. (2017),

 a∫T0Vsds=Y0−σ22∫T0g2(a,T−s)Vsds+σ∫T0g(a,T−s)√VsdBs, (2.11)

with

 Y0=κϕ∫T0g(a,s)ds+V0∫T0[a−κg(a,s)+σ22g2(a,s)]ds. (2.12)

Since is continuous on and therefore bounded, is a martingale by Lemma 7.3 in Abi Jaber et al. (2017). Therefore,

 E[exp(a∫T0Vsds)]=exp(Y0)=exp[κϕ∫T0g(a,s)ds+V0∫T0[a−κg(a,s)+σ22g2(a,s)]ds]. (2.13)

Note that implies

 ∫T0[a−κg(a,s)+σ22g2(a,s)]ds=∫T0g(a,T−s)L(ds). (2.14)

The result follows. ∎

Theorem 2.5 recovers the same expression for in El Euch and Rosenbaum (2018, Theorem 3.2). We stress that the proof circumvents the usage of Hawkes processes. In addition, we want to mention Gerhold et al. (2018) about the moment explosions in rough Heston model as a related reference.

## 3 Mean-variance portfolio selection

Let be the investment strategy, where is the amount of wealth invested in the stock. Then wealth process satisfies

 dXt=(rtXt+θ√Vtut)dt+utdW1t,X0=x0>0. (3.1)
###### Definition 3.1.

An investment strategy is said to be admissible if

2. and ;

3. the wealth process (3.1) has a unique solution in the sense of Yong and Zhou (1999, Chapter 1, Definition 6.15), with -a.s. continuous paths.

The set of all admissible investment strategies is denoted as .

###### Remark 3.2.

In Condition (1), is possibly strictly larger than the Brownian filtration of . It means extra information besides can be used to construct an admissible strategy. In general, can rely on a local -martingale which is strongly -orthogonal to . See hedging strategy (3.6) in Jeanblanc et al. (2012, Theorem 3.1) for such examples. However, we find optimal strategy only depends on variance and Brownian motion in Theorem 4.2.

###### Remark 3.3.

Again, we emphasize that the underlying probability space and Brownian motions are not parts of our control. Therefore, our formulation should still be referred to as strong formulation. Readers may refer to Yong and Zhou (1999), Chapter 2, Section 4 for discussions on the difference between strong and weak formulations for stochastic control problems.

The MV portfolio selection in continuous-time is the following problem222There are several equivalent formulations..

 ⎧⎪ ⎪⎨⎪ ⎪⎩minu(⋅)∈UJ(x0;u(⋅))=E[(XT−c)2], subject to E[XT]=c,(X(⋅),u(⋅)) satisfy (???). (3.2)

The constant is the target wealth level at the terminal time . We assume following Lim and Zhou (2002), Lim (2004) and Shen (2015). Otherwise, a trivial strategy that puts all the wealth into the risk-free asset, can dominate any other admissible strategy. The MV problem is said to be feasible for if there exists a which satisfies . Note that is deterministic and . It is then clear that the feasibility of our problem is guaranteed for any by a slight modification to the proof in Lim (2004, Propsition 6.1).

Problem (3.2) has a constraint. It is well-known that it is equivalent to the following max-min problem, by Luenberger (1968).

 {maxη∈Rminu(⋅)∈UJ(x0;u(⋅))=E[(XT−(c−η))2]−η2,(X(⋅),u(⋅)) satisfy (???). (3.3)

Let and consider the inner Problem (3.4) of (3.3) first.

 {minu(⋅)∈UJ(x0;u(⋅))=E[(XT−ζ)2]−η2,(X(⋅),u(⋅)) satisfy (???). (3.4)

## 4 Optimal investment strategy

To solve Problem (3.4), we introduce a new probability measure together with as the new Brownian motion by Girsanov theorem and Assumption 2.4. Under ,

 Vt=V0+∫t0K(t−s)(κϕ−λVs)ds+∫t0K(t−s)σ√Vsd~Bs, (4.1)

with and .

Denote and as the -expectation and conditional -expectation, respectively. The forward variance under is the conditional -expected variance, that is, . Keller-Ressel et al. (2018, Propsition 3.2) proves the following identity by an application of Abi Jaber et al. (2017, Lemma 4.2).

 ξt(s)=~E[Vs|Ft]=ξ0(s)+∫t01λRλ(s−u)σ√Vud~Bu, (4.2)

where

 ξ0(s)=(1−∫s0Rλ(u)du)V0+κϕλ∫s0Rλ(u)du, (4.3)

and is the resolvent of such that

 λK∗Rλ=Rλ∗(λK)=λK−Rλ. (4.4)

If , interpret and .

