I am trying to solve the problem and come up with a recurrence relation that answers it correctly. Code :

```
 private static int recurse( int[] prices, int index, int[][] dp ) {

    if (index == prices.length - 1) {
        return 0;
    }

    int profit = 0, min = Integer.MAX_VALUE;

    int m = index;
    for (; m < prices.length; m++) {

        min = Math.min(min, prices[m]);//picking at min prices

        int diff = prices[m] - min;//diff. for current min. picked stock profit

        if (dp[m][index] != -1)
            return dp[m][index];

        if (diff > 0) {
            diff += recurse(prices, m + 1, dp);
        }

        profit = Math.max(profit, diff - fees);
    }

    dp[m][index] = profit;

    return profit;
}

```

The above solution returns TLE in test cases like : 

```
arr[] = [1036,2413,2776,825,2640,31,1560,2917,4282,783,3146,2600,1939,694,4284,3881,554,167,372,4620,3037,1175,1075,3845,4981,4495,3406,4228,2807,4774,4526,3914,2633,3762,1570,2334,616,1648,1914,2900,349,2428,4013,1964,4020,1882,629,240,2595,2902,3419,292,224,4437,4918,632,3701,3840,3996,2129,3345,3979,1954,781,1576,1084,3250,4517,3106,2133,309,4520,2225,4366,4628,1303,4373,1266,3181,558,3855,3447,4335,2115,4603,661,1715,3972,2846,342,686,787,273,2575,100,2860,3587,4236,3862,2238,3471,3123,431,4489,1551,596,4037,4986,594,2386,326,628,1363,2377,4986,3780,3853,2670,2852,3519,2998,4083,3392,2394,1083,3958,4082,1506,2322,2715,4901,2555,4097,3748,4717,3901,3329,4616,3334,2603,3705,631,3541,555,508,464,4495,4463,3616,31,2177,3307,1011,2759,751,1537,1000,292,3921,1442,2726,4677,792,82,2580,609,4758,3190,1958,913,955,1259,1634,4729,2672,1761,1467,2347,4295,2049,4708,1452,3411,1428,4078,2627,3785,2432,2916,492,1108,1691,972,3823,4086,2115,1925,1454,291,3266,300,2539,2681,2084,4633,1084,1061,1043,1304,2205,410,4332,2567,703,529,4273,3684,308,3164,4876,3108,4993,4555,1237,4753,549,2795,3426,819,2897,825,2514,3419,1854,3209,3766,2794,4117,4668,2162,1571,2446,1480,974,1090,3903,4655,4452,1451,2953,1241,842,1750,3847,3053,4395,4338,1493,1660,1569,3418,3029,4416,2056,2283,3392,2032,4354,803,4959,3630,2080,1553,873,4050,1986,2328,55,4602,1430,4238,4326,3382,4845,4968,1903,423,4717,2427,4618,2644,4541,380,3404,4880,2577,1640,189,2692,3788,818,4091,4730,611,1776,3594,4746,580,2083,4183,3355,3063,658,4532,3318,3902,556,2249,4653,2118,1529,4793,4935,4259,3542,1705,2839,1436,3918,564,3277,2988,2460,3213,4445,4238,1954,2213,1748,939,1149,1408,2408,1781,1618,1457,2123,3366,826,2094,16,1161,3337,1864,433,1303,4800,4667,4769,1026,3440,1072,4725,6,1263,4184,2728,1315,2091,3032,2071,2672,4557,1916,638,2133,2687,2408,1677,344,697,1699,8,480,655,2656,4983,455,1611,1726,692,392,1921,2555,3549,3740,3840,3062,3420,2428,1169,4570,389,3509,2169,3290,1680,1733,1765,2518,3260,3644,765,4521,269,2501,4014,1743,239,4908,1656,4433,3647,2612,4872,387,3091,4011,564,4421,810,3623,3451,4108,1428,475,3755,4484,3527,3062