Consider the stochastic process,

 Mt=2exp[∫Tt(2rs−θ2ξt(s)+(1−2ρ2)σ22ψ2(T−s)ξt(s))ds], (4.5)

where

 ψ(t)=∫t0K(t−s)[(1−2ρ2)σ22ψ2(s)−λψ(s)−θ2]ds,ψ(0)=0. (4.6)

Existence and uniqueness of the solution to (4.6) are given in Lemma A.4.

is a crucial process that enables us to apply the completing the square technique in Theorem 4.2, like Lim and Zhou (2002), Lim (2004), and Shen (2015). Heuristically speaking, non-Markovian and non-semimartingale characteristics in Volterra Heston model are overcome by considering . We have the following result about .

###### Theorem 4.1.

Assume (4.6) has a unique continuous solution on , then satisfies the following properties:

1. is essentially bounded and , -a.s., . ;

2. Apply Itô’s lemma to on , then

 dMt=[−2rt+θ2Vt]Mtdt+[2θ√VtU1t+U21tMt]dt+U1tdW1t+U2tdW2t, (4.7)

where

 U1t =ρσMt√Vtψ(T−t), (4.8) U2t =√1−ρ2σMt√Vtψ(T−t). (4.9)
3.  M0=2exp[∫T02rsds+κϕ∫T0ψ(s)ds+V0∫T0[(1−2ρ2)σ22ψ2(s)−λψ(s)−θ2]ds]. (4.10)

Furthermore, for fractional kernel , denote the fractional integral as . Then

 M0=2exp[∫T02rsds+κϕI1ψ(T)+V0I1−αψ(T)]. (4.11)
4. for , .

###### Proof.

Property (1).

It is straightforward to see in (4.5). As for the upper bound, if , note forward variance since it is conditional expectation of variance , then . If , we claim

 M1−2ρ2t=21−2ρ2exp[2(1−2ρ2)∫Ttrsds]~E[exp(−θ2(1−2ρ2)∫TtVsds)∣∣Ft]. (4.12)

It is equivalent to show

 ~E[exp(−θ2(1−2ρ2)∫TtVsds)∣∣Ft] (4.13) =exp[∫Tt(−(1−2ρ2)θ2ξt(s)+(1−2ρ2)2σ22ψ2(T−s)ξt(s))ds].

Denote . Then satisfies

 ~ψ=K∗(σ22~ψ2−λ~ψ−(1−2ρ2)θ2). (4.14)

Therefore, (4.13) holds by Abi Jaber et al. (2017, Theorem 4.3) applying to . The martingale assumption in this Theorem 4.3 is verified by Lemma 7.3 in Abi Jaber et al. (2017).

If , then , which implies . can be discussed similarly. Property (1) is proved.

Property (2).

Denote in (4.5) with proper . From (4.2), apply Itô’s lemma to on time and get

 dξt(s)=1λRλ(s−t)σ√Vtd~Bt. (4.15)

Then

 dZt= [−2rt+θ2Vt−(1−2ρ2)σ22ψ2(T−t)Vt]dt −θ2∫Tt1λRλ(s−t)σ√Vtd~Btds+(1−2ρ2)σ22∫Ttψ2(T−s)1λRλ(s−t)σ√Vtd~Btds = [−2rt+θ2Vt−(1−2ρ2)σ22ψ2(T−t)Vt]dt −θ2∫Ttσ1λRλ(s−t)ds√Vtd~Bt+(1−2ρ2)σ22∫Ttσψ2(T−s)1λRλ(s−t)ds√Vtd~Bt.

The second equality is guaranteed by the stochastic Fubini theorem, see Veraar (2012).

For notation simplicity, denote

 U1t =σρMt√Vt∫Tt[(1−2ρ2)σ22ψ2(T−s)−θ2]1λRλ(s−t)ds, (4.16) U2t =σ√1−ρ2Mt√Vt∫Tt[(1−2ρ2)σ22ψ2(T−s)−θ2]1λRλ(s−t)ds. (4.17)

To match (4.16)-(4.17) with (4.8) - (4.9), we show

 ∫Tt[(1−2ρ2)σ22ψ2(T−s)−θ2]1λRλ(s−t)ds=ψ(T−t). (4.18)

Since

 ∫Tt[(1−2ρ2)σ22ψ2(T−s)−θ2]1λRλ(s−t)ds = ∫T−t0[(1−2ρ2)σ22ψ2(T−t−s)−θ2]1λRλ(s)ds =

then

 ∫Tt[(1−2ρ2)σ22ψ2(T−s)−θ2]1λRλ(s−t)ds−ψ(T−t) = = = −Rλ∗K∗[(1−2ρ2)σ22ψ2−θ2](T−t)+λK∗ψ(T−t).

Use (4.6), we have

 Rλ∗ψ=Rλ∗K∗[(1−2ρ2)σ22ψ2−λψ−θ2]. (4.19)

Consequently,

 −Rλ∗K∗[(1−2ρ2)σ22ψ2−θ2](T−t)+λK∗ψ(T−t) =[λK−Rλ−λK∗Rλ]∗ψ(T−t)=0.

Therefore,

 dMt= MtdZt+12MtdZtdZt = Mt[−2rt+θ2Vt−(1−2ρ2)σ22ψ2(T−t)Vt]dt+U21t+U22t2Mtdt +U1td~W1t+U2tdW2t = [−2rt+θ2Vt]Mtdt+[2θ√VtU1t+U21tMt]dt+U1tdW1t+U2tdW2t.

Property (3).

Proof of the property of in Abi Jaber et al. (2017, Theorem 4.3) indicates

 ∫T0[−θ2ξ0(s)+(1−2ρ2)σ22ψ2(T−s)ξ0(s)]ds =∫T0[−θ2V0+(κϕ−λV0)ψ(s)+(1−2ρ2)σ22ψ2(s)V0]ds.

With integration by parts, it can be shown that, under fractional kernel,

 ∫T0[−θ2−λψ(s)+(1−2ρ2)σ22ψ2(s)]ds=I1−αψ(T). (4.20)

We get the desired result.

Property (4).

Since is continuous on and is essentially bounded, by Assumption 2.4,

We are now ready to derive the optimal strategy and efficient frontier.

###### Theorem 4.2.

The optimal investment strategy for Problem (3.2) is

 u∗(t)=(θ+ρσψ(T−t))√Vt(ζ∗e−∫Ttrsds−X∗t), (4.21)

where is the optimal wealth process under and with

 η∗=e−∫T0rsdsM0x0−e−∫T02rsdsM0c2−e−∫T02rsdsM0. (4.22)

Furthermore,

 Var[X∗T]=M02−e−∫T02rsdsM0(ce−∫T0rsds−x0)2. (4.23)
###### Proof.

First, we consider the inner problem (3.4) with an arbitrary . Denote . By Itô’s lemma with property of and completing the square, for any admissible strategy ,

 d12Mt(Xt−ht)2 =12[(Xt−ht)2Mtθ2Vt+2(Xt−ht)2θ√VtU1t+(Xt−ht)2U21tMt+2Mt(Xt−ht)θ√Vtut +2(Xt−ht)utU1t+Mtu2t]dt +12[(Xt−ht)2U1t+2Mt(Xt−ht)ut]dW1t+12(Xt−ht)2U2tdW2t =12Mt[ut+(θ√Vt+U1tMt)(Xt−ht)]2dt +12[(Xt−ht)2U1t+2Mt(Xt−ht)ut]dW1t+12(Xt−ht)2U2tdW2t.

Since are bounded, for , is admissible, and has -a.s. continuous paths, then stochastic integrals

 ∫t0[(Xs−hs)2U1s+2Ms(Xs−hs)us]dW1s,∫t0(Xs−hs)2U2sdW2s

are -local martingales. There is an increasing localizing sequence of stopping times such that when . The local martingales stopped by are true martingales. Consequently,

 12E[Mτk(Xτk−hτk)2]=12M0(x0−h0)2+12E[∫τk0Mt(ut+(θ√Vt+U1tMt)(Xt−ht))2dt]. (4.24)

From (3.1), by Doob’s maximal inequality and admissibility of ,

 E[X2τk]≤C[x20+E[(∫T0|ut√Vt|dt)2]+E[∫T0u2tdt]]<∞. (4.25)

Then is dominated by a non-negative integrable random variable for all . Sending to infinity, by the dominated convergence theorem and the monotone convergence theorem, we derive

 E[(XT−ζ)2]=12M0(x0−h0)2+12E[∫T0Mt(ut+(θ√Vt+U1tMt)(Xt−ht))2dt]. (4.26)

Therefore, the cost functional is minimized when

 ut=−(θ√Vt+U1tMt)(Xt−ht). (4.27)

Then . To solve the outer maximization problem in (3.3), consider

 J(x0;u(⋅))=12M0[x0−(c−η)e−∫T0rsds