,4706,3424,2678,2411,4446,2556,4305,1305,646,1458,4471,1689,4556,3851,1245,1197,3785,1175,2904,302,2422,4302,2148,2338,4288,375,2824,1623,3717,1142,4254,192,783,1963,2225,1209,1746,3072,2737,4640,4919,3614,804,4029,1751,2360,3789,4445,2283,2769,2833,4452,2978,2809,4532,4365,2124,3541,2658,2902,4688,3980,1543,4041,1420,1452,1284,66,19,947,932,3244,3374,1910,2561,3466,4104,1667,589,3048,730,1770,1241,2270,4016,2835,604,4771,514,3854,3427,1875,2038,3067,3216,4732,3735,4440,2855,4958,4569,1685,3539,4589,3512,3143,898,3004,3072,2573,3163,2522,3927,330,3874,363,1900,1629,1156,4259,2747,3445,4513,2867,52,3870,1761,619,3308,4380,1101,2592,4852,4140,174,3997,4617,3500,3028,907,2355,759,374,2429,412,2132,3973,3583,3028,2070,2235,2659,1053,2558,753,1221,1185,2225,1593,3554,3703,332,2843,3349,3871,4389,6,2768,4382,902,417,191,2107,2838,4958,3905,4966,3937,1105,4150,2682,3396,818,2297,2077,2032,3340,2478,127,4379,954,2593,3454,1230,2308,3694,2179,4134,653,3808,4043,2069,660,4515,4189,4876,1784,4166,342,1766,3305,1980,1909,4115,4115,1461,2061,838,3112,122,656,4856,4822,3468,2111,2700,4124,4663,2948,3029,4182,3847,4760,1323,1505,308,128,874,583,2671,1315,747,2682,2841,67,2712,2703,4471,2952,3081,464,655,57,1460,1395,682,2447,2590,4624,1578,64,4060,2975,1236,831,3313,1432,2589,3777,1868,1720,45,3311,4532,2672,454,752,4839,4717,748,4323,2999,3491,631,1407,1453,4611,4263,3366,584,2014,2396,1902,4569,3002,1938,3998,4093,1899,3071,2815,1974,302,1641,2836,565,264,1332,3319,3689,2181,3873,4883,3849,1991,4633,4556,3866,142,2903,3181,740,3311,2071,280,714,2440,3950,290,3580,738,1604,3631,1989,1299,836,1913,224,1066,1741,1551,1735,4601,2024,4570,4192,1723,3949,3696,1419,1760,697,4764,3405,4443,199,717,4568,3252,2016,2151,1741,2613,2736,4053,814,4282,3392,615,1998,3294,3663,559,4278,4626,55,1418,2056,3191,3181,1732,1887,2517,3180,2154,2166,3096,3930,2721,4332,427,4332,4237,3928,2262,4657,2202,922,3711,1921,4728,2236,2441,622,233,293,1466,1891,1222,3693,3261,2605,3486,102,3612,1897,2698,3524,3567,613,3834,1583,1482,4734,2339,752,1428,4121,3267,3518,4652,3119,1818,4596,3181,3159,4069,3375,3762,1386,3054,3052,67,2246,1493,2738,2835,4906,303,1107,3111,1525,1739,437,2941,545,1458,993,1871,640,4047,2017,4971,4917,701,4811,4335,3221,4187,4414,756,3069,3052,812,3135,928,1264,3356,4518,2136,2691,2638,3156,4909,2944,3920,4609,1856,654,4643,2932,309,3613,4479,4173,1848,165,1171,592,3233,3151,4009,3952,2624,38,2616,2056,841,1764,4667,1526,125,3963,933,3951,2151,2110,4666,1000,1985,3868,2735,635,277,1129,572,2136,980,2731,556,3012,2900,2180,1912,2799,1771,4441,2666,3958,4381,3677,4218,1276,3512,4868,4579,2307,3952,3544,651,1300,218,489,2837,3737,509,3421,879,4353,4695]

Fee: 655

```

I am trying to add Memoization considering :
- User Bought at day M and sold at day Index.

Though I did misunderstand the states in this problem and diverted on the wrong track. Please suggest anything better.

Thanks you.